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Theorem lgsdi 24247
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 987 . . . . 5  |-  ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  <->  ( M  e.  ZZ  /\  N  e.  ZZ  /\  A  e.  ZZ ) )
2 lgsdilem 24237 . . . . 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 474 . . . 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 451 . . . . 5  |-  ( ( ( M  x.  N
)  <  0  /\  A  <  0 )  <->  ( A  <  0  /\  ( M  x.  N )  <  0 ) )
5 ifbi 3930 . . . . 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 451 . . . . . 6  |-  ( ( M  <  0  /\  A  <  0 )  <-> 
( A  <  0  /\  M  <  0
) )
8 ifbi 3930 . . . . . 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 451 . . . . . 6  |-  ( ( N  <  0  /\  A  <  0 )  <-> 
( A  <  0  /\  N  <  0
) )
11 ifbi 3930 . . . . . 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 6314 . . . 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 2488 . . 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 9624 . . . . . 6  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  x.  y
)  e.  CC )
1615adantl 467 . . . . 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 9626 . . . . . 6  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  x.  y
)  =  ( y  x.  x ) )
1817adantl 467 . . . . 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 9628 . . . . . 6  |-  ( ( x  e.  CC  /\  y  e.  CC  /\  z  e.  CC )  ->  (
( x  x.  y
)  x.  z )  =  ( x  x.  ( y  x.  z
) ) )
2019adantl 467 . . . . 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 1009 . . . . . . . 8  |-  ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0
) )  ->  M  e.  ZZ )
22 simpl3 1010 . . . . . . . 8  |-  ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0
) )  ->  N  e.  ZZ )
2321, 22zmulcld 11047 . . . . . . 7  |-  ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0
) )  ->  ( M  x.  N )  e.  ZZ )
2421zcnd 11042 . . . . . . . 8  |-  ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0
) )  ->  M  e.  CC )
2522zcnd 11042 . . . . . . . 8  |-  ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0
) )  ->  N  e.  CC )
26 simprl 762 . . . . . . . 8  |-  ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0
) )  ->  M  =/=  0 )
27 simprr 764 . . . . . . . 8  |-  ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0
) )  ->  N  =/=  0 )
2824, 25, 26, 27mulne0d 10265 . . . . . . 7  |-  ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0
) )  ->  ( M  x.  N )  =/=  0 )
29 nnabscl 13377 . . . . . . 7  |-  ( ( ( M  x.  N
)  e.  ZZ  /\  ( M  x.  N
)  =/=  0 )  ->  ( abs `  ( M  x.  N )
)  e.  NN )
3023, 28, 29syl2anc 665 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0
) )  ->  ( abs `  ( M  x.  N ) )  e.  NN )
31 nnuz 11195 . . . . . 6  |-  NN  =  ( ZZ>= `  1 )
3230, 31syl6eleq 2520 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0
) )  ->  ( abs `  ( M  x.  N ) )  e.  ( ZZ>= `  1 )
)
33 simpl1 1008 . . . . . . . 8  |-  ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0
) )  ->  A  e.  ZZ )
34 eqid 2422 . . . . . . . . 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 24232 . . . . . . . 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 1264 . . . . . . 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 11829 . . . . . . 7  |-  ( k  e.  ( 1 ... ( abs `  ( M  x.  N )
) )  ->  k  e.  NN )
