MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mumullem2 Structured version   Unicode version

Theorem mumullem2 23210
Description: Lemma for mumul 23211. The product of two coprime squarefree numbers is squarefree. (Contributed by Mario Carneiro, 3-Oct-2014.)
Assertion
Ref Expression
mumullem2  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  ( ( mmu `  A )  =/=  0  /\  ( mmu `  B
)  =/=  0 ) )  ->  ( mmu `  ( A  x.  B
) )  =/=  0
)

Proof of Theorem mumullem2
Dummy variable  p is distinct from all other variables.
StepHypRef Expression
1 r19.26 2989 . . . 4  |-  ( A. p  e.  Prime  ( ( p  pCnt  A )  <_  1  /\  ( p 
pCnt  B )  <_  1
)  <->  ( A. p  e.  Prime  ( p  pCnt  A )  <_  1  /\  A. p  e.  Prime  (
p  pCnt  B )  <_  1 ) )
2 simpr 461 . . . . . . . . . 10  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  ->  p  e.  Prime )
3 simpl1 999 . . . . . . . . . 10  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  ->  A  e.  NN )
42, 3pccld 14233 . . . . . . . . 9  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  -> 
( p  pCnt  A
)  e.  NN0 )
54nn0red 10853 . . . . . . . 8  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  -> 
( p  pCnt  A
)  e.  RR )
6 simpl2 1000 . . . . . . . . . 10  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  ->  B  e.  NN )
72, 6pccld 14233 . . . . . . . . 9  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  -> 
( p  pCnt  B
)  e.  NN0 )
87nn0red 10853 . . . . . . . 8  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  -> 
( p  pCnt  B
)  e.  RR )
9 1red 9611 . . . . . . . 8  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  -> 
1  e.  RR )
10 le2add 10034 . . . . . . . 8  |-  ( ( ( ( p  pCnt  A )  e.  RR  /\  ( p  pCnt  B )  e.  RR )  /\  ( 1  e.  RR  /\  1  e.  RR ) )  ->  ( (
( p  pCnt  A
)  <_  1  /\  ( p  pCnt  B )  <_  1 )  -> 
( ( p  pCnt  A )  +  ( p 
pCnt  B ) )  <_ 
( 1  +  1 ) ) )
115, 8, 9, 9, 10syl22anc 1229 . . . . . . 7  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  -> 
( ( ( p 
pCnt  A )  <_  1  /\  ( p  pCnt  B
)  <_  1 )  ->  ( ( p 
pCnt  A )  +  ( p  pCnt  B )
)  <_  ( 1  +  1 ) ) )
12 ax-1ne0 9561 . . . . . . . . . . . 12  |-  1  =/=  0
13 simpl3 1001 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  -> 
( A  gcd  B
)  =  1 )
1413oveq2d 6300 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  -> 
( p  pCnt  ( A  gcd  B ) )  =  ( p  pCnt  1 ) )
153nnzd 10965 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  ->  A  e.  ZZ )
166nnzd 10965 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  ->  B  e.  ZZ )
17 pcgcd 14260 . . . . . . . . . . . . . . . 16  |-  ( ( p  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  (
p  pCnt  ( A  gcd  B ) )  =  if ( ( p 
pCnt  A )  <_  (
p  pCnt  B ) ,  ( p  pCnt  A ) ,  ( p 
pCnt  B ) ) )
182, 15, 16, 17syl3anc 1228 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  -> 
( p  pCnt  ( A  gcd  B ) )  =  if ( ( p  pCnt  A )  <_  ( p  pCnt  B
) ,  ( p 
pCnt  A ) ,  ( p  pCnt  B )
) )
19 pc1 14238 . . . . . . . . . . . . . . . 16  |-  ( p  e.  Prime  ->  ( p 
pCnt  1 )  =  0 )
2019adantl 466 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  -> 
( p  pCnt  1
)  =  0 )
2114, 18, 203eqtr3d 2516 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  ->  if ( ( p  pCnt  A )  <_  ( p  pCnt  B ) ,  ( p  pCnt  A ) ,  ( p  pCnt  B ) )  =  0 )
