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Theorem mumullem2 20418
Description: Lemma for mumul 20419. 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 2675 . . . 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 447 . . . . . . . . . 10  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  ->  p  e.  Prime )
3 simpl1 958 . . . . . . . . . 10  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  ->  A  e.  NN )
42, 3pccld 12903 . . . . . . . . 9  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  -> 
( p  pCnt  A
)  e.  NN0 )
54nn0red 10019 . . . . . . . 8  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  -> 
( p  pCnt  A
)  e.  RR )
6 simpl2 959 . . . . . . . . . 10  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  ->  B  e.  NN )
72, 6pccld 12903 . . . . . . . . 9  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  -> 
( p  pCnt  B
)  e.  NN0 )
87nn0red 10019 . . . . . . . 8  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  -> 
( p  pCnt  B
)  e.  RR )
9 1re 8837 . . . . . . . . 9  |-  1  e.  RR
109a1i 10 . . . . . . . 8  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  -> 
1  e.  RR )
11 le2add 9256 . . . . . . . 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 ) ) )
125, 8, 10, 10, 11syl22anc 1183 . . . . . . 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 ) ) )
13 ax-1ne0 8806 . . . . . . . . . . . 12  |-  1  =/=  0
14 simpl3 960 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  -> 
( A  gcd  B
)  =  1 )
1514oveq2d 5874 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  -> 
( p  pCnt  ( A  gcd  B ) )  =  ( p  pCnt  1 ) )
163nnzd 10116 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  ->  A  e.  ZZ )
176nnzd 10116 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  ->  B  e.  ZZ )
18 pcgcd 12930 . . . . . . . . . . . . . . . 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 ) ) )
192, 16, 17, 18syl3anc 1182 . . . . . . . . . . . . . . 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 )
) )
20 pc1 12908 . . . . . . . . . . . . . . . 16  |-  ( p  e.  Prime  ->  ( p 
pCnt  1 )  =  0 )
2120adantl 452 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  -> 
( p  pCnt  1
)  =  0 )
2215, 19, 213eqtr3d 2323 . . . . . . . . . . . . . 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 )
23 ifid 3597 . . . . . . . . . . . . . . . 16  |-  if ( ( p  pCnt  A
)  <_  ( p  pCnt  B ) ,  1 ,  1 )  =  1
24 ifeq12 3578 . . . . . . . . . . . . . . . 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 ) ) )
2523, 24syl5eqr 2329 . . . . . . . . . . . . . . 15  |-  ( ( 1  =  ( p 
pCnt  A )  /\  1  =  ( p  pCnt  B ) )  ->  1  =  if ( ( p 
pCnt  A )  <_  (
p  pCnt  B ) ,  ( p  pCnt  A ) ,  ( p 
pCnt  B ) ) )
2625eqeq1d 2291 . . . . . . . . . . . . . 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 ) )
2722, 26syl5ibrcom 213 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  -> 
( ( 1  =  ( p  pCnt  A
)  /\  1  =  ( p  pCnt  B ) )  ->  1  = 
0 ) )
2827necon3ad 2482 . . . . . . . . . . . 12  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  -> 
( 1  =/=  0  ->  -.  ( 1  =  ( p  pCnt  A
)  /\  1  =  ( p  pCnt  B ) ) ) )
2913, 28mpi 16 . . . . . . . . . . 11  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  ->  -.  ( 1  =  ( p  pCnt  A )  /\  1  =  (
p  pCnt  B )
) )
30 ax-1cn 8795 . . . . . . . . . . . . 13  |-  1  e.  CC
315recnd 8861 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  -> 
( p  pCnt  A
)  e.  CC )
32 subeq0 9073 . . . . . . . . . . . . 13  |-  ( ( 1  e.  CC  /\  ( p  pCnt  A )  e.  CC )  -> 
( ( 1  -  ( p  pCnt  A
) )  =  0  <->  1  =  ( p 
pCnt  A ) ) )
3330, 31, 32sylancr 644 . . . . . . . . . . . 12  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  -> 
( ( 1  -  ( p  pCnt  A
) )  =  0  <->  1  =  ( p 
