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Theorem lgsdi 21108
Description: The Legendre symbol is completely multiplicative in its right argument. (Contributed by Mario Carneiro, 5-Feb-2015.)
Assertion
Ref Expression
lgsdi  |-  ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0
) )  ->  ( A  / L ( M  x.  N ) )  =  ( ( A  / L M )  x.  ( A  / L N ) ) )

Proof of Theorem lgsdi
Dummy variables  k  n  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 3anrot 941 . . . . 5  |-  ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  <->  ( M  e.  ZZ  /\  N  e.  ZZ  /\  A  e.  ZZ ) )
2 lgsdilem 21098 . . . . 5  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  A  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0
) )  ->  if ( ( A  <  0  /\  ( M  x.  N )  <  0 ) ,  -u
1 ,  1 )  =  ( if ( ( A  <  0  /\  M  <  0
) ,  -u 1 ,  1 )  x.  if ( ( A  <  0  /\  N  <  0 ) ,  -u
1 ,  1 ) ) )
31, 2sylanb 459 . . . 4  |-  ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0
) )  ->  if ( ( A  <  0  /\  ( M  x.  N )  <  0 ) ,  -u
1 ,  1 )  =  ( if ( ( A  <  0  /\  M  <  0
) ,  -u 1 ,  1 )  x.  if ( ( A  <  0  /\  N  <  0 ) ,  -u
1 ,  1 ) ) )
4 ancom 438 . . . . 5  |-  ( ( ( M  x.  N
)  <  0  /\  A  <  0 )  <->  ( A  <  0  /\  ( M  x.  N )  <  0 ) )
5 ifbi 3748 . . . . 5  |-  ( ( ( ( M  x.  N )  <  0  /\  A  <  0
)  <->  ( A  <  0  /\  ( M  x.  N )  <  0 ) )  ->  if ( ( ( M  x.  N )  <  0  /\  A  <  0 ) ,  -u
1 ,  1 )  =  if ( ( A  <  0  /\  ( M  x.  N
)  <  0 ) ,  -u 1 ,  1 ) )
64, 5ax-mp 8 . . . 4  |-  if ( ( ( M  x.  N )  <  0  /\  A  <  0
) ,  -u 1 ,  1 )  =  if ( ( A  <  0  /\  ( M  x.  N )  <  0 ) ,  -u
1 ,  1 )
7 ancom 438 . . . . . 6  |-  ( ( M  <  0  /\  A  <  0 )  <-> 
( A  <  0  /\  M  <  0
) )
8 ifbi 3748 . . . . . 6  |-  ( ( ( M  <  0  /\  A  <  0
)  <->  ( A  <  0  /\  M  <  0 ) )  ->  if ( ( M  <  0  /\  A  <  0 ) ,  -u
1 ,  1 )  =  if ( ( A  <  0  /\  M  <  0 ) ,  -u 1 ,  1 ) )
97, 8ax-mp 8 . . . . 5  |-  if ( ( M  <  0  /\  A  <  0
) ,  -u 1 ,  1 )  =  if ( ( A  <  0  /\  M  <  0 ) ,  -u
1 ,  1 )
10 ancom 438 . . . . . 6  |-  ( ( N  <  0  /\  A  <  0 )  <-> 
( A  <  0  /\  N  <  0
) )
11 ifbi 3748 . . . . . 6  |-  ( ( ( N  <  0  /\  A  <  0
)  <->  ( A  <  0  /\  N  <  0 ) )  ->  if ( ( N  <  0  /\  A  <  0 ) ,  -u
1 ,  1 )  =  if ( ( A  <  0  /\  N  <  0 ) ,  -u 1 ,  1 ) )
1210, 11ax-mp 8 . . . . 5  |-  if ( ( N  <  0  /\  A  <  0
) ,  -u 1 ,  1 )  =  if ( ( A  <  0  /\  N  <  0 ) ,  -u
1 ,  1 )
139, 12oveq12i 6085 . . . 4  |-  ( if ( ( M  <  0  /\  A  <  0 ) ,  -u
1 ,  1 )  x.  if ( ( N  <  0  /\  A  <  0 ) ,  -u 1 ,  1 ) )  =  ( if ( ( A  <  0  /\  M  <  0 ) ,  -u
1 ,  1 )  x.  if ( ( A  <  0  /\  N  <  0 ) ,  -u 1 ,  1 ) )
143, 6, 133eqtr4g 2492 . . 3  |-  ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0
) )  ->  if ( ( ( M  x.  N )  <  0  /\  A  <  0 ) ,  -u
1 ,  1 )  =  ( if ( ( M  <  0  /\  A  <  0
) ,  -u 1 ,  1 )  x.  if ( ( N  <  0  /\  A  <  0 ) ,  -u
1 ,  1 ) ) )
15 mulcl 9066 . . . . . 6  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  x.  y
)  e.  CC )
1615adantl 453 . . . . 5  |-  ( ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0 ) )  /\  ( x  e.  CC  /\  y  e.  CC ) )  ->  ( x  x.  y )  e.  CC )
17 mulcom 9068 . . . . . 6  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  x.  y
)  =  ( y  x.  x ) )
1817adantl 453 . . . . 5  |-  ( ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0 ) )  /\  ( x  e.  CC  /\  y  e.  CC ) )  ->  ( x  x.  y )  =  ( y  x.  x ) )
19 mulass 9070 . . . . . 6  |-  ( ( x  e.  CC  /\  y  e.  CC  /\  z  e.  CC )  ->  (
( x  x.  y
)  x.  z )  =  ( x  x.  ( y  x.  z
) ) )
2019adantl 453 . . . . 5  |-  ( ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0 ) )  /\  ( x  e.  CC  /\  y  e.  CC  /\  z  e.  CC )
)  ->  ( (
x  x.  y )  x.  z )  =  ( x  x.  (
y  x.  z ) ) )
