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Theorem lgsdi 23473
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 978 . . . . 5  |-  ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  <->  ( M  e.  ZZ  /\  N  e.  ZZ  /\  A  e.  ZZ ) )
2 lgsdilem 23463 . . . . 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 472 . . . 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 450 . . . . 5  |-  ( ( ( M  x.  N
)  <  0  /\  A  <  0 )  <->  ( A  <  0  /\  ( M  x.  N )  <  0 ) )
5 ifbi 3966 . . . . 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 5 . . . 4  |-  if ( ( ( M  x.  N )  <  0  /\  A  <  0
) ,  -u 1 ,  1 )  =  if ( ( A  <  0  /\  ( M  x.  N )  <  0 ) ,  -u
1 ,  1 )
7 ancom 450 . . . . . 6  |-  ( ( M  <  0  /\  A  <  0 )  <-> 
( A  <  0  /\  M  <  0
) )
8 ifbi 3966 . . . . . 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 5 . . . . 5  |-  if ( ( M  <  0  /\  A  <  0
) ,  -u 1 ,  1 )  =  if ( ( A  <  0  /\  M  <  0 ) ,  -u
1 ,  1 )
10 ancom 450 . . . . . 6  |-  ( ( N  <  0  /\  A  <  0 )  <-> 
( A  <  0  /\  N  <  0
) )
11 ifbi 3966 . . . . . 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 5 . . . . 5  |-  if ( ( N  <  0  /\  A  <  0
) ,  -u 1 ,  1 )  =  if ( ( A  <  0  /\  N  <  0 ) ,  -u
1 ,  1 )
139, 12oveq12i 6307 . . . 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 2533 . . 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 9588 . . . . . 6  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  x.  y
)  e.  CC )
1615adantl 466 . . . . 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 9590 . . . . . 6  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  x.  y
)  =  ( y  x.  x ) )
1817adantl 466 . . . . 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 9592 . . . . . 6  |-  ( ( x  e.  CC  /\  y  e.  CC  /\  z  e.  CC )  ->  (
( x  x.  y
)  x.  z )  =  ( x  x.  ( y  x.  z
) ) )
2019adantl 466 . . . . 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 1000 . . . . . . . 8  |-  ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0
) )  ->  M  e.  ZZ )
22 simpl3 1001 . . . . . . . 8  |-  ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0
) )  ->  N  e.  ZZ )
2321, 22zmulcld 10984 . . . . . . 7  |-  ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0
) )  ->  ( M  x.  N )  e.  ZZ )
2421zcnd 10979 . . . . . . . 8  |-  ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0
) )  ->  M  e.  CC )
2522zcnd 10979 . . . . . . . 8  |-  ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0
) )  ->  N  e.  CC )
26 simprl 755 . . . . . . . 8  |-  ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0
) )  ->  M  =/=  0 )
27 simprr 756 . . . . . . . 8  |-  ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0
) )  ->  N  =/=  0 )
2824, 25, 26, 27mulne0d 10213 . . . . . . 7  |-  ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0
) )  ->  ( M  x.  N )  =/=  0 )
29 nnabscl 13138 . . . . . . 7  |-  ( ( ( M  x.  N
)  e.  ZZ  /\  ( M  x.  N
)  =/=  0 )  ->  ( abs `  ( M  x.  N )
)  e.  NN )
3023, 28, 29syl2anc 661 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0
) )  ->  ( abs `  ( M  x.  N ) )  e.  NN )
31 nnuz 11129 . . . . . 6  |-  NN  =  ( ZZ>= `  1 )
3230, 31syl6eleq 2565 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0
) )  ->  ( abs `  ( M  x.  N ) )  e.  ( ZZ>= `  1 )
)
33 simpl1 999 . . . . . . . 8  |-  ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0
) )  ->  A  e.  ZZ )
34 eqid 2467 . . . . . . . . 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 23458 . . . . . . . 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 1228 . . . . . . 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 11726 . . . . . . 7  |-  ( k  e.  ( 1 ... ( abs `  ( M  x.  N )
) )  ->  k  e.  NN )
