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Theorem lgsdir 23583
Description: The Legendre symbol is completely multiplicative in its left argument. Together with lgsqr 23599 this implies that the product of two quadratic residues or nonresidues is a residue, and the product of a residue and a nonresidue is a nonresidue. (Contributed by Mario Carneiro, 4-Feb-2015.)
Assertion
Ref Expression
lgsdir  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0
) )  ->  (
( A  x.  B
)  /L N )  =  ( ( A  /L N )  x.  ( B  /L N ) ) )

Proof of Theorem lgsdir
Dummy variables  k  n  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ax-1cn 9553 . . . . . . 7  |-  1  e.  CC
2 0cn 9591 . . . . . . 7  |-  0  e.  CC
31, 2keepel 3994 . . . . . 6  |-  if ( ( B ^ 2 )  =  1 ,  1 ,  0 )  e.  CC
43mulid2i 9602 . . . . 5  |-  ( 1  x.  if ( ( B ^ 2 )  =  1 ,  1 ,  0 ) )  =  if ( ( B ^ 2 )  =  1 ,  1 ,  0 )
5 iftrue 3932 . . . . . . 7  |-  ( ( A ^ 2 )  =  1  ->  if ( ( A ^
2 )  =  1 ,  1 ,  0 )  =  1 )
65adantl 466 . . . . . 6  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =  0 )  /\  ( A ^
2 )  =  1 )  ->  if (
( A ^ 2 )  =  1 ,  1 ,  0 )  =  1 )
76oveq1d 6296 . . . . 5  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =  0 )  /\  ( A ^
2 )  =  1 )  ->  ( if ( ( A ^
2 )  =  1 ,  1 ,  0 )  x.  if ( ( B ^ 2 )  =  1 ,  1 ,  0 ) )  =  ( 1  x.  if ( ( B ^ 2 )  =  1 ,  1 ,  0 ) ) )
8 simpl1 1000 . . . . . . . . . . 11  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0
) )  ->  A  e.  ZZ )
98zcnd 10977 . . . . . . . . . 10  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0
) )  ->  A  e.  CC )
109ad2antrr 725 . . . . . . . . 9  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =  0 )  /\  ( A ^
2 )  =  1 )  ->  A  e.  CC )
11 simpl2 1001 . . . . . . . . . . 11  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0
) )  ->  B  e.  ZZ )
1211zcnd 10977 . . . . . . . . . 10  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0
) )  ->  B  e.  CC )
1312ad2antrr 725 . . . . . . . . 9  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =  0 )  /\  ( A ^
2 )  =  1 )  ->  B  e.  CC )
1410, 13sqmuld 12304 . . . . . . . 8  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =  0 )  /\  ( A ^
2 )  =  1 )  ->  ( ( A  x.  B ) ^ 2 )  =  ( ( A ^
2 )  x.  ( B ^ 2 ) ) )
15 simpr 461 . . . . . . . . 9  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =  0 )  /\  ( A ^
2 )  =  1 )  ->  ( A ^ 2 )  =  1 )
1615oveq1d 6296 . . . . . . . 8  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =  0 )  /\  ( A ^
2 )  =  1 )  ->  ( ( A ^ 2 )  x.  ( B ^ 2 ) )  =  ( 1  x.  ( B ^ 2 ) ) )
1712sqcld 12290 . . . . . . . . . 10  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0
) )  ->  ( B ^ 2 )  e.  CC )
1817ad2antrr 725 . . . . . . . . 9  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =  0 )  /\  ( A ^
2 )  =  1 )  ->  ( B ^ 2 )  e.  CC )
1918mulid2d 9617 . . . . . . . 8  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =  0 )  /\  ( A ^
2 )  =  1 )  ->  ( 1  x.  ( B ^
2 ) )  =  ( B ^ 2 ) )
2014, 16, 193eqtrd 2488 . . . . . . 7  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =  0 )  /\  ( A ^
