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Theorem lgsdi 23805
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 976 . . . . 5  |-  ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  <->  ( M  e.  ZZ  /\  N  e.  ZZ  /\  A  e.  ZZ ) )
2 lgsdilem 23795 . . . . 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 470 . . . 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 448 . . . . 5  |-  ( ( ( M  x.  N
)  <  0  /\  A  <  0 )  <->  ( A  <  0  /\  ( M  x.  N )  <  0 ) )
5 ifbi 3950 . . . . 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 448 . . . . . 6  |-  ( ( M  <  0  /\  A  <  0 )  <-> 
( A  <  0  /\  M  <  0
) )
8 ifbi 3950 . . . . . 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 448 . . . . . 6  |-  ( ( N  <  0  /\  A  <  0 )  <-> 
( A  <  0  /\  N  <  0
) )
11 ifbi 3950 . . . . . 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 6282 . . . 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 2520 . . 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 9565 . . . . . 6  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  x.  y
)  e.  CC )
1615adantl 464 . . . . 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 9567 . . . . . 6  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  x.  y
)  =  ( y  x.  x ) )
1817adantl 464 . . . . 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 9569 . . . . . 6  |-  ( ( x  e.  CC  /\  y  e.  CC  /\  z  e.  CC )  ->  (
( x  x.  y
)  x.  z )  =  ( x  x.  ( y  x.  z
) ) )
2019adantl 464 . . . . 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 998 . . . . . . . 8  |-  ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0
) )  ->  M  e.  ZZ )
22 simpl3 999 . . . . . . . 8  |-  ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0
) )  ->  N  e.  ZZ )
2321, 22zmulcld 10971 . . . . . . 7  |-  ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0
) )  ->  ( M  x.  N )  e.  ZZ )
2421zcnd 10966 . . . . . . . 8  |-  ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0
) )  ->  M  e.  CC )
2522zcnd 10966 . . . . . . . 8  |-  ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0
) )  ->  N  e.  CC )
26 simprl 754 . . . . . . . 8  |-  ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0
) )  ->  M  =/=  0 )
27 simprr 755 . . . . . . . 8  |-  ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0
) )  ->  N  =/=  0 )
2824, 25, 26, 27mulne0d 10197 . . . . . . 7  |-  ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0
) )  ->  ( M  x.  N )  =/=  0 )
29 nnabscl 13240 . . . . . . 7  |-  ( ( ( M  x.  N
)  e.  ZZ  /\  ( M  x.  N
)  =/=  0 )  ->  ( abs `  ( M  x.  N )
)  e.  NN )
3023, 28, 29syl2anc 659 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0
) )  ->  ( abs `  ( M  x.  N ) )  e.  NN )
31 nnuz 11117 . . . . . 6  |-  NN  =  ( ZZ>= `  1 )
3230, 31syl6eleq 2552 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0
) )  ->  ( abs `  ( M  x.  N ) )  e.  ( ZZ>= `  1 )
)
33 simpl1 997 . . . . . . . 8  |-  ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0
) )  ->  A  e.  ZZ )
34 eqid 2454 . . . . . . . . 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 23790 . . . . . . . 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 1226 . . . . . . 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 11717 . . . . . . 7  |-  ( k  e.  ( 1 ... ( abs `  ( M  x.  N )
) )  ->  k  e.  NN )
38 ffvelrn 6005 . . . . . . 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 475 . . . . . 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 10966 . . . . 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 2454 . . . . . . . . 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 23790 . . . . . . . 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 1226 . . . . . . 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 6005 . . . . . . 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 475 . . . . . 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 10966 . . . . 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 459 . . . . . . . . . . 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 723 . . . . . . . . . . 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 723 . . . . . . . . . . 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 723 . . . . . . . . . . 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 723 . . . . . . . . . . 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 14459 . . . . . . . . . . 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 1235 . . . . . . . . . 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 6286 . . . . . . . . 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 723 . . . . . . . . . . . 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 14305 . . . . . . . . . . . . 13  |-  ( k  e.  Prime  ->  k  e.  ZZ )