21 simpl2 961 . . . . . . . 8  |-  ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0
) )  ->  M  e.  ZZ )
22 simpl3 962 . . . . . . . 8  |-  ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0
) )  ->  N  e.  ZZ )
2321, 22zmulcld 10373 . . . . . . 7  |-  ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0
) )  ->  ( M  x.  N )  e.  ZZ )
2421zcnd 10368 . . . . . . . 8  |-  ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0
) )  ->  M  e.  CC )
2522zcnd 10368 . . . . . . . 8  |-  ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0
) )  ->  N  e.  CC )
26 simprl 733 . . . . . . . 8  |-  ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0
) )  ->  M  =/=  0 )
27 simprr 734 . . . . . . . 8  |-  ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0
) )  ->  N  =/=  0 )
2824, 25, 26, 27mulne0d 9666 . . . . . . 7  |-  ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0
) )  ->  ( M  x.  N )  =/=  0 )
29 nnabscl 12121 . . . . . . 7  |-  ( ( ( M  x.  N
)  e.  ZZ  /\  ( M  x.  N
)  =/=  0 )  ->  ( abs `  ( M  x.  N )
)  e.  NN )
3023, 28, 29syl2anc 643 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0
) )  ->  ( abs `  ( M  x.  N ) )  e.  NN )
31 nnuz 10513 . . . . . 6  |-  NN  =  ( ZZ>= `  1 )
3230, 31syl6eleq 2525 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0
) )  ->  ( abs `  ( M  x.  N ) )  e.  ( ZZ>= `  1 )
)
33 simpl1 960 . . . . . . . 8  |-  ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0
) )  ->  A  e.  ZZ )
34 eqid 2435 . . . . . . . . 9  |-  ( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( A  / L
n ) ^ (
n  pCnt  M )
) ,  1 ) )  =  ( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( A  / L
n ) ^ (
n  pCnt  M )
) ,  1 ) )
3534lgsfcl3 21093 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  M  =/=  0 )  ->  (
n  e.  NN  |->  if ( n  e.  Prime ,  ( ( A  / L n ) ^
( n  pCnt  M
) ) ,  1 ) ) : NN --> ZZ )
3633, 21, 26, 35syl3anc 1184 . . . . . . 7  |-  ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0
) )  ->  (
n  e.  NN  |->  if ( n  e.  Prime ,  ( ( A  / L n ) ^
( n  pCnt  M
) ) ,  1 ) ) : NN --> ZZ )
37 elfznn 11072 . . . . . . 7  |-  ( k  e.  ( 1 ... ( abs `  ( M  x.  N )
) )  ->  k  e.  NN )
38 ffvelrn 5860 . . . . . . 7  |-  ( ( ( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( A  / L n ) ^
( n  pCnt  M
) ) ,  1 ) ) : NN --> ZZ  /\  k  e.  NN )  ->  ( ( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( A  / L
n ) ^ (
n  pCnt  M )
) ,  1 ) ) `  k )  e.  ZZ )
3936, 37, 38syl2an 464 . . . . . 6  |-  ( ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0 ) )  /\  k  e.  ( 1 ... ( abs `  ( M  x.  N )
) ) )  -> 
( ( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( A  / L n ) ^ ( n 
pCnt  M ) ) ,  1 ) ) `  k )  e.  ZZ )
4039zcnd 10368 . . . . 5  |-  ( ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0 ) )  /\  k  e.  ( 1 ... ( abs `  ( M  x.  N )
) ) )  -> 
( ( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( A  / L n ) ^ ( n 
pCnt  M ) ) ,  1 ) ) `  k )  e.  CC )
41 eqid 2435 . . . . . . . . 9  |-  ( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( A  / L
n ) ^ (
n  pCnt  N )
) ,  1 ) )  =  ( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( A  / L
n ) ^ (
n  pCnt  N )
) ,  1 ) )
4241lgsfcl3 21093 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  ->  (
n  e.  NN  |->  if ( n  e.  Prime ,  ( ( A  / L n ) ^
( n  pCnt  N
) ) ,  1 ) ) : NN --> ZZ )
4333, 22, 27, 42syl3anc 1184 . . . . . . 7  |-  ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0
) )  ->  (
n  e.  NN  |->  if ( n  e.  Prime ,  ( ( A  / L n ) ^
( n  pCnt  N
) ) ,  1 ) ) : NN --> ZZ )
44 ffvelrn 5860 . . . . . . 7  |-  ( ( ( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( A  / L n ) ^
( n  pCnt  N
) ) ,  1 ) ) : NN --> ZZ  /\  k  e.  NN )  ->  ( ( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( A  / L
n ) ^ (
n  pCnt  N )
) ,  1 ) ) `  k )  e.  ZZ )