22 ifid 3976 . . . . . . . . . . . . . . . 16  |-  if ( ( p  pCnt  A
)  <_  ( p  pCnt  B ) ,  1 ,  1 )  =  1
23 ifeq12 3956 . . . . . . . . . . . . . . . 16  |-  ( ( 1  =  ( p 
pCnt  A )  /\  1  =  ( p  pCnt  B ) )  ->  if ( ( p  pCnt  A )  <_  ( p  pCnt  B ) ,  1 ,  1 )  =  if ( ( p 
pCnt  A )  <_  (
p  pCnt  B ) ,  ( p  pCnt  A ) ,  ( p 
pCnt  B ) ) )
2422, 23syl5eqr 2522 . . . . . . . . . . . . . . 15  |-  ( ( 1  =  ( p 
pCnt  A )  /\  1  =  ( p  pCnt  B ) )  ->  1  =  if ( ( p 
pCnt  A )  <_  (
p  pCnt  B ) ,  ( p  pCnt  A ) ,  ( p 
pCnt  B ) ) )
2524eqeq1d 2469 . . . . . . . . . . . . . 14  |-  ( ( 1  =  ( p 
pCnt  A )  /\  1  =  ( p  pCnt  B ) )  ->  (
1  =  0  <->  if ( ( p  pCnt  A )  <_  ( p  pCnt  B ) ,  ( p  pCnt  A ) ,  ( p  pCnt  B ) )  =  0 ) )
2621, 25syl5ibrcom 222 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  -> 
( ( 1  =  ( p  pCnt  A
)  /\  1  =  ( p  pCnt  B ) )  ->  1  = 
0 ) )
2726necon3ad 2677 . . . . . . . . . . . 12  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  -> 
( 1  =/=  0  ->  -.  ( 1  =  ( p  pCnt  A
)  /\  1  =  ( p  pCnt  B ) ) ) )
2812, 27mpi 17 . . . . . . . . . . 11  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  ->  -.  ( 1  =  ( p  pCnt  A )  /\  1  =  (
p  pCnt  B )
) )
29 ax-1cn 9550 . . . . . . . . . . . . 13  |-  1  e.  CC
305recnd 9622 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  -> 
( p  pCnt  A
)  e.  CC )
31 subeq0 9845 . . . . . . . . . . . . 13  |-  ( ( 1  e.  CC  /\  ( p  pCnt  A )  e.  CC )  -> 
( ( 1  -  ( p  pCnt  A
) )  =  0  <->  1  =  ( p 
pCnt  A ) ) )
3229, 30, 31sylancr 663 . . . . . . . . . . . 12  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  -> 
( ( 1  -  ( p  pCnt  A
) )  =  0  <->  1  =  ( p 
pCnt  A ) ) )
338recnd 9622 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  -> 
( p  pCnt  B
)  e.  CC )
34 subeq0 9845 . . . . . . . . . . . . 13  |-  ( ( 1  e.  CC  /\  ( p  pCnt  B )  e.  CC )  -> 
( ( 1  -  ( p  pCnt  B
) )  =  0  <->  1  =  ( p 
pCnt  B ) ) )
3529, 33, 34sylancr 663 . . . . . . . . . . . 12  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  -> 
( ( 1  -  ( p  pCnt  B
) )  =  0  <->  1  =  ( p 
pCnt  B ) ) )
3632, 35anbi12d 710 . . . . . . . . . . 11  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  -> 
( ( ( 1  -  ( p  pCnt  A ) )  =  0  /\  ( 1  -  ( p  pCnt  B
) )  =  0 )  <->  ( 1  =  ( p  pCnt  A
)  /\  1  =  ( p  pCnt  B ) ) ) )
3728, 36mtbird 301 . . . . . . . . . 10  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  ->  -.  ( ( 1  -  ( p  pCnt  A
) )  =  0  /\  ( 1  -  ( p  pCnt  B
) )  =  0 ) )
3837adantr 465 . . . . . . . . 9  |-  ( ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  /\  (
( p  pCnt  A
)  <_  1  /\  ( p  pCnt  B )  <_  1 ) )  ->  -.  ( (
1  -  ( p 
pCnt  A ) )  =  0  /\  ( 1  -  ( p  pCnt  B ) )  =  0 ) )
39 eqcom 2476 . . . . . . . . . . 11  |-  ( ( 1  +  1 )  =  ( ( p 
pCnt  A )  +  ( p  pCnt  B )
)  <->  ( ( p 
pCnt  A )  +  ( p  pCnt  B )
)  =  ( 1  +  1 ) )
40 1re 9595 . . . . . . . . . . . . . . . . . 18  |-  1  e.  RR
4140, 40readdcli 9609 . . . . . . . . . . . . . . . . 17  |-  ( 1  +  1 )  e.  RR
4241recni 9608 . . . . . . . . . . . . . . . 16  |-  ( 1  +  1 )  e.  CC
434, 7nn0addcld 10856 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  -> 
( ( p  pCnt  A )  +  ( p 
pCnt  B ) )  e. 