38 ffvelrn 6030 . . . . . . 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 477 . . . . . 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 10979 . . . . 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 2467 . . . . . . . . 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 23458 . . . . . . . 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 1228 . . . . . . 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 6030 . . . . . . 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 477 . . . . . 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 10979 . . . . 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 461 . . . . . . . . . . 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 725 . . . . . . . . . . 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 725 . . . . . . . . . . 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 725 . . . . . . . . . . 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 725 . . . . . . . . . . 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 14251 . . . . . . . . . . 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 1237 . . . . . . . . . 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 6311 . . . . . . . . 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 725 . . . . . . . . . . . 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 14097 . . . . . . . . . . . . 13  |-  ( k  e.  Prime  ->  k  e.  ZZ )
5756adantl 466 . . . . . . . . . . . 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 23451 . . . . . . . . . . . 12  |-  ( ( A  e.  ZZ  /\  k  e.  ZZ )  ->  ( A  /L
k )  e.  ZZ )
5955, 57, 58syl2anc 661 . . . . . . . . . . 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 10979 . . . . . . . . . 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 14248 . . . . . . . . . . 11  |-  ( ( k  e.  Prime  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( k  pCnt  N
)  e.  NN0 )
6247, 50, 51, 61syl12anc 1226 . . . . . . . . . 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 14248 . . . . . . . . . . 11  |-  ( ( k  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 ) )  -> 
( k  pCnt  M
)  e.  NN0 )
6447, 48, 49, 63syl12anc 1226 . . . . . . . . . 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 12292 . . . . . . . . 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 2508 . . . . . . . 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 3951 . . . . . . . . 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 466 . . . . . . . 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 3951 . . . . . . . . . 10  |-  ( k  e.  Prime  ->  if ( k  e.  Prime ,  ( ( A  /L
k ) ^ (
k  pCnt  M )
) ,  1 )  =  ( ( A  /L k ) ^ ( k  pCnt  M ) ) )
70 iftrue 3951 . . . . . . . . . 10  |-  ( k  e.  Prime  ->  if ( k  e.  Prime ,  ( ( A  /L
k ) ^ (
k  pCnt  N )
) ,  1 )  =  ( ( A  /L k ) ^ ( k  pCnt  N ) ) )
7169, 70oveq12d 6313 . . . . . . . . 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 466 . . . . . . . 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 2519 . . . . . . 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 10695 . . . . . . . . 9  |-  ( 1  x.  1 )  =  1
75 iffalse 3954 . . . . . . . . . 10  |-  ( -.  k  e.  Prime  ->  if ( k  e.  Prime ,  ( ( A  /L k ) ^
( k  pCnt  M
) ) ,  1 )  =  1 )
76 iffalse 3954 . . . . . . . . . 10  |-  ( -.  k  e.  Prime  ->  if ( k  e.  Prime ,  ( ( A  /L k ) ^
( k  pCnt  N
) ) ,  1 )  =  1 )
7775, 76oveq12d 6313 . . . . . . . . 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 3954 . . . . . . . . 9  |-  ( -.  k  e.  Prime  ->  if ( k  e.  Prime ,  ( ( A  /L k ) ^
( k  pCnt  ( M  x.  N )
) ) ,  1 )  =  1 )
7974, 77, 783eqtr4a 2534 . . . . . . . 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 466 . . . . . . 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 789 . . . . . 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 466 . . . . . . 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 2539 . . . . . . . . . 10  |-  ( n  =  k  ->  (
n  e.  Prime  <->  k  e.  Prime ) )
84 oveq2 6303 . . . . . . . . . . 11  |-  ( n  =  k  ->  ( A  /L n )  =  ( A  /L k ) )
85 oveq1 6302 . . . . . . . . . . 11  |-  ( n  =  k  ->  (
n  pCnt  M )  =  ( k  pCnt  M ) )
8684, 85oveq12d 6313 . . . . . . . . . 10  |-  ( n  =  k  ->  (
( A  /L
n ) ^ (
n  pCnt  M )
)  =  ( ( A  /L k ) ^ ( k 
pCnt  M ) ) )
8783, 86ifbieq1d 3968 . . . . . . . . 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 ) )
88 ovex 6320 . . . . . . . . . 10  |-  ( ( A  /L k ) ^ ( k 
pCnt  M ) )  e. 