4543, 37, 44syl2an 464 . . . . . 6  |-  ( ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0 ) )  /\  k  e.  ( 1 ... ( abs `  ( M  x.  N )
) ) )  -> 
( ( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( A  / L n ) ^ ( n 
pCnt  N ) ) ,  1 ) ) `  k )  e.  ZZ )
4645zcnd 10368 . . . . 5  |-  ( ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0 ) )  /\  k  e.  ( 1 ... ( abs `  ( M  x.  N )
) ) )  -> 
( ( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( A  / L n ) ^ ( n 
pCnt  N ) ) ,  1 ) ) `  k )  e.  CC )
47 simpr 448 . . . . . . . . . . 11  |-  ( ( ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0 ) )  /\  k  e.  ( 1 ... ( abs `  ( M  x.  N )
) ) )  /\  k  e.  Prime )  -> 
k  e.  Prime )
4821ad2antrr 707 . . . . . . . . . . 11  |-  ( ( ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0 ) )  /\  k  e.  ( 1 ... ( abs `  ( M  x.  N )
) ) )  /\  k  e.  Prime )  ->  M  e.  ZZ )
4926ad2antrr 707 . . . . . . . . . . 11  |-  ( ( ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0 ) )  /\  k  e.  ( 1 ... ( abs `  ( M  x.  N )
) ) )  /\  k  e.  Prime )  ->  M  =/=  0 )
5022ad2antrr 707 . . . . . . . . . . 11  |-  ( ( ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0 ) )  /\  k  e.  ( 1 ... ( abs `  ( M  x.  N )
) ) )  /\  k  e.  Prime )  ->  N  e.  ZZ )
5127ad2antrr 707 . . . . . . . . . . 11  |-  ( ( ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0 ) )  /\  k  e.  ( 1 ... ( abs `  ( M  x.  N )
) ) )  /\  k  e.  Prime )  ->  N  =/=  0 )
52 pcmul 13217 . . . . . . . . . . 11  |-  ( ( k  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( k  pCnt  ( M  x.  N )
)  =  ( ( k  pCnt  M )  +  ( k  pCnt  N ) ) )
5347, 48, 49, 50, 51, 52syl122anc 1193 . . . . . . . . . 10  |-  ( ( ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0 ) )  /\  k  e.  ( 1 ... ( abs `  ( M  x.  N )
) ) )  /\  k  e.  Prime )  -> 
( k  pCnt  ( M  x.  N )
)  =  ( ( k  pCnt  M )  +  ( k  pCnt  N ) ) )
5453oveq2d 6089 . . . . . . . . 9  |-  ( ( ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0 ) )  /\  k  e.  ( 1 ... ( abs `  ( M  x.  N )
) ) )  /\  k  e.  Prime )  -> 
( ( A  / L k ) ^
( k  pCnt  ( M  x.  N )
) )  =  ( ( A  / L
k ) ^ (
( k  pCnt  M
)  +  ( k 
pCnt  N ) ) ) )
5533ad2antrr 707 . . . . . . . . . . . 12  |-  ( ( ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0 ) )  /\  k  e.  ( 1 ... ( abs `  ( M  x.  N )
) ) )  /\  k  e.  Prime )  ->  A  e.  ZZ )
56 prmz 13075 . . . . . . . . . . . . 13  |-  ( k  e.  Prime  ->  k  e.  ZZ )
5756adantl 453 . . . . . . . . . . . 12  |-  ( ( ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0 ) )  /\  k  e.  ( 1 ... ( abs `  ( M  x.  N )
) ) )  /\  k  e.  Prime )  -> 
k  e.  ZZ )
58 lgscl 21086 . . . . . . . . . . . 12  |-  ( ( A  e.  ZZ  /\  k  e.  ZZ )  ->  ( A  / L
k )  e.  ZZ )
5955, 57, 58syl2anc 643 . . . . . . . . . . 11  |-  ( ( ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0 ) )  /\  k  e.  ( 1 ... ( abs `  ( M  x.  N )
) ) )  /\  k  e.  Prime )  -> 
( A  / L
k )  e.  ZZ )
6059zcnd 10368 . . . . . . . . . 10  |-  ( ( ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0 ) )  /\  k  e.  ( 1 ... ( abs `  ( M  x.  N )
) ) )  /\  k  e.  Prime )  -> 
( A  / L
k )  e.  CC )
61 pczcl 13214 . . . . . . . . . . 11  |-  ( ( k  e.  Prime  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( k  pCnt  N
)  e.  NN0 )
6247, 50, 51, 61syl12anc 1182 . . . . . . . . . 10  |-  ( ( ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0 ) )  /\  k  e.  ( 1 ... ( abs `  ( M  x.  N )
) ) )  /\  k  e.  Prime )  -> 
( k  pCnt  N
)  e.  NN0 )
63 pczcl 13214 . . . . . . . . . . 11  |-  ( ( k  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 ) )  -> 
( k  pCnt  M
)  e.  NN0 )
6447, 48, 49, 63syl12anc 1182 . . . . . . . . . 10  |-  ( ( ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0 ) )  /\  k  e.  ( 1 ... ( abs `  ( M  x.  N )
