:: NAT_D semantic presentation
:: deftheorem Def1 defines div NAT_D: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_D: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 Th46: :: NAT_D:1
for
i,
j being
Nat st 0
< i holds
j mod i < i
theorem Th47: :: NAT_D:2
:: deftheorem Def3 defines divides NAT_D:def 3 :
theorem Th49: :: NAT_D:3
theorem :: NAT_D:4
theorem Th52: :: NAT_D:5
theorem Th53: :: NAT_D:6
theorem Th54: :: NAT_D:7
theorem Th55: :: NAT_D:8
theorem Th56: :: NAT_D:9
theorem Th57: :: NAT_D:10
theorem Th58: :: NAT_D:11
:: deftheorem Def4 defines lcm NAT_D:def 4 :
:: deftheorem Def5 defines hcf NAT_D:def 5 :
theorem :: NAT_D:12
theorem :: NAT_D:13
for
k,
n being
Nat holds
(k * n) mod k = 0
theorem :: NAT_D:14
for
k being
Nat st
k > 1 holds
1
mod k = 1
theorem :: NAT_D:15
for
k,
n,
l,
m being
Nat st
k mod n = 0 &
l = k - (m * n) holds
l mod n = 0
theorem :: NAT_D:16
for
n,
k,
l being
Nat st
n <> 0 &
k mod n = 0 &
l < n holds
(k + l) mod n = l
theorem :: NAT_D:17
theorem :: NAT_D:18
for
k,
n being
Nat st
k <> 0 holds
(k * n) div k = n
theorem :: NAT_D:19
theorem :: NAT_D:20
for
k,
m being
Nat st
k <> 0 holds
(m * k) div k = m
theorem Th73: :: NAT_D:21
theorem Th74: :: NAT_D:22
theorem :: NAT_D:23
theorem Th76: :: NAT_D:24
for
k,
n being
Nat st
k < n holds
k mod n = k
theorem :: NAT_D:25
theorem :: NAT_D:26
theorem :: NAT_D:27
for
i,
j being
Nat st
i < j holds
i div j = 0
theorem :: NAT_D:28