:: MIDSP_1 semantic presentation
:: deftheorem Def1 defines @ MIDSP_1:def 1 :
:: deftheorem Def2 defines op2 MIDSP_1:def 2 :
:: deftheorem Def3 defines Example MIDSP_1:def 3 :
theorem Th1: :: MIDSP_1:1
canceled;
theorem Th2: :: MIDSP_1:2
canceled;
theorem Th3: :: MIDSP_1:3
canceled;
theorem Th4: :: MIDSP_1:4
canceled;
theorem Th5: :: MIDSP_1:5
theorem Th6: :: MIDSP_1:6
theorem Th7: :: MIDSP_1:7
theorem Th8: :: MIDSP_1:8
theorem Th9: :: MIDSP_1:9
canceled;
theorem Th10: :: MIDSP_1:10
:: deftheorem Def4 defines MidSp-like MIDSP_1:def 4 :
theorem Th11: :: MIDSP_1:11
canceled;
theorem Th12: :: MIDSP_1:12
canceled;
theorem Th13: :: MIDSP_1:13
canceled;
theorem Th14: :: MIDSP_1:14
canceled;
theorem Th15: :: MIDSP_1:15
for b
1 being
MidSpfor b
2, b
3, b
4 being
Element of b
1 holds
(b2 @ b3) @ b
4 = (b2 @ b4) @ (b3 @ b4)
theorem Th16: :: MIDSP_1:16
for b
1 being
MidSpfor b
2, b
3, b
4 being
Element of b
1 holds b
2 @ (b3 @ b4) = (b2 @ b3) @ (b2 @ b4)
theorem Th17: :: MIDSP_1:17
for b
1 being
MidSpfor b
2, b
3 being
Element of b
1 holds
( b
2 @ b
3 = b
2 implies b
2 = b
3 )
theorem Th18: :: MIDSP_1:18
for b
1 being
MidSpfor b
2, b
3, b
4 being
Element of b
1 holds
( b
2 @ b
3 = b
4 @ b
3 implies b
2 = b
4 )
theorem Th19: :: MIDSP_1:19
for b
1 being
MidSpfor b
2, b
3, b
4 being
Element of b
1 holds
( b
2 @ b
3 = b
2 @ b
4 implies b
3 = b
4 )
by Th18;
:: deftheorem Def5 defines @@ MIDSP_1:def 5 :
for b
1 being
MidSpfor b
2, b
3, b
4, b
5 being
Element of b
1 holds
( b
2,b
3 @@ b
4,b
5 iff b
2 @ b
5 = b
3 @ b
4 );
theorem Th20: :: MIDSP_1:20
canceled;
theorem Th21: :: MIDSP_1:21
theorem Th22: :: MIDSP_1:22
for b
1 being
MidSpfor b
2, b
3, b
4, b
5 being
Element of b
1 holds
( b
2,b
3 @@ b
4,b
5 implies b
4,b
5 @@ b
2,b
3 )
theorem Th23: :: MIDSP_1:23
for b
1 being
MidSpfor b
2, b
3, b
4 being
Element of b
1 holds
( b
2,b
2 @@ b
3,b
4 implies b
3 = b
4 )
theorem Th24: :: MIDSP_1:24
for b
1 being
MidSpfor b
2, b
3, b
4 being
Element of b
1 holds
( b
2,b
3 @@ b
4,b
4 implies b
2 = b
3 )
theorem Th25: :: MIDSP_1:25
theorem Th26: :: MIDSP_1:26
theorem Th27: :: MIDSP_1:27
for b
1 being
MidSpfor b
2, b
3, b
4, b
5, b
6 being
Element of b
1 holds
( b
2,b
3 @@ b
4,b
5 & b
2,b
3 @@ b
4,b
6 implies b
5 = b
6 )
theorem Th28: :: MIDSP_1:28
for b
1 being
MidSpfor b
2, b
3, b
4, b
5, b
6, b
7 being
Element of b
1 holds
( b
2,b
3 @@ b
4,b
5 & b
2,b
3 @@ b
6,b
7 implies b
4,b
5 @@ b
6,b
7 )