38 ffvelrn 6032 . . . . . . 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 479 . . . . . 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 11042 . . . . 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 2422 . . . . . . . . 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 24232 . . . . . . . 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 1264 . . . . . . 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 6032 . . . . . . 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 479 . . . . . 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 11042 . . . . 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 462 . . . . . . . . . . 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 730 . . . . . . . . . . 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 730 . . . . . . . . . . 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 730 . . . . . . . . . . 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 730 . . . . . . . . . . 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 14789 . . . . . . . . . . 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 1273 . . . . . . . . . 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 6318 . . . . . . . . 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 730 . . . . . . . . . . . 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 14614 . . . . . . . . . . . . 13  |-  ( k  e.  Prime  ->  k  e.  ZZ )
5756adantl 467 . . . . . . . . . . . 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 24225 . . . . . . . . . . . 12  |-  ( ( A  e.  ZZ  /\  k  e.  ZZ )  ->  ( A  /L
k )  e.  ZZ )
5955, 57, 58syl2anc 665 . . . . . . . . . . 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 11042 . . . . . . . . . 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 14786 . . . . . . . . . . 11  |-  ( ( k  e.  Prime  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( k  pCnt  N
)  e.  NN0 )
6247, 50, 51, 61syl12anc 1262 . . . . . . . . . 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 14786 . . . . . . . . . . 11  |-  ( ( k  e.  Prime  /\  ( M  e.  ZZ  /\  M  =/=  0 ) )  -> 
( k  pCnt  M
)  e.  NN0 )
6447, 48, 49, 63syl12anc 1262 . . . . . . . . . 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 12418 . . . . . . . . 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 2463 . . . . . . . 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 3915 . . . . . . . . 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 467 . . . . . . . 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 3915 . . . . . . . . . 10  |-  ( k  e.  Prime  ->  if ( k  e.  Prime ,  ( ( A  /L
k ) ^ (
k  pCnt  M )
) ,  1 )  =  ( ( A  /L k ) ^ ( k  pCnt  M ) ) )
70 iftrue 3915 . . . . . . . . . 10  |-  ( k  e.  Prime  ->  if ( k  e.  Prime ,  ( ( A  /L
k ) ^ (
k  pCnt  N )
) ,  1 )  =  ( ( A  /L k ) ^ ( k  pCnt  N ) ) )
7169, 70oveq12d 6320 . . . . . . . . 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 467 . . . . . . . 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 2474 . . . . . . 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 10758 . . . . . . . . 9  |-  ( 1  x.  1 )  =  1
75 iffalse 3918 . . . . . . . . . 10  |-  ( -.  k  e.  Prime  ->  if ( k  e.  Prime ,  ( ( A  /L k ) ^
( k  pCnt  M
) ) ,  1 )  =  1 )
76 iffalse 3918 . . . . . . . . . 10  |-  ( -.  k  e.  Prime  ->  if ( k  e.  Prime ,  ( ( A  /L k ) ^
( k  pCnt  N
) ) ,  1 )  =  1 )
7775, 76oveq12d 6320 . . . . . . . . 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 3918 . . . . . . . . 9  |-  ( -.  k  e.  Prime  ->  if ( k  e.  Prime ,  ( ( A  /L k ) ^
( k  pCnt  ( M  x.  N )
) ) ,  1 )  =  1 )
7974, 77, 783eqtr4a 2489 . . . . . . . 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 467 . . . . . . 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 798 . . . . . 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 467 . . . . . . 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 2494 . . . . . . . . . 10  |-  ( n  =  k  ->  (
n  e.  Prime  <->  k  e.  Prime ) )
84 oveq2 6310 . . . . . . . . . . 11  |-  ( n  =  k  ->  ( A  /L n )  =  ( A  /L k ) )
85 oveq1 6309 . . . . . . . . . . 11  |-  ( n  =  k  ->  (
n  pCnt  M )  =  ( k  pCnt  M ) )
8684, 85oveq12d 6320 . . . . . . . . . 10  |-  ( n  =  k  ->  (
( A  /L
n ) ^ (
n  pCnt  M )
)  =  ( ( A  /L k ) ^ ( k 
pCnt  M ) ) )
8783, 86ifbieq1d 3932 . . . . . . . . 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 6330 . . . . . . . . . 10  |-  ( ( A  /L k ) ^ ( k 
pCnt  M ) )  e. 