) ) )  /\  k  e.  Prime )  -> 
( k  pCnt  M
)  e.  NN0 )
6560, 62, 64expaddd 11517 . . . . . . . . 9  |-  ( ( ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0 ) )  /\  k  e.  ( 1 ... ( abs `  ( M  x.  N )
) ) )  /\  k  e.  Prime )  -> 
( ( A  / L k ) ^
( ( k  pCnt  M )  +  ( k 
pCnt  N ) ) )  =  ( ( ( A  / L k ) ^ ( k 
pCnt  M ) )  x.  ( ( A  / L k ) ^
( k  pCnt  N
) ) ) )
6654, 65eqtrd 2467 . . . . . . . 8  |-  ( ( ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0 ) )  /\  k  e.  ( 1 ... ( abs `  ( M  x.  N )
) ) )  /\  k  e.  Prime )  -> 
( ( A  / L k ) ^
( k  pCnt  ( M  x.  N )
) )  =  ( ( ( A  / L k ) ^
( k  pCnt  M
) )  x.  (
( A  / L
k ) ^ (
k  pCnt  N )
) ) )
67 iftrue 3737 . . . . . . . . 9  |-  ( k  e.  Prime  ->  if ( k  e.  Prime ,  ( ( A  / L
k ) ^ (
k  pCnt  ( M  x.  N ) ) ) ,  1 )  =  ( ( A  / L k ) ^
( k  pCnt  ( M  x.  N )
) ) )
6867adantl 453 . . . . . . . 8  |-  ( ( ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0 ) )  /\  k  e.  ( 1 ... ( abs `  ( M  x.  N )
) ) )  /\  k  e.  Prime )  ->  if ( k  e.  Prime ,  ( ( A  / L k ) ^
( k  pCnt  ( M  x.  N )
) ) ,  1 )  =  ( ( A  / L k ) ^ ( k 
pCnt  ( M  x.  N ) ) ) )
69 iftrue 3737 . . . . . . . . . 10  |-  ( k  e.  Prime  ->  if ( k  e.  Prime ,  ( ( A  / L
k ) ^ (
k  pCnt  M )
) ,  1 )  =  ( ( A  / L k ) ^ ( k  pCnt  M ) ) )
70 iftrue 3737 . . . . . . . . . 10  |-  ( k  e.  Prime  ->  if ( k  e.  Prime ,  ( ( A  / L
k ) ^ (
k  pCnt  N )
) ,  1 )  =  ( ( A  / L k ) ^ ( k  pCnt  N ) ) )
7169, 70oveq12d 6091 . . . . . . . . 9  |-  ( k  e.  Prime  ->  ( if ( k  e.  Prime ,  ( ( A  / L k ) ^
( k  pCnt  M
) ) ,  1 )  x.  if ( k  e.  Prime ,  ( ( A  / L
k ) ^ (
k  pCnt  N )
) ,  1 ) )  =  ( ( ( A  / L
k ) ^ (
k  pCnt  M )
)  x.  ( ( A  / L k ) ^ ( k 
pCnt  N ) ) ) )
7271adantl 453 . . . . . . . 8  |-  ( ( ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0 ) )  /\  k  e.  ( 1 ... ( abs `  ( M  x.  N )
) ) )  /\  k  e.  Prime )  -> 
( if ( k  e.  Prime ,  ( ( A  / L k ) ^ ( k 
pCnt  M ) ) ,  1 )  x.  if ( k  e.  Prime ,  ( ( A  / L k ) ^
( k  pCnt  N
) ) ,  1 ) )  =  ( ( ( A  / L k ) ^
( k  pCnt  M
) )  x.  (
( A  / L
k ) ^ (
k  pCnt  N )
) ) )
7366, 68, 723eqtr4rd 2478 . . . . . . 7  |-  ( ( ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0 ) )  /\  k  e.  ( 1 ... ( abs `  ( M  x.  N )
) ) )  /\  k  e.  Prime )  -> 
( if ( k  e.  Prime ,  ( ( A  / L k ) ^ ( k 
pCnt  M ) ) ,  1 )  x.  if ( k  e.  Prime ,  ( ( A  / L k ) ^
( k  pCnt  N
) ) ,  1 ) )  =  if ( k  e.  Prime ,  ( ( A  / L k ) ^
( k  pCnt  ( M  x.  N )
) ) ,  1 ) )
74 1t1e1 10118 . . . . . . . . 9  |-  ( 1  x.  1 )  =  1
75 iffalse 3738 . . . . . . . . . 10  |-  ( -.  k  e.  Prime  ->  if ( k  e.  Prime ,  ( ( A  / L k ) ^
( k  pCnt  M
) ) ,  1 )  =  1 )
76 iffalse 3738 . . . . . . . . . 10  |-  ( -.  k  e.  Prime  ->  if ( k  e.  Prime ,  ( ( A  / L k ) ^
( k  pCnt  N
) ) ,  1 )  =  1 )
7775, 76oveq12d 6091 . . . . . . . . 9  |-  ( -.  k  e.  Prime  ->  ( if ( k  e. 
Prime ,  ( ( A  / L k ) ^ ( k  pCnt  M ) ) ,  1 )  x.  if ( k  e.  Prime ,  ( ( A  / L
k ) ^ (
k  pCnt  N )
) ,  1 ) )  =  ( 1  x.  1 ) )
78 iffalse 3738 . . . . . . . . 9  |-  ( -.  k  e.  Prime  ->  if ( k  e.  Prime ,  ( ( A  / L k ) ^
( k  pCnt  ( M  x.  N )
) ) ,  1 )  =  1 )
7974, 77, 783eqtr4a 2493 . . . . . . . 8  |-  ( -.  k  e.  Prime  ->  ( if ( k  e. 
Prime ,  ( ( A  / L k ) ^ ( k  pCnt  M ) ) ,  1 )  x.  if ( k  e.  Prime ,  ( ( A  / L
k ) ^ (
k  pCnt  N )
) ,  1 ) )  =  if ( k  e.  Prime ,  ( ( A  / L
k ) ^ (
k  pCnt  ( M  x.  N ) ) ) ,  1 ) )
8079adantl 453 . . . . . . 7  |-  ( ( ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0 ) )  /\  k  e.  ( 1 ... ( abs `  ( M  x.  N )
) ) )  /\  -.  k  e.  Prime )  ->  ( if ( k  e.  Prime ,  ( ( A  / L
k ) ^ (
k  pCnt  M )
) ,  1 )  x.  if ( k  e.  Prime ,  ( ( A  / L k ) ^ ( k 
pCnt  N ) ) ,  1 ) )  =  if ( k  e. 