NN0 )
4443nn0red 10853 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  -> 
( ( p  pCnt  A )  +  ( p 
pCnt  B ) )  e.  RR )
4544recnd 9622 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  -> 
( ( p  pCnt  A )  +  ( p 
pCnt  B ) )  e.  CC )
46 subeq0 9845 . . . . . . . . . . . . . . . 16  |-  ( ( ( 1  +  1 )  e.  CC  /\  ( ( p  pCnt  A )  +  ( p 
pCnt  B ) )  e.  CC )  ->  (
( ( 1  +  1 )  -  (
( p  pCnt  A
)  +  ( p 
pCnt  B ) ) )  =  0  <->  ( 1  +  1 )  =  ( ( p  pCnt  A )  +  ( p 
pCnt  B ) ) ) )
4742, 45, 46sylancr 663 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  -> 
( ( ( 1  +  1 )  -  ( ( p  pCnt  A )  +  ( p 
pCnt  B ) ) )  =  0  <->  ( 1  +  1 )  =  ( ( p  pCnt  A )  +  ( p 
pCnt  B ) ) ) )
4847, 39syl6bb 261 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  -> 
( ( ( 1  +  1 )  -  ( ( p  pCnt  A )  +  ( p 
pCnt  B ) ) )  =  0  <->  ( (
p  pCnt  A )  +  ( p  pCnt  B ) )  =  ( 1  +  1 ) ) )
499recnd 9622 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  -> 
1  e.  CC )
5049, 49, 30, 33addsub4d 9977 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  -> 
( ( 1  +  1 )  -  (
( p  pCnt  A
)  +  ( p 
pCnt  B ) ) )  =  ( ( 1  -  ( p  pCnt  A ) )  +  ( 1  -  ( p 
pCnt  B ) ) ) )
5150eqeq1d 2469 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  -> 
( ( ( 1  +  1 )  -  ( ( p  pCnt  A )  +  ( p 
pCnt  B ) ) )  =  0  <->  ( (
1  -  ( p 
pCnt  A ) )  +  ( 1  -  (
p  pCnt  B )
) )  =  0 ) )
5248, 51bitr3d 255 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  -> 
( ( ( p 
pCnt  A )  +  ( p  pCnt  B )
)  =  ( 1  +  1 )  <->  ( (
1  -  ( p 
pCnt  A ) )  +  ( 1  -  (
p  pCnt  B )
) )  =  0 ) )
5352adantr 465 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  /\  (
( p  pCnt  A
)  <_  1  /\  ( p  pCnt  B )  <_  1 ) )  ->  ( ( ( p  pCnt  A )  +  ( p  pCnt  B ) )  =  ( 1  +  1 )  <-> 
( ( 1  -  ( p  pCnt  A
) )  +  ( 1  -  ( p 
pCnt  B ) ) )  =  0 ) )
54 subge0 10065 . . . . . . . . . . . . . . . 16  |-  ( ( 1  e.  RR  /\  ( p  pCnt  A )  e.  RR )  -> 
( 0  <_  (
1  -  ( p 
pCnt  A ) )  <->  ( p  pCnt  A )  <_  1
) )
5540, 5, 54sylancr 663 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  -> 
( 0  <_  (
1  -  ( p 
pCnt  A ) )  <->  ( p  pCnt  A )  <_  1
) )
56 subge0 10065 . . . . . . . . . . . . . . . 16  |-  ( ( 1  e.  RR  /\  ( p  pCnt  B )  e.  RR )  -> 
( 0  <_  (
1  -  ( p 
pCnt  B ) )  <->  ( p  pCnt  B )  <_  1
) )
5740, 8, 56sylancr 663 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  -> 