theorem Th29: :: MIDSP_1:29
for b
1 being
MidSpfor b
2, b
3, b
4, b
5, b
6, b
7 being
Element of b
1 holds
( b
2,b
3 @@ b
4,b
5 & b
3,b
6 @@ b
5,b
7 implies b
2,b
6 @@ b
4,b
7 )
definition
let c
1 be
MidSp;
let c
2, c
3 be
Element of
[:the carrier of c1,the carrier of c1:];
pred c
2 ## c
3 means :
Def6:
:: MIDSP_1:def 6
a
2 `1 ,a
2 `2 @@ a
3 `1 ,a
3 `2 ;
reflexivity
for b1 being Element of [:the carrier of c1,the carrier of c1:] holds b1 `1 ,b1 `2 @@ b1 `1 ,b1 `2
by Th25;
symmetry
for b1, b2 being Element of [:the carrier of c1,the carrier of c1:] holds
( b1 `1 ,b1 `2 @@ b2 `1 ,b2 `2 implies b2 `1 ,b2 `2 @@ b1 `1 ,b1 `2 )
by Th22;
end;
:: deftheorem Def6 defines ## MIDSP_1:def 6 :
theorem Th30: :: MIDSP_1:30
canceled;
theorem Th31: :: MIDSP_1:31
for b
1 being
MidSpfor b
2, b
3, b
4, b
5 being
Element of b
1 holds
( b
2,b
3 @@ b
4,b
5 implies
[b2,b3] ## [b4,b5] )
theorem Th32: :: MIDSP_1:32
for b
1 being
MidSpfor b
2, b
3, b
4, b
5 being
Element of b
1 holds
(
[b2,b3] ## [b4,b5] implies b
2,b
3 @@ b
4,b
5 )
theorem Th33: :: MIDSP_1:33
canceled;
theorem Th34: :: MIDSP_1:34
canceled;
theorem Th35: :: MIDSP_1:35
theorem Th36: :: MIDSP_1:36
theorem Th37: :: MIDSP_1:37
theorem Th38: :: MIDSP_1:38
theorem Th39: :: MIDSP_1:39
:: deftheorem Def7 defines ~ MIDSP_1:def 7 :
theorem Th40: :: MIDSP_1:40
canceled;
theorem Th41: :: MIDSP_1:41
theorem Th42: :: MIDSP_1:42
theorem Th43: :: MIDSP_1:43
theorem Th44: :: MIDSP_1:44
for b
1 being
MidSpfor b
2, b
3, b
4, b
5 being
Element of b
1 holds
(
[b2,b3] ~ = [b4,b5] ~ implies b
2 @ b
5 = b
3 @ b
4 )
theorem Th45: :: MIDSP_1:45
:: deftheorem Def8 defines Vector MIDSP_1:def 8 :
theorem Th46: :: MIDSP_1:46
canceled;
theorem Th47: :: MIDSP_1:47
canceled;
theorem Th48: :: MIDSP_1:48
:: deftheorem Def9 defines ID MIDSP_1:def 9 :
theorem Th49: :: MIDSP_1:49
canceled;
theorem Th50: :: MIDSP_1:50
theorem Th51: :: MIDSP_1:51
theorem Th52: :: MIDSP_1:52
definition
let c
1 be
MidSp;
let c
2, c
3 be
Vector of c
1;
func c
2 + c
3 -> Vector of a
1 means :
Def10:
:: MIDSP_1:def 10
ex b
1, b
2 being
Element of
[:the carrier of a1,the carrier of a1:] st
( a
2 = b
1 ~ & a
3 = b
2 ~ & b
1 `2 = b
2 `1 & a
4 = [(b1 `1 ),(b2 `2 )] ~ );
existence
ex b1 being Vector of c1ex b2, b3 being Element of [:the carrier of c1,the carrier of c1:] st
( c2 = b2 ~ & c3 = b3 ~ & b2 `2 = b3 `1 & b1 = [(b2 `1 ),(b3 `2 )] ~ )
by Th51;
uniqueness
for b1, b2 being Vector of c1 holds