Prime ,  ( ( A  / L k ) ^ ( k  pCnt  ( M  x.  N ) ) ) ,  1 ) )
8173, 80pm2.61dan 767 . . . . . 6  |-  ( ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0 ) )  /\  k  e.  ( 1 ... ( abs `  ( M  x.  N )
) ) )  -> 
( if ( k  e.  Prime ,  ( ( A  / L k ) ^ ( k 
pCnt  M ) ) ,  1 )  x.  if ( k  e.  Prime ,  ( ( A  / L k ) ^
( k  pCnt  N
) ) ,  1 ) )  =  if ( k  e.  Prime ,  ( ( A  / L k ) ^
( k  pCnt  ( M  x.  N )
) ) ,  1 ) )
8237adantl 453 . . . . . . 7  |-  ( ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0 ) )  /\  k  e.  ( 1 ... ( abs `  ( M  x.  N )
) ) )  -> 
k  e.  NN )
83 eleq1 2495 . . . . . . . . . 10  |-  ( n  =  k  ->  (
n  e.  Prime  <->  k  e.  Prime ) )
84 oveq2 6081 . . . . . . . . . . 11  |-  ( n  =  k  ->  ( A  / L n )  =  ( A  / L k ) )
85 oveq1 6080 . . . . . . . . . . 11  |-  ( n  =  k  ->  (
n  pCnt  M )  =  ( k  pCnt  M ) )
8684, 85oveq12d 6091 . . . . . . . . . 10  |-  ( n  =  k  ->  (
( A  / L
n ) ^ (
n  pCnt  M )
)  =  ( ( A  / L k ) ^ ( k 
pCnt  M ) ) )
87 eqidd 2436 . . . . . . . . . 10  |-  ( n  =  k  ->  1  =  1 )
8883, 86, 87ifbieq12d 3753 . . . . . . . . 9  |-  ( n  =  k  ->  if ( n  e.  Prime ,  ( ( A  / L n ) ^
( n  pCnt  M
) ) ,  1 )  =  if ( k  e.  Prime ,  ( ( A  / L
k ) ^ (
k  pCnt  M )
) ,  1 ) )
89 ovex 6098 . . . . . . . . . 10  |-  ( ( A  / L k ) ^ ( k 
pCnt  M ) )  e. 
_V
90 1ex 9078 . . . . . . . . . 10  |-  1  e.  _V
9189, 90ifex 3789 . . . . . . . . 9  |-  if ( k  e.  Prime ,  ( ( A  / L
k ) ^ (
k  pCnt  M )
) ,  1 )  e.  _V
9288, 34, 91fvmpt 5798 . . . . . . . 8  |-  ( k  e.  NN  ->  (
( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( A  / L n ) ^
( n  pCnt  M
) ) ,  1 ) ) `  k
)  =  if ( k  e.  Prime ,  ( ( A  / L
k ) ^ (
k  pCnt  M )
) ,  1 ) )
93 oveq1 6080 . . . . . . . . . . 11  |-  ( n  =  k  ->  (
n  pCnt  N )  =  ( k  pCnt  N ) )
9484, 93oveq12d 6091 . . . . . . . . . 10  |-  ( n  =  k  ->  (
( A  / L
n ) ^ (
n  pCnt  N )
)  =  ( ( A  / L k ) ^ ( k 
pCnt  N ) ) )
9583, 94, 87ifbieq12d 3753 . . . . . . . . 9  |-  ( n  =  k  ->  if ( n  e.  Prime ,  ( ( A  / L n ) ^
( n  pCnt  N
) ) ,  1 )  =  if ( k  e.  Prime ,  ( ( A  / L
k ) ^ (
k  pCnt  N )
) ,  1 ) )
96 ovex 6098 . . . . . . . . . 10  |-  ( ( A  / L k ) ^ ( k 
pCnt  N ) )  e. 
_V
9796, 90ifex 3789 . . . . . . . . 9  |-  if ( k  e.  Prime ,  ( ( A  / L
k ) ^ (
k  pCnt  N )
) ,  1 )  e.  _V
9895, 41, 97fvmpt 5798 . . . . . . . 8  |-  ( k  e.  NN  ->  (
( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( A  / L n ) ^
( n  pCnt  N
) ) ,  1 ) ) `  k
)  =  if ( k  e.  Prime ,  ( ( A  / L
k ) ^ (
k  pCnt  N )
) ,  1 ) )
9992, 98oveq12d 6091 . . . . . . 7  |-  ( k  e.  NN  ->  (
( ( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( A  / L n ) ^ ( n 
pCnt  M ) ) ,  1 ) ) `  k )  x.  (
( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( A  / L n ) ^
( n  pCnt  N
) ) ,  1 ) ) `  k
) )  =  ( if ( k  e. 
Prime ,  ( ( A  / L k ) ^ ( k  pCnt  M ) ) ,  1 )  x.  if ( k  e.  Prime ,  ( ( A  / L
k ) ^ (
k  pCnt  N )
) ,  1 ) ) )
10082, 99syl 16 . . . . . 6  |-  ( ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0 ) )  /\  k  e.  ( 1 ... ( abs `  ( M  x.  N )
) ) )  -> 
( ( ( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( A  / L
n ) ^ (
n  pCnt  M )
) ,  1 ) ) `  k )  x.  ( ( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( A  / L
n ) ^ (
n  pCnt  N )
) ,  1 ) ) `  k ) )  =  ( if ( k  e.  Prime ,  ( ( A  / L k ) ^
( k  pCnt  M
) ) ,  1 )  x.  if ( k  e.  Prime ,  ( ( A  / L
k ) ^ (
k  pCnt  N )
) ,  1 ) ) )
101 oveq1 6080 . . . . . . . . . 10  |-  ( n  =  k  ->  (
n  pCnt  ( M  x.  N ) )  =  ( k  pCnt  ( M  x.  N )
) )
10284, 101oveq12d 6091 . . . . . . . . 9  |-  ( n  =  k  ->  (
( A  / L
n ) ^ (
n  pCnt  ( M  x.  N ) ) )  =  ( ( A  / L k ) ^ ( k  pCnt  ( M  x.  N ) ) ) )
10383, 102, 87ifbieq12d 3753 . . . . . . . 8  |-  ( n  =  k  ->  if ( n  e.  Prime ,  ( ( A  / L n ) ^
( n  pCnt  ( M  x.  N )
) ) ,  1 )  =  if ( k  e.  Prime ,  ( ( A  / L
k ) ^ (
k  pCnt  ( M  x.  N ) ) ) ,  1 ) )
104 eqid 2435 . . . . . . . 8  |-  ( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( A  / L
n ) ^ (
n  pCnt  ( M  x.  N ) ) ) ,  1 ) )  =  ( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( A  / L n ) ^ ( n 
pCnt  ( M  x.  N ) ) ) ,  1 ) )
105 ovex 6098 . . . . . . . . 9  |-  ( ( A  / L k ) ^ ( k 
pCnt  ( M  x.  N ) ) )  e.  _V
106105, 90ifex 3789 . . . . . . . 8  |-  if ( k  e.  Prime ,  ( ( A  / L
k ) ^ (
k  pCnt  ( M  x.  N ) ) ) ,  1 )  e. 