pCnt  A ) ) )
348recnd 8861 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  -> 
( p  pCnt  B
)  e.  CC )
35 subeq0 9073 . . . . . . . . . . . . 13  |-  ( ( 1  e.  CC  /\  ( p  pCnt  B )  e.  CC )  -> 
( ( 1  -  ( p  pCnt  B
) )  =  0  <->  1  =  ( p 
pCnt  B ) ) )
3630, 34, 35sylancr 644 . . . . . . . . . . . 12  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  -> 
( ( 1  -  ( p  pCnt  B
) )  =  0  <->  1  =  ( p 
pCnt  B ) ) )
3733, 36anbi12d 691 . . . . . . . . . . 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 ) ) ) )
3829, 37mtbird 292 . . . . . . . . . 10  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  ->  -.  ( ( 1  -  ( p  pCnt  A
) )  =  0  /\  ( 1  -  ( p  pCnt  B
) )  =  0 ) )
3938adantr 451 . . . . . . . . 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 ) )
40 eqcom 2285 . . . . . . . . . . 11  |-  ( ( 1  +  1 )  =  ( ( p 
pCnt  A )  +  ( p  pCnt  B )
)  <->  ( ( p 
pCnt  A )  +  ( p  pCnt  B )
)  =  ( 1  +  1 ) )
419, 9readdcli 8850 . . . . . . . . . . . . . . . . 17  |-  ( 1  +  1 )  e.  RR
4241recni 8849 . . . . . . . . . . . . . . . 16  |-  ( 1  +  1 )  e.  CC
434, 7nn0addcld 10022 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  -> 
( ( p  pCnt  A )  +  ( p 
pCnt  B ) )  e. 
NN0 )
4443nn0red 10019 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  -> 
( ( p  pCnt  A )  +  ( p 
pCnt  B ) )  e.  RR )
4544recnd 8861 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  -> 
( ( p  pCnt  A )  +  ( p 
pCnt  B ) )  e.  CC )
46 subeq0 9073 . . . . . . . . . . . . . . . 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 644 . . . . . . . . . . . . . . 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, 40syl6bb 252 . . . . . . . . . . . . . 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 ) ) )
4910recnd 8861 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  -> 
1  e.  CC )
5049, 49, 31, 34addsub4d 9204 . . . . . . . . . . . . . . 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 2291 . . . . . . . . . . . . . 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 246 . . . . . . . . . . . . 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 451 . . . . . . . . . . . 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 9287 . . . . . . . . . . . . . . . 16  |-  ( ( 1  e.  RR  /\  ( p  pCnt  A )  e.  RR )  -> 
( 0  <_  (
1  -  ( p 
pCnt  A ) )  <->  ( p  pCnt  A )  <_  1
) )
559, 5, 54sylancr 644 . . . . . . . . . . . . . . 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 9287 . . . . . . . . . . . . . . . 16  |-  ( ( 1  e.  RR  /\  ( p  pCnt  B )  e.  RR )  -> 
( 0  <_  (
1  -  ( p 
pCnt  B ) )  <->  ( p  pCnt  B )  <_  1
) )
579, 8, 56sylancr 644 . . . . . . . . . . . . . . 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 691 . . . . . . . . . . . . . 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 9111 . . . . . . . . . . . . . . . 16  |-  ( ( 1  e.  RR  /\  ( p  pCnt  A )  e.  RR )  -> 
( 1  -  (
p  pCnt  A )
)  e.  RR )
609, 5, 59sylancr 644 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  -> 
( 1  -  (
p  pCnt  A )
)  e.  RR )
61 resubcl 9111 . . . . . . . . . . . . . . . 16  |-  ( ( 1  e.  RR  /\  ( p  pCnt  B )  e.  RR )  -> 
( 1  -  (
p  pCnt  B )
)  e.  RR )
629, 8, 61sylancr 644 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  -> 
( 1  -  (
p  pCnt  B )
)  e.  RR )
63 add20 9286 . . . . . . . . . . . . . . . . 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 799 . . . . . . . . . . . . . . . 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 423 . . . . . . . . . . . . . . 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 642 . . . . . . . . . . . . . 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 226 . . . . . . . . . . . . 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 418 . . . . . . . . . . . 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 244 . . . . . . . . . . 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 ) ) )