( 0  <_  (
1  -  ( p 
pCnt  B ) )  <->  ( p  pCnt  B )  <_  1
) )
5855, 57anbi12d 710 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  -> 
( ( 0  <_ 
( 1  -  (
p  pCnt  A )
)  /\  0  <_  ( 1  -  ( p 
pCnt  B ) ) )  <-> 
( ( p  pCnt  A )  <_  1  /\  ( p  pCnt  B )  <_  1 ) ) )
59 resubcl 9883 . . . . . . . . . . . . . . . 16  |-  ( ( 1  e.  RR  /\  ( p  pCnt  A )  e.  RR )  -> 
( 1  -  (
p  pCnt  A )
)  e.  RR )
6040, 5, 59sylancr 663 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  -> 
( 1  -  (
p  pCnt  A )
)  e.  RR )
61 resubcl 9883 . . . . . . . . . . . . . . . 16  |-  ( ( 1  e.  RR  /\  ( p  pCnt  B )  e.  RR )  -> 
( 1  -  (
p  pCnt  B )
)  e.  RR )
6240, 8, 61sylancr 663 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  -> 
( 1  -  (
p  pCnt  B )
)  e.  RR )
63 add20 10064 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( 1  -  ( p  pCnt  A
) )  e.  RR  /\  0  <_  ( 1  -  ( p  pCnt  A ) ) )  /\  ( ( 1  -  ( p  pCnt  B
) )  e.  RR  /\  0  <_  ( 1  -  ( p  pCnt  B ) ) ) )  ->  ( ( ( 1  -  ( p 
pCnt  A ) )  +  ( 1  -  (
p  pCnt  B )
) )  =  0  <-> 
( ( 1  -  ( p  pCnt  A
) )  =  0  /\  ( 1  -  ( p  pCnt  B
) )  =  0 ) ) )
6463an4s 824 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( 1  -  ( p  pCnt  A
) )  e.  RR  /\  ( 1  -  (
p  pCnt  B )
)  e.  RR )  /\  ( 0  <_ 
( 1  -  (
p  pCnt  A )
)  /\  0  <_  ( 1  -  ( p 
pCnt  B ) ) ) )  ->  ( (
( 1  -  (
p  pCnt  A )
)  +  ( 1  -  ( p  pCnt  B ) ) )  =  0  <->  ( ( 1  -  ( p  pCnt  A ) )  =  0  /\  ( 1  -  ( p  pCnt  B
) )  =  0 ) ) )
6564ex 434 . . . . . . . . . . . . . . 15  |-  ( ( ( 1  -  (
p  pCnt  A )
)  e.  RR  /\  ( 1  -  (
p  pCnt  B )
)  e.  RR )  ->  ( ( 0  <_  ( 1  -  ( p  pCnt  A
) )  /\  0  <_  ( 1  -  (
p  pCnt  B )
) )  ->  (
( ( 1  -  ( p  pCnt  A
) )  +  ( 1  -  ( p 
pCnt  B ) ) )  =  0  <->  ( (
1  -  ( p 
pCnt  A ) )  =  0  /\  ( 1  -  ( p  pCnt  B ) )  =  0 ) ) ) )
6660, 62, 65syl2anc 661 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  -> 
( ( 0  <_ 
( 1  -  (
p  pCnt  A )
)  /\  0  <_  ( 1  -  ( p 
pCnt  B ) ) )  ->  ( ( ( 1  -  ( p 
pCnt  A ) )  +  ( 1  -  (
p  pCnt  B )
) )  =  0  <-> 
( ( 1  -  ( p  pCnt  A
) )  =  0  /\  ( 1  -  ( p  pCnt  B
) )  =  0 ) ) ) )
6758, 66sylbird 235 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  -> 
( ( ( p 
pCnt  A )  <_  1  /\  ( p  pCnt  B
)  <_  1 )  ->  ( ( ( 1  -  ( p 
pCnt  A ) )  +  ( 1  -  (
p  pCnt  B )
) )  =  0  <-> 
( ( 1  -  ( p  pCnt  A
) )  =  0  /\  ( 1  -  ( p  pCnt  B
