:: COMPLSP1 semantic presentation
theorem :: COMPLSP1:1
canceled;
theorem :: COMPLSP1:2
canceled;
theorem Th3: :: COMPLSP1:3
Lm1:
the_unity_wrt addcomplex = 0c
by BINOP_2:1;
theorem :: COMPLSP1:4
canceled;
theorem :: COMPLSP1:5
canceled;
theorem Th6: :: COMPLSP1:6
theorem Th7: :: COMPLSP1:7
theorem Th8: :: COMPLSP1:8
:: deftheorem COMPLSP1:def 1 :
canceled;
:: deftheorem COMPLSP1:def 2 :
canceled;
:: deftheorem defines diffcomplex COMPLSP1:def 3 :
Lm2:
for c1, c2 being Element of COMPLEX holds diffcomplex . c1,c2 = c1 - c2
by BINOP_2:def 4;
theorem :: COMPLSP1:9
canceled;
theorem :: COMPLSP1:10
canceled;
theorem :: COMPLSP1:11
canceled;
theorem :: COMPLSP1:12
Lm3:
the_unity_wrt multcomplex = 1r
by BINOP_2:6;
theorem :: COMPLSP1:13
canceled;
theorem :: COMPLSP1:14
canceled;
theorem Th15: :: COMPLSP1:15
:: deftheorem COMPLSP1:def 4 :
canceled;
:: deftheorem defines multcomplex COMPLSP1:def 5 :
Lm4:
for c, c' being Element of COMPLEX holds (multcomplex [;] c,(id COMPLEX )) . c' = c * c'
theorem Th16: :: COMPLSP1:16
theorem Th17: :: COMPLSP1:17
:: deftheorem Def6 defines abscomplex COMPLSP1:def 6 :
:: deftheorem defines + COMPLSP1:def 7 :
:: deftheorem defines - COMPLSP1:def 8 :
:: deftheorem defines - COMPLSP1:def 9 :
:: deftheorem defines * COMPLSP1:def 10 :
:: deftheorem defines abs COMPLSP1:def 11 :
:: deftheorem defines COMPLEX COMPLSP1:def 12 :
Lm5:
for n being Element of NAT
for z being Element of COMPLEX n holds len z = n
by FINSEQ_2:109;
Lm6:
for n being Element of NAT
for z being Element of COMPLEX n holds dom z = Seg n
theorem :: COMPLSP1:18
canceled;
theorem :: COMPLSP1:19
canceled;
theorem :: COMPLSP1:20
canceled;
theorem Th21: :: COMPLSP1:21
Lm7:
for n being Element of NAT
for z1, z2 being Element of COMPLEX n st ( for k being Element of NAT st k in Seg n holds
z1 . k = z2 . k ) holds
z1 = z2
by FINSEQ_2:139;
theorem :: COMPLSP1:22
canceled;
theorem :: COMPLSP1:23
canceled;
theorem Th24: :: COMPLSP1:24
Lm8:
for n being Element of NAT
for z1, z2 being Element of COMPLEX n holds z1 + z2 = z2 + z1
by FINSEQOP:34;
Lm9:
for n being Element of NAT
for z1, z2, z3 being Element of COMPLEX n holds z1 + (z2 + z3) = (z1 + z2) + z3
by FINSEQOP:29;
:: deftheorem defines 0c COMPLSP1:def 13 :
Lm10:
for n being Element of NAT
for z being Element of COMPLEX n holds z + (0c n) = z
by Lm1, FINSEQOP:57;
theorem :: COMPLSP1:25
canceled;
theorem :: COMPLSP1:26
canceled;
theorem :: COMPLSP1:27
canceled;
theorem Th28: :: COMPLSP1:28
theorem Th29: :: COMPLSP1:29
Lm11:
for n being Element of NAT
for z being Element of COMPLEX n holds z + (- z) = 0c n
by Lm1, Th7, Th8, FINSEQOP:77;
Lm12:
for n being Element of NAT holds - (0c n) = 0c n
theorem Th30: :: COMPLSP1:30
theorem Th31: :: COMPLSP1:31
theorem Th32: :: COMPLSP1:32
theorem :: COMPLSP1:33
Lm13:
for n being Element of NAT
for z1, z, z2 being Element of COMPLEX n st z1 + z = z2 + z holds
z1 = z2
theorem :: COMPLSP1:34
theorem Th35: :: COMPLSP1:35
theorem :: COMPLSP1:36