( ex b3, b4 being Element of [:the carrier of c1,the carrier of c1:] st
( c2 = b3 ~ & c3 = b4 ~ & b3 `2 = b4 `1 & b1 = [(b3 `1 ),(b4 `2 )] ~ ) & ex b3, b4 being Element of [:the carrier of c1,the carrier of c1:] st
( c2 = b3 ~ & c3 = b4 ~ & b3 `2 = b4 `1 & b2 = [(b3 `1 ),(b4 `2 )] ~ ) implies b1 = b2 )
by Th52;
end;
:: deftheorem Def10 defines + MIDSP_1:def 10 :
theorem Th53: :: MIDSP_1:53
:: deftheorem Def11 defines vect MIDSP_1:def 11 :
theorem Th54: :: MIDSP_1:54
canceled;
theorem Th55: :: MIDSP_1:55
theorem Th56: :: MIDSP_1:56
theorem Th57: :: MIDSP_1:57
theorem Th58: :: MIDSP_1:58
theorem Th59: :: MIDSP_1:59
theorem Th60: :: MIDSP_1:60
theorem Th61: :: MIDSP_1:61
theorem Th62: :: MIDSP_1:62
theorem Th63: :: MIDSP_1:63
for b
1 being
MidSpfor b
2, b
3, b
4 being
Vector of b
1 holds
(b2 + b3) + b
4 = b
2 + (b3 + b4)
theorem Th64: :: MIDSP_1:64
theorem Th65: :: MIDSP_1:65
theorem Th66: :: MIDSP_1:66
for b
1 being
MidSpfor b
2, b
3 being
Vector of b
1 holds b
2 + b
3 = b
3 + b
2
theorem Th67: :: MIDSP_1:67
for b
1 being
MidSpfor b
2, b
3, b
4 being
Vector of b
1 holds
( b
2 + b
3 = b
2 + b
4 implies b
3 = b
4 )
:: deftheorem Def12 defines - MIDSP_1:def 12 :
for b
1 being
MidSpfor b
2, b
3 being
Vector of b
1 holds
( b
3 = - b
2 iff b
2 + b
3 = ID b
1 );
:: deftheorem Def13 defines setvect MIDSP_1:def 13 :
theorem Th68: :: MIDSP_1:68
canceled;
theorem Th69: :: MIDSP_1:69
canceled;
theorem Th70: :: MIDSP_1:70
canceled;
theorem Th71: :: MIDSP_1:71
:: deftheorem Def14 defines + MIDSP_1:def 14 :
for b
1 being
MidSpfor b
2, b
3, b
4 being
Element of
setvect b
1 holds
( b
4 = b
2 + b
3 iff for b
5, b
6 being
Vector of b
1 holds
( b
2 = b
5 & b
3 = b
6 implies b
4 = b
5 + b
6 ) );
theorem Th72: :: MIDSP_1:72
canceled;
theorem Th73: :: MIDSP_1:73
canceled;
theorem Th74: :: MIDSP_1:74
theorem Th75: :: MIDSP_1:75
:: deftheorem Def15 defines addvect MIDSP_1:def 15 :
theorem Th76: :: MIDSP_1:76
canceled;
theorem Th77: :: MIDSP_1:77
theorem Th78: :: MIDSP_1:78
:: deftheorem Def16 defines complvect MIDSP_1:def 16 :
:: deftheorem Def17 defines zerovect MIDSP_1:def 17 :
:: deftheorem Def18 defines vectgroup MIDSP_1:def 18 :
theorem Th79: :: MIDSP_1:79
canceled;
theorem Th80: :: MIDSP_1:80
canceled;
theorem Th81: :: MIDSP_1:81
canceled;
theorem Th82: :: MIDSP_1:82
theorem Th83: :: MIDSP_1:83
theorem Th84: :: MIDSP_1:84
canceled;
theorem Th85: :: MIDSP_1:85
theorem Th86: :: MIDSP_1:86