5756adantl 464 . . . . . . . . . . . 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 23783 . . . . . . . . . . . 12  |-  ( ( A  e.  ZZ  /\  k  e.  ZZ )  ->  ( A  /L
k )  e.  ZZ )
5955, 57, 58syl2anc 659 . . . . . . . . . . 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 10966 . . . . . . . . . 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 14456 . . . . . . . . . . 11  |-  ( ( k  e.  Prime  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( k  pCnt  N
)  e.  NN0 )
6247, 50, 51, 61syl12anc 1224 . . . . . . . . . 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 14456 . . . . . . . . . . 11  |-  ( ( k  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 ) )  -> 
( k  pCnt  M
)  e.  NN0 )
6447, 48, 49, 63syl12anc 1224 . . . . . . . . . 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 12294 . . . . . . . . 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 2495 . . . . . . . 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 3935 . . . . . . . . 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 464 . . . . . . . 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 3935 . . . . . . . . . 10  |-  ( k  e.  Prime  ->  if ( k  e.  Prime ,  ( ( A  /L
k ) ^ (
k  pCnt  M )
) ,  1 )  =  ( ( A  /L k ) ^ ( k  pCnt  M ) ) )
70 iftrue 3935 . . . . . . . . . 10  |-  ( k  e.  Prime  ->  if ( k  e.  Prime ,  ( ( A  /L
k ) ^ (
k  pCnt  N )
) ,  1 )  =  ( ( A  /L k ) ^ ( k  pCnt  N ) ) )
7169, 70oveq12d 6288 . . . . . . . . 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 464 . . . . . . . 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 2506 . . . . . . 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 10679 . . . . . . . . 9  |-  ( 1  x.  1 )  =  1
75 iffalse 3938 . . . . . . . . . 10  |-  ( -.  k  e.  Prime  ->  if ( k  e.  Prime ,  ( ( A  /L k ) ^
( k  pCnt  M
) ) ,  1 )  =  1 )
76 iffalse 3938 . . . . . . . . . 10  |-  ( -.  k  e.  Prime  ->  if ( k  e.  Prime ,  ( ( A  /L k ) ^
( k  pCnt  N
) ) ,  1 )  =  1 )
7775, 76oveq12d 6288 . . . . . . . . 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 3938 . . . . . . . . 9  |-  ( -.  k  e.  Prime  ->  if ( k  e.  Prime ,  ( ( A  /L k ) ^
( k  pCnt  ( M  x.  N )
) ) ,  1 )  =  1 )
7974, 77, 783eqtr4a 2521 . . . . . . . 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 464 . . . . . . 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 464 . . . . . . 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 2526 . . . . . . . . . 10  |-  ( n  =  k  ->  (
n  e.  Prime  <->  k  e.  Prime ) )
84 oveq2 6278 . . . . . . . . . . 11  |-  ( n  =  k  ->  ( A  /L n )  =  ( A  /L k ) )
85 oveq1 6277 . . . . . . . . . . 11  |-  ( n  =  k  ->  (
n  pCnt  M )  =  ( k  pCnt  M ) )
8684, 85oveq12d 6288 . . . . . . . . . 10  |-  ( n  =  k  ->  (
( A  /L
n ) ^ (
n  pCnt  M )
)  =  ( ( A  /L k ) ^ ( k 
pCnt  M ) ) )
8783, 86ifbieq1d 3952 . . . . . . . . 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 6298 . . . . . . . . . 10  |-  ( ( A  /L k ) ^ ( k 
pCnt  M ) )  e. 
_V
89 1ex 9580 . . . . . . . . . 10  |-  1  e.  _V
9088, 89ifex 3997 . . . . . . . . 9  |-  if ( k  e.  Prime ,  ( ( A  /L
k ) ^ (
k  pCnt  M )
) ,  1 )  e.  _V
9187, 34, 90fvmpt 5931 . . . . . . . 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 6277 . . . . . . . . . . 11  |-  ( n  =  k  ->  (
n  pCnt  N )  =  ( k  pCnt  N ) )
9384, 92oveq12d 6288 . . . . . . . . . 10  |-  ( n  =  k  ->  (
( A  /L
n ) ^ (
n  pCnt  N )
)  =  ( ( A  /L k ) ^ ( k 
pCnt  N ) ) )
9483, 93ifbieq1d 3952 . . . . . . . . 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 6298 . . . . . . . . . 10  |-  ( ( A  /L k ) ^ ( k 
pCnt  N ) )  e. 