_V
89 1ex 9639 . . . . . . . . . 10  |-  1  e.  _V
9088, 89ifex 3977 . . . . . . . . 9  |-  if ( k  e.  Prime ,  ( ( A  /L
k ) ^ (
k  pCnt  M )
) ,  1 )  e.  _V
9187, 34, 90fvmpt 5961 . . . . . . . 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 6309 . . . . . . . . . . 11  |-  ( n  =  k  ->  (
n  pCnt  N )  =  ( k  pCnt  N ) )
9384, 92oveq12d 6320 . . . . . . . . . 10  |-  ( n  =  k  ->  (
( A  /L
n ) ^ (
n  pCnt  N )
)  =  ( ( A  /L k ) ^ ( k 
pCnt  N ) ) )
9483, 93ifbieq1d 3932 . . . . . . . . 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 6330 . . . . . . . . . 10  |-  ( ( A  /L k ) ^ ( k 
pCnt  N ) )  e. 
_V
9695, 89ifex 3977 . . . . . . . . 9  |-  if ( k  e.  Prime ,  ( ( A  /L
k ) ^ (
k  pCnt  N )
) ,  1 )  e.  _V
9794, 41, 96fvmpt 5961 . . . . . . . 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 6320 . . . . . . 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 17 . . . . . 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 6309 . . . . . . . . . 10  |-  ( n  =  k  ->  (
n  pCnt  ( M  x.  N ) )  =  ( k  pCnt  ( M  x.  N )
) )
10184, 100oveq12d 6320 . . . . . . . . 9  |-  ( n  =  k  ->  (
( A  /L
n ) ^ (
n  pCnt  ( M  x.  N ) ) )  =  ( ( A  /L k ) ^ ( k  pCnt  ( M  x.  N ) ) ) )
10283, 101ifbieq1d 3932 . . . . . . . 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 2422 . . . . . . . 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 6330 . . . . . . . . 9  |-  ( ( A  /L k ) ^ ( k 
pCnt  ( M  x.  N ) ) )  e.  _V
105104, 89ifex 3977 . . . . . . . 8  |-  if ( k  e.  Prime ,  ( ( A  /L
k ) ^ (
k  pCnt  ( M  x.  N ) ) ) ,  1 )  e. 
_V
106102, 103, 105fvmpt 5961 . . . . . . 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 17 . . . . . 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 2474 . . . . 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 12250 . . . 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 24246 . . . . 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 24246 . . . . . 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 9665 . . . . . . . 8  |-  ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0
) )  ->  ( M  x.  N )  =  ( N  x.  M ) )
113112fveq2d 5882 . . . . . . 7  |-  ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0
) )  ->  ( abs `  ( M  x.  N ) )  =  ( abs `  ( N  x.  M )
) )
114113fveq2d 5882 . . . . . 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 2466 . . . . 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 6320 . . . 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 2466 . . 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 6320 . 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 24231 . . 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 1264 . 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 24231 . . . . 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 1264 . . . 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 24231 . . . . 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 1264 . . . 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 6320 . . 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 10714 . . . . . 6  |-  -u 1  e.  CC
127 ax-1cn 9598 . . . . . 6  |-  1  e.  CC
128126, 127keepel 3976 . . . . 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 13377 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  M  =/=  0 )  -> 
( abs `  M
)  e.  NN )
13121, 26, 130syl2anc 665 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0
) )  ->  ( abs `  M )  e.  NN )
132131, 31syl6eleq 2520 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0
) )  ->  ( abs `  M )  e.  ( ZZ>= `  1 )
)
133 elfznn 11829 . . . . . . 7  |-  ( k  e.  ( 1 ... ( abs `  M
) )  ->  k  e.  NN )