2 )  =  1 )  ->  ( ( A  x.  B ) ^ 2 )  =  ( B ^ 2 ) )
2120eqeq1d 2445 . . . . . 6  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =  0 )  /\  ( A ^
2 )  =  1 )  ->  ( (
( A  x.  B
) ^ 2 )  =  1  <->  ( B ^ 2 )  =  1 ) )
2221ifbid 3948 . . . . 5  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =  0 )  /\  ( A ^
2 )  =  1 )  ->  if (
( ( A  x.  B ) ^ 2 )  =  1 ,  1 ,  0 )  =  if ( ( B ^ 2 )  =  1 ,  1 ,  0 ) )
234, 7, 223eqtr4a 2510 . . . 4  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =  0 )  /\  ( A ^
2 )  =  1 )  ->  ( if ( ( A ^
2 )  =  1 ,  1 ,  0 )  x.  if ( ( B ^ 2 )  =  1 ,  1 ,  0 ) )  =  if ( ( ( A  x.  B ) ^ 2 )  =  1 ,  1 ,  0 ) )
243mul02i 9772 . . . . 5  |-  ( 0  x.  if ( ( B ^ 2 )  =  1 ,  1 ,  0 ) )  =  0
25 iffalse 3935 . . . . . . 7  |-  ( -.  ( A ^ 2 )  =  1  ->  if ( ( A ^
2 )  =  1 ,  1 ,  0 )  =  0 )
2625adantl 466 . . . . . 6  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =  0 )  /\  -.  ( A ^ 2 )  =  1 )  ->  if ( ( A ^
2 )  =  1 ,  1 ,  0 )  =  0 )
2726oveq1d 6296 . . . . 5  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =  0 )  /\  -.  ( A ^ 2 )  =  1 )  ->  ( if ( ( A ^
2 )  =  1 ,  1 ,  0 )  x.  if ( ( B ^ 2 )  =  1 ,  1 ,  0 ) )  =  ( 0  x.  if ( ( B ^ 2 )  =  1 ,  1 ,  0 ) ) )
28 dvdsmul1 13987 . . . . . . . . . . . 12  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  A  ||  ( A  x.  B ) )
298, 11, 28syl2anc 661 . . . . . . . . . . 11  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0
) )  ->  A  ||  ( A  x.  B
) )
308, 11zmulcld 10982 . . . . . . . . . . . 12  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0
) )  ->  ( A  x.  B )  e.  ZZ )
31 dvdssq 14180 . . . . . . . . . . . 12  |-  ( ( A  e.  ZZ  /\  ( A  x.  B
)  e.  ZZ )  ->  ( A  ||  ( A  x.  B
)  <->  ( A ^
2 )  ||  (
( A  x.  B
) ^ 2 ) ) )
328, 30, 31syl2anc 661 . . . . . . . . . . 11  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0
) )  ->  ( A  ||  ( A  x.  B )  <->  ( A ^ 2 )  ||  ( ( A  x.  B ) ^ 2 ) ) )
3329, 32mpbid 210 . . . . . . . . . 10  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0
) )  ->  ( A ^ 2 )  ||  ( ( A  x.  B ) ^ 2 ) )
3433adantr 465 . . . . . . . . 9  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =  0 )  ->  ( A ^
2 )  ||  (
( A  x.  B
) ^ 2 ) )
35 breq2 4441 . . . . . . . . 9  |-  ( ( ( A  x.  B
) ^ 2 )  =  1  ->  (
( A ^ 2 )  ||  ( ( A  x.  B ) ^ 2 )  <->  ( A ^ 2 )  ||  1 ) )
3634, 35syl5ibcom 220 . . . . . . . 8  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =  0 )  ->  ( ( ( A  x.  B ) ^ 2 )  =  1  ->  ( A ^ 2 )  ||  1 ) )
37 simprl 756 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0
) )  ->  A  =/=  0 )
3837neneqd 2645 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0
) )  ->  -.  A  =  0 )
39 sqeq0 12214 . . . . . . . . . . . . . . . 16  |-  ( A  e.  CC  ->  (
( A ^ 2 )  =  0  <->  A  =  0 ) )
409, 39syl 16 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0
) )  ->  (
( A ^ 2 )  =  0  <->  A  =  0 ) )
4138, 40mtbird 301 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0
) )  ->  -.  ( A ^ 2 )  =  0 )
42 zsqcl2 12227 . . . . . . . . . . . . . . . . 17  |-  ( A  e.  ZZ  ->  ( A ^ 2 )  e. 
