:: NAT_1 semantic presentation
theorem Th1: :: NAT_1:1
canceled;
theorem Th2: :: NAT_1:2
theorem Th3: :: NAT_1:3
canceled;
theorem Th4: :: NAT_1:4
canceled;
theorem Th5: :: NAT_1:5
canceled;
theorem Th6: :: NAT_1:6
canceled;
theorem Th7: :: NAT_1:7
canceled;
theorem Th8: :: NAT_1:8
canceled;
theorem Th9: :: NAT_1:9
canceled;
theorem Th10: :: NAT_1:10
canceled;
theorem Th11: :: NAT_1:11
canceled;
theorem Th12: :: NAT_1:12
canceled;
theorem Th13: :: NAT_1:13
canceled;
theorem Th14: :: NAT_1:14
canceled;
theorem Th15: :: NAT_1:15
canceled;
theorem Th16: :: NAT_1:16
canceled;
theorem Th17: :: NAT_1:17
canceled;
theorem Th18: :: NAT_1:18
for
i being
Nat holds 0
<= i
theorem Th19: :: NAT_1:19
for
i being
Nat st 0
<> i holds
0
< i by ;
theorem Th20: :: NAT_1:20
for
i,
j,
h being
Nat st
i <= j holds
i * h <= j * h
theorem Th21: :: NAT_1:21
for
i being
Nat holds 0
< i + 1
theorem Th22: :: NAT_1:22
for
i being
Nat holds
(
i = 0 or ex
k being
Nat st
i = k + 1 )
theorem Th23: :: NAT_1:23
for
i,
j being
Nat st
i + j = 0 holds
(
i = 0 &
j = 0 )
theorem Th24: :: NAT_1:24
canceled;
theorem Th25: :: NAT_1:25
canceled;
theorem Th26: :: NAT_1:26
for
i,
j being
Nat holds
( not
i <= j + 1 or
i <= j or
i = j + 1 )
theorem Th27: :: NAT_1:27
for
i,
j being
Nat st
i <= j &
j <= i + 1 & not
i = j holds
j = i + 1
theorem Th28: :: NAT_1:28
for
i,
j being
Nat st
i <= j holds
ex
k being
Nat st
j = i + k
theorem Th29: :: NAT_1:29
for
i,
j being
Nat holds
i <= i + j
theorem Th30: :: NAT_1:30
canceled;
theorem Th31: :: NAT_1:31
canceled;
theorem Th32: :: NAT_1:32
canceled;
theorem Th33: :: NAT_1:33
canceled;
theorem Th34: :: NAT_1:34
canceled;
theorem Th35: :: NAT_1:35
canceled;
theorem Th36: :: NAT_1:36
canceled;
theorem Th37: :: NAT_1:37
for
i,
j,
h being
Nat st
i <= j holds
i <= j + h
theorem Th38: :: NAT_1:38
for
i,
j being
Nat holds
(
i < j + 1 iff
i <= j )
theorem Th39: :: NAT_1:39
for
j being
Nat st
j < 1 holds
j = 0
theorem Th40: :: NAT_1:40
for
i,
j being
Nat st
i * j = 1 holds
(
i = 1 &
j = 1 )
theorem Th41: :: NAT_1:41
for
k,
n being
Nat st
k <> 0 holds
n < n + k
theorem Th42: :: NAT_1:42
for
m being
Nat st 0
< m holds
for
n being
Nat ex
k,
t being
Nat st
(
n = (m * k) + t &
t < m )
theorem Th43: :: NAT_1:43
for
n,
m,
k,
t,
k1,
t1 being
Nat st
n = (m * k) + t &
t < m &
n = (m * k1) + t1 &
t1 < m holds
(
k = k1 &
t = t1 )
:: deftheorem Def1 defines div NAT_1:def 1 :
for
k,
l,
b3 being
Nat holds
(
b3 = k div l iff ( ex
t being
Nat st
(
k = (l * b3) + t &
t < l ) or (
b3 = 0 &
l = 0 ) ) );
:: deftheorem Def2 defines mod NAT_1:def 2 :
for
k,
l,
b3 being
Nat holds
(
b3 = k mod l iff ( ex
t being
Nat st
(
k = (l * t) + b3 &
b3 < l ) or (
b3 = 0 &
l = 0 ) ) );
theorem Th44: :: NAT_1:44
canceled;
theorem Th45: :: NAT_1:45
canceled;
theorem Th46: :: NAT_1:46
for
i,
j being
Nat st 0
< i holds
j mod i < i
theorem Th47: :: NAT_1:47
for
i,
j being