_V
107103, 104, 106fvmpt 5798 . . . . . . 7  |-  ( k  e.  NN  ->  (
( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( A  / L n ) ^
( n  pCnt  ( M  x.  N )
) ) ,  1 ) ) `  k
)  =  if ( k  e.  Prime ,  ( ( A  / L
k ) ^ (
k  pCnt  ( M  x.  N ) ) ) ,  1 ) )
10882, 107syl 16 . . . . . 6  |-  ( ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0 ) )  /\  k  e.  ( 1 ... ( abs `  ( M  x.  N )
) ) )  -> 
( ( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( A  / L n ) ^ ( n 
pCnt  ( M  x.  N ) ) ) ,  1 ) ) `
 k )  =  if ( k  e. 
Prime ,  ( ( A  / L k ) ^ ( k  pCnt  ( M  x.  N ) ) ) ,  1 ) )
10981, 100, 1083eqtr4rd 2478 . . . . 5  |-  ( ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0 ) )  /\  k  e.  ( 1 ... ( abs `  ( M  x.  N )
) ) )  -> 
( ( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( A  / L n ) ^ ( n 
pCnt  ( M  x.  N ) ) ) ,  1 ) ) `
 k )  =  ( ( ( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( A  / L
n ) ^ (
n  pCnt  M )
) ,  1 ) ) `  k )  x.  ( ( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( A  / L
n ) ^ (
n  pCnt  N )
) ,  1 ) ) `  k ) ) )
11016, 18, 20, 32, 40, 46, 109seqcaopr 11352 . . . 4  |-  ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0
) )  ->  (  seq  1 (  x.  , 
( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( A  / L n ) ^
( n  pCnt  ( M  x.  N )
) ) ,  1 ) ) ) `  ( abs `  ( M  x.  N ) ) )  =  ( (  seq  1 (  x.  ,  ( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( A  / L n ) ^ ( n 
pCnt  M ) ) ,  1 ) ) ) `
 ( abs `  ( M  x.  N )
) )  x.  (  seq  1 (  x.  , 
( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( A  / L n ) ^
( n  pCnt  N
) ) ,  1 ) ) ) `  ( abs `  ( M  x.  N ) ) ) ) )
11133, 21, 22, 26, 27, 34lgsdilem2 21107 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0
) )  ->  (  seq  1 (  x.  , 
( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( A  / L n ) ^
( n  pCnt  M
) ) ,  1 ) ) ) `  ( abs `  M ) )  =  (  seq  1 (  x.  , 
( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( A  / L n ) ^
( n  pCnt  M
) ) ,  1 ) ) ) `  ( abs `  ( M  x.  N ) ) ) )
11233, 22, 21, 27, 26, 41lgsdilem2 21107 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0
) )  ->  (  seq  1 (  x.  , 
( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( A  / L n ) ^
( n  pCnt  N
) ) ,  1 ) ) ) `  ( abs `  N ) )  =  (  seq  1 (  x.  , 
( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( A  / L n ) ^
( n  pCnt  N
) ) ,  1 ) ) ) `  ( abs `  ( N  x.  M ) ) ) )
11324, 25mulcomd 9101 . . . . . . . 8  |-  ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0
) )  ->  ( M  x.  N )  =  ( N  x.  M ) )
114113fveq2d 5724 . . . . . . 7  |-  ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0
) )  ->  ( abs `  ( M  x.  N ) )  =  ( abs `  ( N  x.  M )
) )
115114fveq2d 5724 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0
) )  ->  (  seq  1 (  x.  , 
( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( A  / L n ) ^
( n  pCnt  N
) ) ,  1 ) ) ) `  ( abs `  ( M  x.  N ) ) )  =  (  seq  1 (  x.  , 
( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( A  / L n ) ^
( n  pCnt  N
) ) ,  1 ) ) ) `  ( abs `  ( N  x.  M ) ) ) )
116112, 115eqtr4d 2470 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0
) )  ->  (  seq  1 (  x.  , 
( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( A  / L n ) ^
( n  pCnt  N
) ) ,  1 ) ) ) `  ( abs `  N ) )  =  (  seq  1 (  x.  , 
( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( A  / L n ) ^
( n  pCnt  N
) ) ,  1 ) ) ) `  ( abs `  ( M  x.  N ) ) ) )
117111, 116oveq12d 6091 . . . 4  |-  ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0
) )  ->  (
(  seq  1 (  x.  ,  ( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( A  / L
n ) ^ (
n  pCnt  M )
) ,  1 ) ) ) `  ( abs `  M ) )  x.  (  seq  1
(  x.  ,  ( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( A  / L n ) ^
( n  pCnt  N
) ) ,  1 ) ) ) `  ( abs `  N ) ) )  =  ( (  seq  1 (  x.  ,  ( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( A  / L
n ) ^ (
n  pCnt  M )
) ,  1 ) ) ) `  ( abs `  ( M  x.  N ) ) )  x.  (  seq  1
(  x.  ,  ( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( A  / L n ) ^
( n  pCnt  N
) ) ,  1 ) ) ) `  ( abs `  ( M  x.  N ) ) ) ) )
118110, 117eqtr4d 2470 . . 3  |-  ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0
) )  ->  (  seq  1 (  x.  , 
( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( A  / L n ) ^
( n  pCnt  ( M  x.  N )
) ) ,  1 ) ) ) `  ( abs `  ( M  x.  N ) ) )  =  ( (  seq  1 (  x.  ,  ( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( A  / L n ) ^ ( n 
pCnt  M ) ) ,  1 ) ) ) `
 ( abs `  M
