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Theorem mumullem2 24186
Description: Lemma for mumul 24187. 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 2904 . . . 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 468 . . . . . . . . . 10  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  ->  p  e.  Prime )
3 simpl1 1033 . . . . . . . . . 10  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  ->  A  e.  NN )
42, 3pccld 14879 . . . . . . . . 9  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  -> 
( p  pCnt  A
)  e.  NN0 )
54nn0red 10950 . . . . . . . 8  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  -> 
( p  pCnt  A
)  e.  RR )
6 simpl2 1034 . . . . . . . . . 10  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  ->  B  e.  NN )
72, 6pccld 14879 . . . . . . . . 9  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  -> 
( p  pCnt  B
)  e.  NN0 )
87nn0red 10950 . . . . . . . 8  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  -> 
( p  pCnt  B
)  e.  RR )
9 1red 9676 . . . . . . . 8  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  -> 
1  e.  RR )
10 le2add 10117 . . . . . . . 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 1293 . . . . . . 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 9626 . . . . . . . . . . . 12  |-  1  =/=  0
13 simpl3 1035 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  -> 
( A  gcd  B
)  =  1 )
1413oveq2d 6324 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  -> 
( p  pCnt  ( A  gcd  B ) )  =  ( p  pCnt  1 ) )
153nnzd 11062 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  ->  A  e.  ZZ )
166nnzd 11062 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  ->  B  e.  ZZ )
17 pcgcd 14906 . . . . . . . . . . . . . . . 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 1292 . . . . . . . . . . . . . . 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 14884 . . . . . . . . . . . . . . . 16  |-  ( p  e.  Prime  ->  ( p 
pCnt  1 )  =  0 )
2019adantl 473 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  -> 
( p  pCnt  1
)  =  0 )
2114, 18, 203eqtr3d 2513 . . . . . . . . . . . . . 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 3909 . . . . . . . . . . . . . . . 16  |-  if ( ( p  pCnt  A
)  <_  ( p  pCnt  B ) ,  1 ,  1 )  =  1
23 ifeq12 3889 . . . . . . . . . . . . . . . 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 2519 . . . . . . . . . . . . . . 15  |-  ( ( 1  =  ( p 
pCnt  A )  /\  1  =  ( p  pCnt  B ) )  ->  1  =  if ( ( p 
pCnt  A )  <_  (
p  pCnt  B ) ,  ( p  pCnt  A ) ,  ( p 
pCnt  B ) ) )
2524eqeq1d 2473 . . . . . . . . . . . . . 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 230 . . . . . . . . . . . . 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 2656 . . . . . . . . . . . 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 20 . . . . . . . . . . 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 9615 . . . . . . . . . . . . 13  |-  1  e.  CC
305recnd 9687 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  -> 
( p  pCnt  A
)  e.  CC )
31 subeq0 9920 . . . . . . . . . . . . 13  |-  ( ( 1  e.  CC  /\  ( p  pCnt  A )  e.  CC )  -> 
( ( 1  -  ( p  pCnt  A
) )  =  0  <->  1  =  ( p 
pCnt  A ) ) )
3229, 30, 31sylancr 676 . . . . . . . . . . . 12  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  -> 
( ( 1  -  ( p  pCnt  A
) )  =  0  <->  1  =  ( p 
pCnt  A ) ) )
338recnd 9687 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  -> 
( p  pCnt  B
)  e.  CC )
34 subeq0 9920 . . . . . . . . . . . . 13  |-  ( ( 1  e.  CC  /\  ( p  pCnt  B )  e.  CC )  -> 
( ( 1  -  ( p  pCnt  B
) )  =  0  <->  1  =  ( p 
pCnt  B ) ) )
3529, 33, 34sylancr 676 . . . . . . . . . . . 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 725 . . . . . . . . . . 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 308 . . . . . . . . . 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 472 . . . . . . . . 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 2478 . . . . . . . . . . 11  |-  ( ( 1  +  1 )  =  ( ( p 
pCnt  A )  +  ( p  pCnt  B )
)  <->  ( ( p 
pCnt  A )  +  ( p  pCnt  B )
)  =  ( 1  +  1 ) )
40 1re 9660 . . . . . . . . . . . . . . . . . 18  |-  1  e.  RR
4140, 40readdcli 9674 . . . . . . . . . . . . . . . . 17  |-  ( 1  +  1 )  e.  RR
4241recni 9673 . . . . . . . . . . . . . . . 16  |-  ( 1  +  1 )  e.  CC
434, 7nn0addcld 10953 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  -> 
( ( p  pCnt  A )  +  ( p 
pCnt  B ) )  e. 