_V
89 1ex 9603 . . . . . . . . . 10  |-  1  e.  _V
9088, 89ifex 4014 . . . . . . . . 9  |-  if ( k  e.  Prime ,  ( ( A  /L
k ) ^ (
k  pCnt  M )
) ,  1 )  e.  _V
9187, 34, 90fvmpt 5957 . . . . . . . 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 ) )
92 oveq1 6302 . . . . . . . . . . 11  |-  ( n  =  k  ->  (
n  pCnt  N )  =  ( k  pCnt  N ) )
9384, 92oveq12d 6313 . . . . . . . . . 10  |-  ( n  =  k  ->  (
( A  /L
n ) ^ (
n  pCnt  N )
)  =  ( ( A  /L k ) ^ ( k 
pCnt  N ) ) )
9483, 93ifbieq1d 3968 . . . . . . . . 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 ) )
95 ovex 6320 . . . . . . . . . 10  |-  ( ( A  /L k ) ^ ( k 
pCnt  N ) )  e. 
_V
9695, 89ifex 4014 . . . . . . . . 9  |-  if ( k  e.  Prime ,  ( ( A  /L
k ) ^ (
k  pCnt  N )
) ,  1 )  e.  _V
9794, 41, 96fvmpt 5957 . . . . . . . 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 ) )
9891, 97oveq12d 6313 . . . . . . 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 ) ) )
9982, 98syl 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 ) ) )
100 oveq1 6302 . . . . . . . . . 10  |-  ( n  =  k  ->  (
n  pCnt  ( M  x.  N ) )  =  ( k  pCnt  ( M  x.  N )
) )
10184, 100oveq12d 6313 . . . . . . . . 9  |-  ( n  =  k  ->  (
( A  /L
n ) ^ (
n  pCnt  ( M  x.  N ) ) )  =  ( ( A  /L k ) ^ ( k  pCnt  ( M  x.  N ) ) ) )
10283, 101ifbieq1d 3968 . . . . . . . 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 ) )
103 eqid 2467 . . . . . . . 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 ) )
104 ovex 6320 . . . . . . . . 9  |-  ( ( A  /L k ) ^ ( k 
pCnt  ( M  x.  N ) ) )  e.  _V
105104, 89ifex 4014 . . . . . . . 8  |-  if ( k  e.  Prime ,  ( ( A  /L
k ) ^ (
k  pCnt  ( M  x.  N ) ) ) ,  1 )  e. 
_V
106102, 103, 105fvmpt 5957 . . . . . . 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 ) )
10782, 106syl 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 ) )
10881, 99, 1073eqtr4rd 2519 . . . . 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 ) ) )
10916, 18, 20, 32, 40, 46, 108seqcaopr 12124 . . . 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 ) ) ) ) )
11033, 21, 22, 26, 27, 34lgsdilem2 23472 . . . . 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 ) ) ) )
11133, 22, 21, 27, 26, 41lgsdilem2 23472 . . . . . 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 ) ) ) )
11224, 25mulcomd 9629 . . . . . . . 8  |-  ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0
) )  ->  ( M  x.  N )  =  ( N  x.  M ) )
113112fveq2d 5876 . . . . . . 7  |-  ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0
) )  ->  ( abs `  ( M  x.  N ) )  =  ( abs `  ( N  x.  M )
) )
114113fveq2d 5876 . . . . . 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 ) ) ) )
115111, 114eqtr4d 2511 . . . . 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 ) ) ) )
116110, 115oveq12d 6313 . . . 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 ) ) ) ) )
117109, 116eqtr4d 2511 . . 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 ) ) ) )
11814, 117oveq12d 6313 . 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 ) ) ) ) )
119103lgsval4 23457 . . 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 ) ) ) ) )
12033, 23, 28, 119syl3anc 1228 . 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
) ) ) ) )
12134lgsval4 23457 . . . . 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 ) ) ) )
12233, 21, 26, 121syl3anc 1228 . . . 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 ) ) ) )
12341lgsval4 23457 . . . . 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 ) ) ) )
12433, 22, 27, 123syl3anc 1228 . . . 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 ) ) ) )