) )  x.  (  seq  1 (  x.  , 
( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( A  / L n ) ^
( n  pCnt  N
) ) ,  1 ) ) ) `  ( abs `  N ) ) ) )
11914, 118oveq12d 6091 . 2  |-  ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0
) )  ->  ( if ( ( ( M  x.  N )  <  0  /\  A  <  0 ) ,  -u
1 ,  1 )  x.  (  seq  1
(  x.  ,  ( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( A  / L n ) ^
( n  pCnt  ( M  x.  N )
) ) ,  1 ) ) ) `  ( abs `  ( M  x.  N ) ) ) )  =  ( ( if ( ( M  <  0  /\  A  <  0 ) ,  -u 1 ,  1 )  x.  if ( ( N  <  0  /\  A  <  0
) ,  -u 1 ,  1 ) )  x.  ( (  seq  1 (  x.  , 
( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( A  / L n ) ^
( n  pCnt  M
) ) ,  1 ) ) ) `  ( abs `  M ) )  x.  (  seq  1 (  x.  , 
( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( A  / L n ) ^
( n  pCnt  N
) ) ,  1 ) ) ) `  ( abs `  N ) ) ) ) )
120104lgsval4 21092 . . 3  |-  ( ( A  e.  ZZ  /\  ( M  x.  N
)  e.  ZZ  /\  ( M  x.  N
)  =/=  0 )  ->  ( A  / L ( M  x.  N ) )  =  ( if ( ( ( M  x.  N
)  <  0  /\  A  <  0 ) , 
-u 1 ,  1 )  x.  (  seq  1 (  x.  , 
( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( A  / L n ) ^
( n  pCnt  ( M  x.  N )
) ) ,  1 ) ) ) `  ( abs `  ( M  x.  N ) ) ) ) )
12133, 23, 28, 120syl3anc 1184 . 2  |-  ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0
) )  ->  ( A  / L ( M  x.  N ) )  =  ( if ( ( ( M  x.  N )  <  0  /\  A  <  0
) ,  -u 1 ,  1 )  x.  (  seq  1 (  x.  ,  ( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( A  / L
n ) ^ (
n  pCnt  ( M  x.  N ) ) ) ,  1 ) ) ) `  ( abs `  ( M  x.  N
) ) ) ) )
12234lgsval4 21092 . . . . 5  |-  ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  M  =/=  0 )  ->  ( A  / L M )  =  ( if ( ( M  <  0  /\  A  <  0
) ,  -u 1 ,  1 )  x.  (  seq  1 (  x.  ,  ( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( A  / L
n ) ^ (
n  pCnt  M )
) ,  1 ) ) ) `  ( abs `  M ) ) ) )
12333, 21, 26, 122syl3anc 1184 . . . 4  |-  ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0
) )  ->  ( A  / L M )  =  ( if ( ( M  <  0  /\  A  <  0
) ,  -u 1 ,  1 )  x.  (  seq  1 (  x.  ,  ( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( A  / L
n ) ^ (
n  pCnt  M )
) ,  1 ) ) ) `  ( abs `  M ) ) ) )
12441lgsval4 21092 . . . . 5  |-  ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  ->  ( A  / L N )  =  ( if ( ( N  <  0  /\  A  <  0
) ,  -u 1 ,  1 )  x.  (  seq  1 (  x.  ,  ( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( A  / L
n ) ^ (
n  pCnt  N )
) ,  1 ) ) ) `  ( abs `  N ) ) ) )
12533, 22, 27, 124syl3anc 1184 . . . 4  |-  ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0
) )  ->  ( A  / L N )  =  ( if ( ( N  <  0  /\  A  <  0
) ,  -u 1 ,  1 )  x.  (  seq  1 (  x.  ,  ( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( A  / L
n ) ^ (
n  pCnt  N )
) ,  1 ) ) ) `  ( abs `  N ) ) ) )
126123, 125oveq12d 6091 . . 3  |-  ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0
) )  ->  (
( A  / L M )  x.  ( A  / L N ) )  =  ( ( if ( ( M  <  0  /\  A  <  0 ) ,  -u
1 ,  1 )  x.  (  seq  1
(  x.  ,  ( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( A  / L n ) ^
( n  pCnt  M
) ) ,  1 ) ) ) `  ( abs `  M ) ) )  x.  ( if ( ( N  <  0  /\  A  <  0 ) ,  -u
1 ,  1 )  x.  (  seq  1
(  x.  ,  ( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( A  / L n ) ^
( n  pCnt  N
) ) ,  1 ) ) ) `  ( abs `  N ) ) ) ) )
127 neg1cn 10059 . . . . . 6  |-  -u 1  e.  CC
128 ax-1cn 9040 . . . . . 6  |-  1  e.  CC
129127, 128keepel 3788 . . . . 5  |-  if ( ( M  <  0  /\  A  <  0
) ,  -u 1 ,  1 )  e.  CC
130129a1i 11 . . . 4  |-  ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0
) )  ->  if ( ( M  <  0  /\  A  <  0 ) ,  -u
1 ,  1 )  e.  CC )
131 nnabscl 12121 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  M  =/=  0 )  -> 
( abs `  M
)  e.  NN )
13221, 26, 131syl2anc 643 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0
) )  ->  ( abs `  M )  e.  NN )
133132, 31syl6eleq 2525 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0
) )  ->  ( abs `  M )  e.  ( ZZ>= `  1 )
)
134 elfznn 11072 . . . . . . 7  |-  ( k  e.  ( 1 ... ( abs `  M
) )  ->  k  e.  NN )
13536, 134, 38syl2an 464 . . . . . 6  |-  ( ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0 ) )  /\  k  e.  ( 1 ... ( abs `  M
) ) )  -> 
( ( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( A  / L n ) ^ ( n 
pCnt  M ) ) ,  1 ) ) `  k )  e.  ZZ )
136135zcnd 10368 . . . . 5  |-  ( ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0 ) )  /\  k  e.  ( 1 ... ( abs `  M
) ) )  -> 
( ( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( A  / L n ) ^ ( n 
pCnt  M ) ) ,  1 ) ) `  k )  e.  CC )