) )  =  0 ) ) ) )
6867imp 429 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  /\  (
( p  pCnt  A
)  <_  1  /\  ( p  pCnt  B )  <_  1 ) )  ->  ( ( ( 1  -  ( p 
pCnt  A ) )  +  ( 1  -  (
p  pCnt  B )
) )  =  0  <-> 
( ( 1  -  ( p  pCnt  A
) )  =  0  /\  ( 1  -  ( p  pCnt  B
) )  =  0 ) ) )
6953, 68bitrd 253 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  /\  (
( p  pCnt  A
)  <_  1  /\  ( p  pCnt  B )  <_  1 ) )  ->  ( ( ( p  pCnt  A )  +  ( p  pCnt  B ) )  =  ( 1  +  1 )  <-> 
( ( 1  -  ( p  pCnt  A
) )  =  0  /\  ( 1  -  ( p  pCnt  B
) )  =  0 ) ) )
7039, 69syl5bb 257 . . . . . . . . . 10  |-  ( ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  /\  (
( p  pCnt  A
)  <_  1  /\  ( p  pCnt  B )  <_  1 ) )  ->  ( ( 1  +  1 )  =  ( ( p  pCnt  A )  +  ( p 
pCnt  B ) )  <->  ( (
1  -  ( p 
pCnt  A ) )  =  0  /\  ( 1  -  ( p  pCnt  B ) )  =  0 ) ) )
7170necon3abid 2713 . . . . . . . . 9  |-  ( ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  /\  (
( p  pCnt  A
)  <_  1  /\  ( p  pCnt  B )  <_  1 ) )  ->  ( ( 1  +  1 )  =/=  ( ( p  pCnt  A )  +  ( p 
pCnt  B ) )  <->  -.  (
( 1  -  (
p  pCnt  A )
)  =  0  /\  ( 1  -  (
p  pCnt  B )
)  =  0 ) ) )
7238, 71mpbird 232 . . . . . . . 8  |-  ( ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  /\  (
( p  pCnt  A
)  <_  1  /\  ( p  pCnt  B )  <_  1 ) )  ->  ( 1  +  1 )  =/=  (
( p  pCnt  A
)  +  ( p 
pCnt  B ) ) )
7372ex 434 . . . . . . 7  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  -> 
( ( ( p 
pCnt  A )  <_  1  /\  ( p  pCnt  B
)  <_  1 )  ->  ( 1  +  1 )  =/=  (
( p  pCnt  A
)  +  ( p 
pCnt  B ) ) ) )
7411, 73jcad 533 . . . . . 6  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  -> 
( ( ( p 
pCnt  A )  <_  1  /\  ( p  pCnt  B
)  <_  1 )  ->  ( ( ( p  pCnt  A )  +  ( p  pCnt  B ) )  <_  (
1  +  1 )  /\  ( 1  +  1 )  =/=  (
( p  pCnt  A
)  +  ( p 
pCnt  B ) ) ) ) )
75 nnz 10886 . . . . . . . . . . 11  |-  ( A  e.  NN  ->  A  e.  ZZ )
76 nnne0 10568 . . . . . . . . . . 11  |-  ( A  e.  NN  ->  A  =/=  0 )
7775, 76jca 532 . . . . . . . . . 10  |-  ( A  e.  NN  ->  ( A  e.  ZZ  /\  A  =/=  0 ) )
783, 77syl 16 . . . . . . . . 9  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  -> 
( A  e.  ZZ  /\  A  =/=  0 ) )
79 nnz 10886 . . . . . . . . . . 11  |-  ( B  e.  NN  ->  B  e.  ZZ )
80 nnne0 10568 . . . . . . . . . . 11  |-  ( B  e.  NN  ->  B  =/=  0 )
8179, 80jca 532 . . . . . . . . . 10  |-  ( B  e.  NN  ->  ( B  e.  ZZ  /\  B  =/=  0 ) )