Nat st 0
< i holds
j = (i * (j div i)) + (j mod i)
:: deftheorem Def3 defines divides NAT_1:def 3 :
theorem Th48: :: NAT_1:48
canceled;
theorem Th49: :: NAT_1:49
theorem Th50: :: NAT_1:50
canceled;
theorem Th51: :: NAT_1:51
theorem Th52: :: NAT_1:52
theorem Th53: :: NAT_1:53
theorem Th54: :: NAT_1:54
theorem Th55: :: NAT_1:55
theorem Th56: :: NAT_1:56
theorem Th57: :: NAT_1:57
theorem Th58: :: NAT_1:58
:: deftheorem Def4 defines lcm NAT_1:def 4 :
:: deftheorem Def5 defines hcf NAT_1:def 5 :
theorem Th59: :: NAT_1:59
for
k,
n being
Nat holds
(
k < k + n iff 1
<= n )
theorem Th60: :: NAT_1:60
theorem Th61: :: NAT_1:61
theorem Th62: :: NAT_1:62
theorem Th63: :: NAT_1:63
for
k,
n being
Nat holds
(k * n) mod k = 0
theorem Th64: :: NAT_1:64
for
k being
Nat st
k > 1 holds
1
mod k = 1
theorem Th65: :: NAT_1:65
for
k,
n,
l,
m being
Nat st
k mod n = 0 &
l = k - (m * n) holds
l mod n = 0
theorem Th66: :: NAT_1:66
for
n,
k,
l being
Nat st
n <> 0 &
k mod n = 0 &
l < n holds
(k + l) mod n = l
theorem Th67: :: NAT_1:67
theorem Th68: :: NAT_1:68
for
k,
n being
Nat st
k <> 0 holds
(k * n) div k = n
theorem Th69: :: NAT_1:69
theorem Th70: :: NAT_1:70
for
m,
n being
Nat holds
( not
m < n + 1 or
m < n or
m = n )
theorem Th71: :: NAT_1:71
for
k being
Nat holds
( not
k < 2 or
k = 0 or
k = 1 )
theorem Th72: :: NAT_1:72
for
k,
m being
Nat st
k <> 0 holds
(m * k) div k = m
theorem Th73: :: NAT_1:73
theorem Th74: :: NAT_1:74
theorem Th75: :: NAT_1:75
theorem Th76: :: NAT_1:76
for
k,
n being
Nat st
k < n holds
k mod n = k
theorem Th77: :: NAT_1:77
theorem Th78: :: NAT_1:78
theorem Th79: :: NAT_1:79
for
i,
h,
j being
Nat st
i <> 0 &
h = j * i holds
j <= h
theorem Th80: :: NAT_1:80
for
i,
j being
Nat st
i < j holds
i div j = 0
theorem Th81: :: NAT_1:81
for
k,
n being
Nat holds
(
k < n iff ( 1
<= k + 1 &
k + 1
<= n ) )
theorem Th82: :: NAT_1:82
for
n being
Nat holds
( not
n <= 1 or
n = 0 or
n = 1 )
theorem Th83: :: NAT_1:83
for
n being
Nat holds
( not
n <= 2 or
n = 0 or
n = 1 or
n = 2 )
theorem Th84: :: NAT_1:84
for
n being
Nat holds
( not
n <= 3 or
n = 0 or
n = 1 or
n = 2 or
n = 3 )
theorem Th85: :: NAT_1:85
for
n being
Nat holds
( not
n <= 4 or
n = 0 or
n = 1 or
n = 2 or
n = 3 or
n = 4 )
theorem Th86: :: NAT_1:86
for
n being
Nat holds
( not
n <= 5 or
n = 0 or
n = 1 or
n = 2 or
n = 3 or
n = 4 or
n = 5 )
theorem Th87: :: NAT_1:87
for
n being
Nat holds
( not
n <= 6 or
n = 0 or
n = 1 or
n = 2 or
n = 3 or
n = 4 or
n = 5 or
n = 6 )
theorem Th88: :: NAT_1:88
for
n being
Nat holds
( not
n <= 7 or
n = 0 or
n = 1 or
n = 2 or
n = 3 or
n = 4 or
n = 5 or
n = 6 or
n = 7 )
theorem Th89: :: NAT_1:89
for
n being
Nat holds
( not
n <= 8 or
n = 0 or
n = 1 or
n = 2 or
n = 3 or
n = 4 or
n = 5 or
n = 6 or
n = 7 or
n = 8 )
theorem Th90: :: NAT_1:90
for
n being
Nat holds
( not
n <= 9 or
n = 0 or
n = 1 or
n = 2 or
n = 3 or
n = 4 or
n = 5 or
n = 6 or
n = 7 or
n = 8 or
n = 9 )