NN0 )
4443nn0red 10950 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  -> 
( ( p  pCnt  A )  +  ( p 
pCnt  B ) )  e.  RR )
4544recnd 9687 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  -> 
( ( p  pCnt  A )  +  ( p 
pCnt  B ) )  e.  CC )
46 subeq0 9920 . . . . . . . . . . . . . . . 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 676 . . . . . . . . . . . . . . 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 269 . . . . . . . . . . . . . 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 9687 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  -> 
1  e.  CC )
5049, 49, 30, 33addsub4d 10052 . . . . . . . . . . . . . . 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 2473 . . . . . . . . . . . . . 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 263 . . . . . . . . . . . . 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 472 . . . . . . . . . . . 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 10148 . . . . . . . . . . . . . . . 16  |-  ( ( 1  e.  RR  /\  ( p  pCnt  A )  e.  RR )  -> 
( 0  <_  (
1  -  ( p 
pCnt  A ) )  <->  ( p  pCnt  A )  <_  1
) )
5540, 5, 54sylancr 676 . . . . . . . . . . . . . . 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 10148 . . . . . . . . . . . . . . . 16  |-  ( ( 1  e.  RR  /\  ( p  pCnt  B )  e.  RR )  -> 
( 0  <_  (
1  -  ( p 
pCnt  B ) )  <->  ( p  pCnt  B )  <_  1
) )
5740, 8, 56sylancr 676 . . . . . . . . . . . . . . 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 725 . . . . . . . . . . . . . 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 9958 . . . . . . . . . . . . . . . 16  |-  ( ( 1  e.  RR  /\  ( p  pCnt  A )  e.  RR )  -> 
( 1  -  (
p  pCnt  A )
)  e.  RR )
6040, 5, 59sylancr 676 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  -> 
( 1  -  (
p  pCnt  A )
)  e.  RR )
61 resubcl 9958 . . . . . . . . . . . . . . . 16  |-  ( ( 1  e.  RR  /\  ( p  pCnt  B )  e.  RR )  -> 
( 1  -  (
p  pCnt  B )
)  e.  RR )
6240, 8, 61sylancr 676 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  -> 
( 1  -  (
p  pCnt  B )
)  e.  RR )
63 add20 10147 . . . . . . . . . . . . . . . . 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 842 . . . . . . . . . . . . . . . 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 441 . . . . . . . . . . . . . . 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 673 . . . . . . . . . . . . . 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 243 . . . . . . . . . . . . 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 436 . . . . . . . . . . . 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 261 . . . . . . . . . . 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 265 . . . . . . . . . 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 2679 . . . . . . . . 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 240 . . . . . . . 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 441 . . . . . . 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 542 . . . . . 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 10983 . . . . . . . . . . 11  |-  ( A  e.  NN  ->  A  e.  ZZ )
76 nnne0 10664 . . . . . . . . . . 11  |-  ( A  e.  NN  ->  A  =/=  0 )
7775, 76jca 541 . . . . . . . . . 10  |-  ( A  e.  NN  ->  ( A  e.  ZZ  /\  A  =/=  0 ) )
783, 77syl 17 . . . . . . . . 9  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  -> 
( A  e.  ZZ  /\  A  =/=  0 ) )
79 nnz 10983 . . . . . . . . . . 11  |-  ( B  e.  NN  ->  B  e.  ZZ )
80 nnne0 10664 . . . . . . . . . . 11  |-  ( B  e.  NN  ->  B  =/=  0 )
8179, 80jca 541 . . . . . . . . . 10  |-  ( B  e.  NN  ->  ( B  e.  ZZ  /\  B  =/=  0 ) )
826, 81syl 17 . . . . . . . . 9  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  -> 
( B  e.  ZZ  /\  B  =/=  0 ) )
83 pcmul 14880 . . . . . . . . 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 1292 . . . . . . . 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 4405 . . . . . . 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 10909 . . . . . . . 8  |-  1  e.  NN0
87 nn0leltp1 11019 . . . . . . . 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 675 . . . . . . 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 9753 . . . . . . . 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 675 . . . . . . 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 287 . . . . . 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 242 . . . . 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 2805 . . . 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 226 . . 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 24142 . . . . 5  |-  ( A  e.  NN  ->  (
( mmu `  A
)  =/=  0  <->  A. p  e.  Prime  ( p 
pCnt  A )  <_  1
) )
96 issqf 24142 . . . . 5  |-  ( B  e.  NN  ->  (
( mmu `  B
)  =/=  0  <->  A. p  e.  Prime  ( p 
pCnt  B )  <_  1
) )
9795, 96bi2anan9 890 . . . 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 1050 . . 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 10654 . . . . 5  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( A  x.  B
)  e.  NN )
100993adant3 1050 . . . 4  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  ->  ( A  x.  B )  e.  NN )
101 issqf 24142 . . . 4  |-  ( ( A  x.  B )  e.  NN  ->  (
( mmu `  ( A  x.  B )
)  =/=  0  <->  A. p  e.  Prime  ( p 
pCnt  ( A  x.  B ) )  <_ 
1 ) )
102100, 101syl 17 . . 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 276 . 2  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  ->  (
( ( mmu `  A )  =/=  0  /\  ( mmu `  B
)  =/=  0 )  ->  ( mmu `  ( A  x.  B
) )  =/=  0
) )
104103imp 436 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 189    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904    =/= wne 2641   A.wral 2756   ifcif 3872   class class class wbr 4395   ` cfv 5589  (class class class)co 6308   CCcc 9555   RRcr 9556   0cc0 9557   1c1 9558    + caddc 9560    x. cmul 9562    < clt 9693    <_ cle 9694    - cmin 9880   NNcn 10631   NN0cn0 10893   ZZcz 10961    gcd cgcd 14547   Primecprime 14701    pCnt cpc 14865   mmucmu 24100
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-2o 7201  df-oadd 7204  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-sup 7974  df-inf 7975  df-card 8391  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-n0 10894  df-z 10962  df-uz 11183  df-q 11288  df-rp 11326  df-fz 11811  df-fl 12061  df-mod 12130  df-seq 12252  df-exp 12311  df-hash 12554  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-dvds 14383  df-gcd 14548  df-prm 14702  df-pc 14866  df-mu 24106
This theorem is referenced by:  mumul  24187
  Copyright terms: Public domain W3C validator