13436, 133, 38syl2an 479 . . . . . 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 11042 . . . . 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 9624 . . . . . 6  |-  ( ( k  e.  CC  /\  x  e.  CC )  ->  ( k  x.  x
)  e.  CC )
137136adantl 467 . . . . 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 12233 . . . 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 3976 . . . . 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 13377 . . . . . . 7  |-  ( ( N  e.  ZZ  /\  N  =/=  0 )  -> 
( abs `  N
)  e.  NN )
14222, 27, 141syl2anc 665 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0
) )  ->  ( abs `  N )  e.  NN )
143142, 31syl6eleq 2520 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0
) )  ->  ( abs `  N )  e.  ( ZZ>= `  1 )
)
144 elfznn 11829 . . . . . . 7  |-  ( k  e.  ( 1 ... ( abs `  N
) )  ->  k  e.  NN )
14543, 144, 44syl2an 479 . . . . . 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 11042 . . . . 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 12233 . . . 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 9846 . . 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 2463 . 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 2473 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 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1868    =/= wne 2618   ifcif 3909   class class class wbr 4420    |-> cmpt 4479   -->wf 5594   ` cfv 5598  (class class class)co 6302   CCcc 9538   0cc0 9540   1c1 9541    + caddc 9543    x. cmul 9545    < clt 9676   -ucneg 9862   NNcn 10610   NN0cn0 10870   ZZcz 10938   ZZ>=cuz 11160   ...cfz 11785    seqcseq 12213   ^cexp 12272   abscabs 13286   Primecprime 14610    pCnt cpc 14774    /Lclgs 24209
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-rep 4533  ax-sep 4543  ax-nul 4552  ax-pow 4599  ax-pr 4657  ax-un 6594  ax-cnex 9596  ax-resscn 9597  ax-1cn 9598  ax-icn 9599  ax-addcl 9600  ax-addrcl 9601  ax-mulcl 9602  ax-mulrcl 9603  ax-mulcom 9604  ax-addass 9605  ax-mulass 9606  ax-distr 9607  ax-i2m1 9608  ax-1ne0 9609  ax-1rid 9610  ax-rnegex 9611  ax-rrecex 9612  ax-cnre 9613  ax-pre-lttri 9614  ax-pre-lttrn 9615  ax-pre-ltadd 9616  ax-pre-mulgt0 9617  ax-pre-sup 9618
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-nel 2621  df-ral 2780  df-rex 2781  df-reu 2782  df-rmo 2783  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-tp 4001  df-op 4003  df-uni 4217  df-int 4253  df-iun 4298  df-br 4421  df-opab 4480  df-mpt 4481  df-tr 4516  df-eprel 4761  df-id 4765  df-po 4771  df-so 4772  df-fr 4809  df-we 4811  df-xp 4856  df-rel 4857  df-cnv 4858  df-co 4859  df-dm 4860  df-rn 4861  df-res 4862  df-ima 4863  df-pred 5396  df-ord 5442  df-on 5443  df-lim 5444  df-suc 5445  df-iota 5562  df-fun 5600  df-fn 5601  df-f 5602  df-f1 5603  df-fo 5604  df-f1o 5605  df-fv 5606  df-riota 6264  df-ov 6305  df-oprab 6306  df-mpt2 6307  df-om 6704  df-1st 6804  df-2nd 6805  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-1o 7187  df-2o 7188  df-oadd 7191  df-er 7368  df-map 7479  df-en 7575  df-dom 7576  df-sdom 7577  df-fin 7578  df-sup 7959  df-inf 7960  df-card 8375  df-cda 8599  df-pnf 9678  df-mnf 9679  df-xr 9680  df-ltxr 9681  df-le 9682  df-sub 9863  df-neg 9864  df-div 10271  df-nn 10611  df-2 10669  df-3 10670  df-n0 10871  df-z 10939  df-uz 11161  df-q 11266  df-rp 11304  df-fz 11786  df-fzo 11917  df-fl 12028  df-mod 12097  df-seq 12214  df-exp 12273  df-hash 12516  df-cj 13151  df-re 13152  df-im 13153  df-sqrt 13287  df-abs 13288  df-dvds 14294  df-gcd 14457  df-prm 14611  df-phi 14702  df-pc 14775  df-lgs 24210
This theorem is referenced by:  lgssq2  24251  lgsdinn0  24255  lgsquad2lem1  24273
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