NN0 )
438, 42syl 16 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0
) )  ->  ( A ^ 2 )  e. 
NN0 )
44 elnn0 10804 . . . . . . . . . . . . . . . 16  |-  ( ( A ^ 2 )  e.  NN0  <->  ( ( A ^ 2 )  e.  NN  \/  ( A ^ 2 )  =  0 ) )
4543, 44sylib 196 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0
) )  ->  (
( A ^ 2 )  e.  NN  \/  ( A ^ 2 )  =  0 ) )
4645ord 377 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0
) )  ->  ( -.  ( A ^ 2 )  e.  NN  ->  ( A ^ 2 )  =  0 ) )
4741, 46mt3d 125 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0
) )  ->  ( A ^ 2 )  e.  NN )
4847adantr 465 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =  0 )  ->  ( A ^
2 )  e.  NN )
4948nnzd 10975 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =  0 )  ->  ( A ^
2 )  e.  ZZ )
50 1nn 10554 . . . . . . . . . . 11  |-  1  e.  NN
51 dvdsle 14013 . . . . . . . . . . 11  |-  ( ( ( A ^ 2 )  e.  ZZ  /\  1  e.  NN )  ->  ( ( A ^
2 )  ||  1  ->  ( A ^ 2 )  <_  1 ) )
5249, 50, 51sylancl 662 . . . . . . . . . 10  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =  0 )  ->  ( ( A ^ 2 )  ||  1  ->  ( A ^
2 )  <_  1
) )
5348nnge1d 10585 . . . . . . . . . 10  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =  0 )  ->  1  <_  ( A ^ 2 ) )
5452, 53jctird 544 . . . . . . . . 9  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =  0 )  ->  ( ( A ^ 2 )  ||  1  ->  ( ( A ^ 2 )  <_ 
1  /\  1  <_  ( A ^ 2 ) ) ) )
5548nnred 10558 . . . . . . . . . 10  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =  0 )  ->  ( A ^
2 )  e.  RR )
56 1re 9598 . . . . . . . . . 10  |-  1  e.  RR
57 letri3 9673 . . . . . . . . . 10  |-  ( ( ( A ^ 2 )  e.  RR  /\  1  e.  RR )  ->  ( ( A ^
2 )  =  1  <-> 
( ( A ^
2 )  <_  1  /\  1  <_  ( A ^ 2 ) ) ) )
5855, 56, 57sylancl 662 . . . . . . . . 9  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =  0 )  ->  ( ( A ^ 2 )  =  1  <->  ( ( A ^ 2 )  <_ 
1  /\  1  <_  ( A ^ 2 ) ) ) )
5954, 58sylibrd 234 . . . . . . . 8  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =  0 )  ->  ( ( A ^ 2 )  ||  1  ->  ( A ^
2 )  =  1 ) )
6036, 59syld 44 . . . . . . 7  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =  0 )  ->  ( ( ( A  x.  B ) ^ 2 )  =  1  ->  ( A ^ 2 )  =  1 ) )
6160con3dimp 441 . . . . . 6  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =  0 )  /\  -.  ( A ^ 2 )  =  1 )  ->  -.  ( ( A  x.  B ) ^ 2 )  =  1 )
6261iffalsed 3937 . . . . 5  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =  0 )  /\  -.  ( A ^ 2 )  =  1 )  ->  if ( ( ( A  x.  B ) ^
2 )  =  1 ,  1 ,  0 )  =  0 )
6324, 27, 623eqtr4a 2510 . . . 4  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =  0 )  /\  -.  ( A ^ 2 )  =  1 )  ->  ( if ( ( A ^