826, 81syl 16 . . . . . . . . 9  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  -> 
( B  e.  ZZ  /\  B  =/=  0 ) )
83 pcmul 14234 . . . . . . . . 9  |-  ( ( p  e.  Prime  /\  ( A  e.  ZZ  /\  A  =/=  0 )  /\  ( B  e.  ZZ  /\  B  =/=  0 ) )  -> 
( p  pCnt  ( A  x.  B )
)  =  ( ( p  pCnt  A )  +  ( p  pCnt  B ) ) )
842, 78, 82, 83syl3anc 1228 . . . . . . . 8  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  -> 
( p  pCnt  ( A  x.  B )
)  =  ( ( p  pCnt  A )  +  ( p  pCnt  B ) ) )
8584breq1d 4457 . . . . . . 7  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  -> 
( ( p  pCnt  ( A  x.  B ) )  <_  1  <->  ( (
p  pCnt  A )  +  ( p  pCnt  B ) )  <_  1
) )
86 1nn0 10811 . . . . . . . 8  |-  1  e.  NN0
87 nn0leltp1 10921 . . . . . . . 8  |-  ( ( ( ( p  pCnt  A )  +  ( p 
pCnt  B ) )  e. 
NN0  /\  1  e.  NN0 )  ->  ( (
( p  pCnt  A
)  +  ( p 
pCnt  B ) )  <_ 
1  <->  ( ( p 
pCnt  A )  +  ( p  pCnt  B )
)  <  ( 1  +  1 ) ) )
8843, 86, 87sylancl 662 . . . . . . 7  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  -> 
( ( ( p 
pCnt  A )  +  ( p  pCnt  B )
)  <_  1  <->  ( (
p  pCnt  A )  +  ( p  pCnt  B ) )  <  (
1  +  1 ) ) )
89 ltlen 9686 . . . . . . . 8  |-  ( ( ( ( p  pCnt  A )  +  ( p 
pCnt  B ) )  e.  RR  /\  ( 1  +  1 )  e.  RR )  ->  (
( ( p  pCnt  A )  +  ( p 
pCnt  B ) )  < 
( 1  +  1 )  <->  ( ( ( p  pCnt  A )  +  ( p  pCnt  B ) )  <_  (
1  +  1 )  /\  ( 1  +  1 )  =/=  (
( p  pCnt  A
)  +  ( p 
pCnt  B ) ) ) ) )
9044, 41, 89sylancl 662 . . . . . . 7  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  -> 
( ( ( p 
pCnt  A )  +  ( p  pCnt  B )
)  <  ( 1  +  1 )  <->  ( (
( p  pCnt  A
)  +  ( p 
pCnt  B ) )  <_ 
( 1  +  1 )  /\  ( 1  +  1 )  =/=  ( ( p  pCnt  A )  +  ( p 
pCnt  B ) ) ) ) )
9185, 88, 903bitrd 279 . . . . . 6  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  -> 
( ( p  pCnt  ( A  x.  B ) )  <_  1  <->  ( (
( p  pCnt  A
)  +  ( p 
pCnt  B ) )  <_ 
( 1  +  1 )  /\  ( 1  +  1 )  =/=  ( ( p  pCnt  A )  +  ( p 
pCnt  B ) ) ) ) )
9274, 91sylibrd 234 . . . . 5  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  -> 
( ( ( p 
pCnt  A )  <_  1  /\  ( p  pCnt  B
)  <_  1 )  ->  ( p  pCnt  ( A  x.  B ) )  <_  1 ) )
9392ralimdva 2872 . . . 4  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  ->  ( A. p  e.  Prime  ( ( p  pCnt  A
)  <_  1  /\  ( p  pCnt  B )  <_  1 )  ->  A. p  e.  Prime  ( p  pCnt  ( A  x.  B ) )  <_ 
1 ) )
941, 93syl5bir 218 . . 3  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  ->  (
( A. p  e. 