_V
9695, 89ifex 3997 . . . . . . . . 9  |-  if ( k  e.  Prime ,  ( ( A  /L
k ) ^ (
k  pCnt  N )
) ,  1 )  e.  _V
9794, 41, 96fvmpt 5931 . . . . . . . 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 6288 . . . . . . 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 6277 . . . . . . . . . 10  |-  ( n  =  k  ->  (
n  pCnt  ( M  x.  N ) )  =  ( k  pCnt  ( M  x.  N )
) )
10184, 100oveq12d 6288 . . . . . . . . 9  |-  ( n  =  k  ->  (
( A  /L
n ) ^ (
n  pCnt  ( M  x.  N ) ) )  =  ( ( A  /L k ) ^ ( k  pCnt  ( M  x.  N ) ) ) )
10283, 101ifbieq1d 3952 . . . . . . . 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 2454 . . . . . . . 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 6298 . . . . . . . . 9  |-  ( ( A  /L k ) ^ ( k 
pCnt  ( M  x.  N ) ) )  e.  _V
105104, 89ifex 3997 . . . . . . . 8  |-  if ( k  e.  Prime ,  ( ( A  /L
k ) ^ (
k  pCnt  ( M  x.  N ) ) ) ,  1 )  e. 
_V
106102, 103, 105fvmpt 5931 . . . . . . 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 2506 . . . . 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 12126 . . . 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 23804 . . . . 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 23804 . . . . . 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 9606 . . . . . . . 8  |-  ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0
) )  ->  ( M  x.  N )  =  ( N  x.  M ) )
113112fveq2d 5852 . . . . . . 7  |-  ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0
) )  ->  ( abs `  ( M  x.  N ) )  =  ( abs `  ( N  x.  M )
) )
114113fveq2d 5852 . . . . . 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 2498 . . . . 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 6288 . . . 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 2498 . . 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 6288 . 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 23789 . . 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 1226 . 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 23789 . . . . 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 1226 . . . 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 23789 . . . . 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 1226 . . . 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 6288 . . 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 10635 . . . . . 6  |-  -u 1  e.  CC
127 ax-1cn 9539 . . . . . 6  |-  1  e.  CC
128126, 127keepel 3996 . . . . 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 13240 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  M  =/=  0 )  -> 
( abs `  M
)  e.  NN )
13121, 26, 130syl2anc 659 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0
) )  ->  ( abs `  M )  e.  NN )
132131, 31syl6eleq 2552 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0
) )  ->  ( abs `  M )  e.  ( ZZ>= `  1 )
)
133 elfznn 11717 . . . . . . 7  |-  ( k  e.  ( 1 ... ( abs `  M
) )  ->  k  e.  NN )
13436, 133, 38syl2an 475 . . . . . 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 10966 . . . . 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 9565 . . . . . 6  |-  ( ( k  e.  CC  /\  x  e.  CC )  ->  ( k  x.  x
)  e.  CC )
137136adantl 464 . . . . 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 12109 . . . 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 3996 . . . . 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 13240 . . . . . . 7  |-  ( ( N  e.  ZZ  /\  N  =/=  0 )  -> 
( abs `  N
)  e.  NN )
14222, 27, 141syl2anc 659 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0
) )  ->  ( abs `  N )  e.  NN )
143142, 31syl6eleq 2552 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0
) )  ->  ( abs `  N )  e.  ( ZZ>= `  1 )
)
144 elfznn 11717 . . . . . . 7  |-  ( k  e.  ( 1 ... ( abs `  N
) )  ->  k  e.  NN )
14543, 144, 44syl2an 475 . . . . . 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 10966 . . . . 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 12109 . . . 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 9781 . . 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 2495 . 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 2505 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 367    /\ w3a 971    = wceq 1398    e. wcel 1823    =/= wne 2649   ifcif 3929   class class class wbr 4439    |-> cmpt 4497   -->wf 5566   ` cfv 5570  (class class class)co 6270   CCcc 9479   0cc0 9481   1c1 9482    + caddc 9484    x. cmul 9486    < clt 9617   -ucneg 9797   NNcn 10531   NN0cn0 10791   ZZcz 10860   ZZ>=cuz 11082   ...cfz 11675    seqcseq 12089   ^cexp 12148   abscabs 13149   Primecprime 14301    pCnt cpc 14444    /Lclgs 23767
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-2o 7123  df-oadd 7126  df-er 7303  df-map 7414  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-sup 7893  df-card 8311  df-cda 8539  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  df-2 10590  df-3 10591  df-n0 10792  df-z 10861  df-uz 11083  df-q 11184  df-rp 11222  df-fz 11676  df-fzo 11800  df-fl 11910  df-mod 11979  df-seq 12090  df-exp 12149  df-hash 12388  df-cj 13014  df-re 13015  df-im 13016  df-sqrt 13150  df-abs 13151  df-dvds 14071  df-gcd 14229  df-prm 14302  df-phi 14380  df-pc 14445  df-lgs 23768
This theorem is referenced by:  lgssq2  23809  lgsdinn0  23813  lgsquad2lem1  23831
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