7040, 69syl5bb 248 . . . . . . . . . 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 2479 . . . . . . . . 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 ) ) )
7239, 71mpbird 223 . . . . . . . 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 423 . . . . . . 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 ) ) ) )
7412, 73jcad 519 . . . . . 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 10045 . . . . . . . . . . 11  |-  ( A  e.  NN  ->  A  e.  ZZ )
76 nnne0 9778 . . . . . . . . . . 11  |-  ( A  e.  NN  ->  A  =/=  0 )
7775, 76jca 518 . . . . . . . . . 10  |-  ( A  e.  NN  ->  ( A  e.  ZZ  /\  A  =/=  0 ) )
783, 77syl 15 . . . . . . . . 9  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  -> 
( A  e.  ZZ  /\  A  =/=  0 ) )
79 nnz 10045 . . . . . . . . . . 11  |-  ( B  e.  NN  ->  B  e.  ZZ )
80 nnne0 9778 . . . . . . . . . . 11  |-  ( B  e.  NN  ->  B  =/=  0 )
8179, 80jca 518 . . . . . . . . . 10  |-  ( B  e.  NN  ->  ( B  e.  ZZ  /\  B  =/=  0 ) )
826, 81syl 15 . . . . . . . . 9  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  -> 
( B  e.  ZZ  /\  B  =/=  0 ) )
83 pcmul 12904 . . . . . . . . 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 1182 . . . . . . . 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 4033 . . . . . . 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 9981 . . . . . . . 8  |-  1  e.  NN0
87 nn0leltp1 10075 . . . . . . . 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 643 . . . . . . 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 8922 . . . . . . . 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 643 . . . . . . 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 270 . . . . . 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 225 . . . . 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 2621 . . . 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 209 . . 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 20374 . . . . 5  |-  ( A  e.  NN  ->  (
( mmu `  A
)  =/=  0  <->  A. p  e.  Prime  ( p 
pCnt  A )  <_  1
) )
96 issqf 20374 . . . . 5  |-  ( B  e.  NN  ->  (
( mmu `  B
)  =/=  0  <->  A. p  e.  Prime  ( p 
pCnt  B )  <_  1
) )
9795, 96bi2anan9 843 . . . 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 975 . . 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 9769 . . . . 5  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( A  x.  B
)  e.  NN )
100993adant3 975 . . . 4  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  ->  ( A  x.  B )  e.  NN )
101 issqf 20374 . . . 4  |-  ( ( A  x.  B )  e.  NN  ->  (
( mmu `  ( A  x.  B )
)  =/=  0  <->  A. p  e.  Prime  ( p 
pCnt  ( A  x.  B ) )  <_ 
1 ) )
102100, 101syl 15 . . 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 259 . 2  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  ->  (
( ( mmu `  A )  =/=  0  /\  ( mmu `  B
)  =/=  0 )  ->  ( mmu `  ( A  x.  B
) )  =/=  0
) )
104103imp 418 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 set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   ifcif 3565   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738    + caddc 8740    x. cmul 8742    < clt 8867    <_ cle 8868    - cmin 9037   NNcn 9746   NN0cn0 9965   ZZcz 10024    gcd cgcd 12685   Primecprime 12758    pCnt cpc 12889   mmucmu 20332
This theorem is referenced by:  mumul  20419
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-q 10317  df-rp 10355  df-fz 10783  df-fl 10925  df-mod 10974  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-dvds 12532  df-gcd 12686  df-prm 12759  df-pc 12890  df-mu 20338
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