Prime  ( p  pCnt  A
)  <_  1  /\  A. p  e.  Prime  (
p  pCnt  B )  <_  1 )  ->  A. p  e.  Prime  ( p  pCnt  ( A  x.  B ) )  <_  1 ) )
95 issqf 23166 . . . . 5  |-  ( A  e.  NN  ->  (
( mmu `  A
)  =/=  0  <->  A. p  e.  Prime  ( p 
pCnt  A )  <_  1
) )
96 issqf 23166 . . . . 5  |-  ( B  e.  NN  ->  (
( mmu `  B
)  =/=  0  <->  A. p  e.  Prime  ( p 
pCnt  B )  <_  1
) )
9795, 96bi2anan9 871 . . . 4  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( ( ( mmu `  A )  =/=  0  /\  ( mmu `  B
)  =/=  0 )  <-> 
( A. p  e. 
Prime  ( p  pCnt  A
)  <_  1  /\  A. p  e.  Prime  (
p  pCnt  B )  <_  1 ) ) )
98973adant3 1016 . . 3  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  ->  (
( ( mmu `  A )  =/=  0  /\  ( mmu `  B
)  =/=  0 )  <-> 
( A. p  e. 
Prime  ( p  pCnt  A
)  <_  1  /\  A. p  e.  Prime  (
p  pCnt  B )  <_  1 ) ) )
99 nnmulcl 10559 . . . . 5  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( A  x.  B
)  e.  NN )
100993adant3 1016 . . . 4  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  ->  ( A  x.  B )  e.  NN )
101 issqf 23166 . . . 4  |-  ( ( A  x.  B )  e.  NN  ->  (
( mmu `  ( A  x.  B )
)  =/=  0  <->  A. p  e.  Prime  ( p 
pCnt  ( A  x.  B ) )  <_ 
1 ) )
102100, 101syl 16 . . 3  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  ->  (
( mmu `  ( A  x.  B )
)  =/=  0  <->  A. p  e.  Prime  ( p 
pCnt  ( A  x.  B ) )  <_ 
1 ) )
10394, 98, 1023imtr4d 268 . 2  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  ->  (
( ( mmu `  A )  =/=  0  /\  ( mmu `  B
)  =/=  0 )  ->  ( mmu `  ( A  x.  B
) )  =/=  0
) )
104103imp 429 1  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  ( ( mmu `  A )  =/=  0  /\  ( mmu `  B
)  =/=  0 ) )  ->  ( mmu `  ( A  x.  B
) )  =/=  0
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2814   ifcif 3939   class class class wbr 4447   ` cfv 5588  (class class class)co 6284   CCcc 9490   RRcr 9491   0cc0 9492   1c1 9493    + caddc 9495    x. cmul 9497    < clt 9628    <_ cle 9629    - cmin 9805   NNcn 10536   NN0cn0 10795   ZZcz 10864    gcd cgcd 14003   Primecprime 14076    pCnt cpc 14219   mmucmu 23124
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569  ax-pre-sup 9570
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-1o 7130  df-2o 7131  df-oadd 7134  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-sup 7901  df-card 8320  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-div 10207  df-nn 10537  df-2 10594  df-3 10595  df-n0 10796  df-z 10865  df-uz 11083  df-q 11183  df-rp 11221  df-fz 11673  df-fl 11897  df-mod 11965  df-seq 12076  df-exp 12135  df-hash 12374  df-cj 12895  df-re 12896  df-im 12897  df-sqrt 13031  df-abs 13032  df-dvds 13848  df-gcd 14004  df-prm 14077  df-pc 14220  df-mu 23130
This theorem is referenced by:  mumul  23211
  Copyright terms: Public domain W3C validator