137 mulcl 9066 . . . . . 6  |-  ( ( k  e.  CC  /\  x  e.  CC )  ->  ( k  x.  x
)  e.  CC )
138137adantl 453 . . . . 5  |-  ( ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0 ) )  /\  ( k  e.  CC  /\  x  e.  CC ) )  ->  ( k  x.  x )  e.  CC )
139133, 136, 138seqcl 11335 . . . 4  |-  ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0
) )  ->  (  seq  1 (  x.  , 
( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( A  / L n ) ^
( n  pCnt  M
) ) ,  1 ) ) ) `  ( abs `  M ) )  e.  CC )
140127, 128keepel 3788 . . . . 5  |-  if ( ( N  <  0  /\  A  <  0
) ,  -u 1 ,  1 )  e.  CC
141140a1i 11 . . . 4  |-  ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0
) )  ->  if ( ( N  <  0  /\  A  <  0 ) ,  -u
1 ,  1 )  e.  CC )
142 nnabscl 12121 . . . . . . 7  |-  ( ( N  e.  ZZ  /\  N  =/=  0 )  -> 
( abs `  N
)  e.  NN )
14322, 27, 142syl2anc 643 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0
) )  ->  ( abs `  N )  e.  NN )
144143, 31syl6eleq 2525 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0
) )  ->  ( abs `  N )  e.  ( ZZ>= `  1 )
)
145 elfznn 11072 . . . . . . 7  |-  ( k  e.  ( 1 ... ( abs `  N
) )  ->  k  e.  NN )
14643, 145, 44syl2an 464 . . . . . 6  |-  ( ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0 ) )  /\  k  e.  ( 1 ... ( abs `  N
) ) )  -> 
( ( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( A  / L n ) ^ ( n 
pCnt  N ) ) ,  1 ) ) `  k )  e.  ZZ )
147146zcnd 10368 . . . . 5  |-  ( ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0 ) )  /\  k  e.  ( 1 ... ( abs `  N
) ) )  -> 
( ( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( A  / L n ) ^ ( n 
pCnt  N ) ) ,  1 ) ) `  k )  e.  CC )
148144, 147, 138seqcl 11335 . . . 4  |-  ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0
) )  ->  (  seq  1 (  x.  , 
( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( A  / L n ) ^
( n  pCnt  N
) ) ,  1 ) ) ) `  ( abs `  N ) )  e.  CC )
149130, 139, 141, 148mul4d 9270 . . 3  |-  ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0
) )  ->  (
( if ( ( M  <  0  /\  A  <  0 ) ,  -u 1 ,  1 )  x.  (  seq  1 (  x.  , 
( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( A  / L n ) ^
( n  pCnt  M
) ) ,  1 ) ) ) `  ( abs `  M ) ) )  x.  ( if ( ( N  <  0  /\  A  <  0 ) ,  -u
1 ,  1 )  x.  (  seq  1
(  x.  ,  ( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( A  / L n ) ^
( n  pCnt  N
) ) ,  1 ) ) ) `  ( abs `  N ) ) ) )  =  ( ( if ( ( M  <  0  /\  A  <  0
) ,  -u 1 ,  1 )  x.  if ( ( N  <  0  /\  A  <  0 ) ,  -u
1 ,  1 ) )  x.  ( (  seq  1 (  x.  ,  ( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( A  / L n ) ^ ( n 
pCnt  M ) ) ,  1 ) ) ) `
 ( abs `  M
) )  x.  (  seq  1 (  x.  , 
( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( A  / L n ) ^
( n  pCnt  N
) ) ,  1 ) ) ) `  ( abs `  N ) ) ) ) )
150126, 149eqtrd 2467 . 2  |-  ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0
) )  ->  (
( A  / L M )  x.  ( A  / L N ) )  =  ( ( if ( ( M  <  0  /\  A  <  0 ) ,  -u
1 ,  1 )  x.  if ( ( N  <  0  /\  A  <  0 ) ,  -u 1 ,  1 ) )  x.  (
(  seq  1 (  x.  ,  ( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( A  / L
n ) ^ (
n  pCnt  M )
) ,  1 ) ) ) `  ( abs `  M ) )  x.  (  seq  1
(  x.  ,  ( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( A  / L n ) ^
( n  pCnt  N
) ) ,  1 ) ) ) `  ( abs `  N ) ) ) ) )
151119, 121, 1503eqtr4d 2477 1  |-  ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0
) )  ->  ( A  / L ( M  x.  N ) )  =  ( ( A  / L M )  x.  ( A  / L N ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598   ifcif 3731   class class class wbr 4204    e. cmpt 4258   -->wf 5442   ` cfv 5446  (class class class)co 6073   CCcc 8980   0cc0 8982   1c1 8983    + caddc 8985    x. cmul 8987    < clt 9112   -ucneg 9284   NNcn 9992   NN0cn0 10213   ZZcz 10274   ZZ>=cuz 10480   ...cfz 11035    seq cseq 11315   ^cexp 11374   abscabs 12031   Primecprime 13071    pCnt cpc 13202    / Lclgs 21070
This theorem is referenced by:  lgssq2  21112  lgsdinn0  21116  lgsquad2lem1  21134
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-2o 6717  df-oadd 6720  df-er 6897  df-map 7012  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-sup 7438  df-card 7818  df-cda 8040  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-n0 10214  df-z 10275  df-uz 10481  df-q 10567  df-rp 10605  df-fz 11036  df-fzo 11128  df-fl 11194  df-mod 11243  df-seq 11316  df-exp 11375  df-hash 11611  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-dvds 12845  df-gcd 12999  df-prm 13072  df-phi 13147  df-pc 13203  df-lgs 21071
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