125122, 124oveq12d 6313 . . 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 ) ) ) ) )
126 neg1cn 10651 . . . . . 6  |-  -u 1  e.  CC
127 ax-1cn 9562 . . . . . 6  |-  1  e.  CC
128126, 127keepel 4013 . . . . 5  |-  if ( ( M  <  0  /\  A  <  0
) ,  -u 1 ,  1 )  e.  CC
129128a1i 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 )
130 nnabscl 13138 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  M  =/=  0 )  -> 
( abs `  M
)  e.  NN )
13121, 26, 130syl2anc 661 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0
) )  ->  ( abs `  M )  e.  NN )
132131, 31syl6eleq 2565 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0
) )  ->  ( abs `  M )  e.  ( ZZ>= `  1 )
)
133 elfznn 11726 . . . . . . 7  |-  ( k  e.  ( 1 ... ( abs `  M
) )  ->  k  e.  NN )
13436, 133, 38syl2an 477 . . . . . 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 )
135134zcnd 10979 . . . . 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 )
136 mulcl 9588 . . . . . 6  |-  ( ( k  e.  CC  /\  x  e.  CC )  ->  ( k  x.  x
)  e.  CC )
137136adantl 466 . . . . 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 )
138132, 135, 137seqcl 12107 . . . 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 )
139126, 127keepel 4013 . . . . 5  |-  if ( ( N  <  0  /\  A  <  0
) ,  -u 1 ,  1 )  e.  CC
140139a1i 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 )
141 nnabscl 13138 . . . . . . 7  |-  ( ( N  e.  ZZ  /\  N  =/=  0 )  -> 
( abs `  N
)  e.  NN )
14222, 27, 141syl2anc 661 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0
) )  ->  ( abs `  N )  e.  NN )
143142, 31syl6eleq 2565 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0
) )  ->  ( abs `  N )  e.  ( ZZ>= `  1 )
)
144 elfznn 11726 . . . . . . 7  |-  ( k  e.  ( 1 ... ( abs `  N
) )  ->  k  e.  NN )
14543, 144, 44syl2an 477 . . . . . 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 )
146145zcnd 10979 . . . . 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 )
147143, 146, 137seqcl 12107 . . . 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 )
148129, 138, 140, 147mul4d 9803 . . 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 ) ) ) ) )
149125, 148eqtrd 2508 . 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 ) ) ) ) )
150118, 120, 1493eqtr4d 2518 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   ifcif 3945   class class class wbr 4453    |-> cmpt 4511   -->wf 5590   ` cfv 5594  (class class class)co 6295   CCcc 9502   0cc0 9504   1c1 9505    + caddc 9507    x. cmul 9509    < clt 9640   -ucneg 9818   NNcn 10548   NN0cn0 10807   ZZcz 10876   ZZ>=cuz 11094   ...cfz 11684    seqcseq 12087   ^cexp 12146   abscabs 13047   Primecprime 14093    pCnt cpc 14236    /Lclgs 23435
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-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-pre-sup 9582
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 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-2o 7143  df-oadd 7146  df-er 7323  df-map 7434  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-sup 7913  df-card 8332  df-cda 8560  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-nn 10549  df-2 10606  df-3 10607  df-n0 10808  df-z 10877  df-uz 11095  df-q 11195  df-rp 11233  df-fz 11685  df-fzo 11805  df-fl 11909  df-mod 11977  df-seq 12088  df-exp 12147  df-hash 12386  df-cj 12912  df-re 12913  df-im 12914  df-sqrt 13048  df-abs 13049  df-dvds 13865  df-gcd 14021  df-prm 14094  df-phi 14172  df-pc 14237  df-lgs 23436
This theorem is referenced by:  lgssq2  23477  lgsdinn0  23481  lgsquad2lem1  23499
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