Lm14:
for n being Element of NAT
for z1, z2 being Element of COMPLEX n holds z1 - z2 = z1 + (- z2)
by FINSEQOP:89;
theorem :: COMPLSP1:37
canceled;
theorem :: COMPLSP1:38
theorem :: COMPLSP1:39
theorem :: COMPLSP1:40
theorem Th41: :: COMPLSP1:41
theorem Th42: :: COMPLSP1:42
theorem Th43: :: COMPLSP1:43
theorem Th44: :: COMPLSP1:44
theorem Th45: :: COMPLSP1:45
theorem Th46: :: COMPLSP1:46
theorem Th47: :: COMPLSP1:47
theorem Th48: :: COMPLSP1:48
theorem Th49: :: COMPLSP1:49
theorem Th50: :: COMPLSP1:50
theorem Th51: :: COMPLSP1:51
theorem Th52: :: COMPLSP1:52
theorem :: COMPLSP1:53
theorem :: COMPLSP1:54
theorem :: COMPLSP1:55
theorem :: COMPLSP1:56
theorem :: COMPLSP1:57
theorem :: COMPLSP1:58
theorem Th59: :: COMPLSP1:59
theorem Th60: :: COMPLSP1:60
theorem Th61: :: COMPLSP1:61
theorem Th62: :: COMPLSP1:62
:: deftheorem defines |. COMPLSP1:def 14 :
theorem Th63: :: COMPLSP1:63
theorem Th64: :: COMPLSP1:64
theorem Th65: :: COMPLSP1:65
theorem Th66: :: COMPLSP1:66
theorem :: COMPLSP1:67
Lm15:
for i, j being Element of NAT
for t being Element of i -tuples_on REAL st j in Seg i holds
t . j is Real
theorem Th68: :: COMPLSP1:68
theorem Th69: :: COMPLSP1:69
theorem :: COMPLSP1:70
theorem :: COMPLSP1:71
theorem Th72: :: COMPLSP1:72
theorem Th73: :: COMPLSP1:73
theorem Th74: :: COMPLSP1:74
theorem Th75: :: COMPLSP1:75
:: deftheorem Def15 defines open COMPLSP1:def 15 :
:: deftheorem defines closed COMPLSP1:def 16 :
theorem :: COMPLSP1:76
theorem Th77: :: COMPLSP1:77
theorem Th78: :: COMPLSP1:78
theorem Th79: :: COMPLSP1:79
:: deftheorem defines Ball COMPLSP1:def 17 :
theorem Th80: :: COMPLSP1:80
theorem Th81: :: COMPLSP1:81
theorem Th82: :: COMPLSP1:82
:: deftheorem Def18 defines dist COMPLSP1:def 18 :
:: deftheorem defines Ball COMPLSP1:def 19 :
Lm16:
for r, r1 being Real st ( for r' being real number st r' > 0 holds
r + r' >= r1 ) holds
r >= r1
by XREAL_1:43;
theorem :: COMPLSP1:83
canceled;
theorem Th84: :: COMPLSP1:84
theorem Th85: :: COMPLSP1:85
theorem Th86: :: COMPLSP1:86
theorem Th87: :: COMPLSP1:87
theorem Th88: :: COMPLSP1:88
theorem Th89: :: COMPLSP1:89
theorem Th90: :: COMPLSP1:90
theorem Th91: :: COMPLSP1:91
theorem Th92: :: COMPLSP1:92
theorem Th93: :: COMPLSP1:93
:: deftheorem Def20 defines dist COMPLSP1:def 20 :
:: deftheorem defines + COMPLSP1:def 21 :
theorem Th94: :: COMPLSP1:94
theorem Th95: :: COMPLSP1:95
theorem Th96: :: COMPLSP1:96
theorem Th97: :: COMPLSP1:97
theorem Th98: :: COMPLSP1:98
theorem :: COMPLSP1:99
theorem Th100: :: COMPLSP1:100
theorem :: COMPLSP1:101
:: deftheorem defines ComplexOpenSets COMPLSP1:def 22 :
theorem Th102: :: COMPLSP1:102
:: deftheorem defines the_Complex_Space COMPLSP1:def 23 :
theorem :: COMPLSP1:103
theorem :: COMPLSP1:104
theorem :: COMPLSP1:105
theorem :: COMPLSP1:106
canceled;
theorem :: COMPLSP1:107
canceled;
theorem Th108: :: COMPLSP1:108
theorem Th109: :: COMPLSP1:109
theorem Th110: :: COMPLSP1:110
theorem :: COMPLSP1:111
theorem :: COMPLSP1:112