2 )  =  1 ,  1 ,  0 )  x.  if ( ( B ^ 2 )  =  1 ,  1 ,  0 ) )  =  if ( ( ( A  x.  B ) ^ 2 )  =  1 ,  1 ,  0 ) )
6423, 63pm2.61dan 791 . . 3  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =  0 )  ->  ( if ( ( A ^ 2 )  =  1 ,  1 ,  0 )  x.  if ( ( B ^ 2 )  =  1 ,  1 ,  0 ) )  =  if ( ( ( A  x.  B
) ^ 2 )  =  1 ,  1 ,  0 ) )
65 oveq2 6289 . . . . 5  |-  ( N  =  0  ->  ( A  /L N )  =  ( A  /L 0 ) )
66 lgs0 23562 . . . . . 6  |-  ( A  e.  ZZ  ->  ( A  /L 0 )  =  if ( ( A ^ 2 )  =  1 ,  1 ,  0 ) )
678, 66syl 16 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0
) )  ->  ( A  /L 0 )  =  if ( ( A ^ 2 )  =  1 ,  1 ,  0 ) )
6865, 67sylan9eqr 2506 . . . 4  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =  0 )  ->  ( A  /L N )  =  if ( ( A ^ 2 )  =  1 ,  1 ,  0 ) )
69 oveq2 6289 . . . . 5  |-  ( N  =  0  ->  ( B  /L N )  =  ( B  /L 0 ) )
70 lgs0 23562 . . . . . 6  |-  ( B  e.  ZZ  ->  ( B  /L 0 )  =  if ( ( B ^ 2 )  =  1 ,  1 ,  0 ) )
7111, 70syl 16 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0
) )  ->  ( B  /L 0 )  =  if ( ( B ^ 2 )  =  1 ,  1 ,  0 ) )
7269, 71sylan9eqr 2506 . . . 4  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =  0 )  ->  ( B  /L N )  =  if ( ( B ^ 2 )  =  1 ,  1 ,  0 ) )
7368, 72oveq12d 6299 . . 3  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =  0 )  ->  ( ( A  /L N )  x.  ( B  /L N ) )  =  ( if ( ( A ^ 2 )  =  1 ,  1 ,  0 )  x.  if ( ( B ^ 2 )  =  1 ,  1 ,  0 ) ) )
74 oveq2 6289 . . . 4  |-  ( N  =  0  ->  (
( A  x.  B
)  /L N )  =  ( ( A  x.  B )  /L 0 ) )
75 lgs0 23562 . . . . 5  |-  ( ( A  x.  B )  e.  ZZ  ->  (
( A  x.  B
)  /L 0 )  =  if ( ( ( A  x.  B ) ^ 2 )  =  1 ,  1 ,  0 ) )
7630, 75syl 16 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0
) )  ->  (
( A  x.  B
)  /L 0 )  =  if ( ( ( A  x.  B ) ^ 2 )  =  1 ,  1 ,  0 ) )
7774, 76sylan9eqr 2506 . . 3  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =  0 )  ->  ( ( A  x.  B )  /L N )  =  if ( ( ( A  x.  B ) ^ 2 )  =  1 ,  1 ,  0 ) )
7864, 73, 773eqtr4rd 2495 . 2  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =  0 )  ->  ( ( A  x.  B )  /L N )  =  ( ( A  /L N )  x.  ( B  /L
N ) ) )
79 lgsdilem 23575 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0
) )  ->  if ( ( N  <  0  /\  ( A  x.  B )  <  0 ) ,  -u
1 ,  1 )  =  ( if ( ( N  <  0  /\  A  <  0
) ,  -u 1 ,  1 )  x.  if ( ( N  <  0  /\  B  <  0 ) ,  -u
1 ,  1 ) ) )
8079adantr 465 . . . 4  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =/=  0 )  ->  if ( ( N  <  0  /\  ( A  x.  B )  <  0 ) ,  -u
1 ,  1 )  =  ( if ( ( N  <  0  /\  A  <  0
) ,  -u 1 ,  1 )  x.  if ( ( N  <  0  /\  B  <  0 ) ,  -u
1 ,  1 ) ) )
81 mulcl 9579 . . . . . 6  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  x.  y
)  e.  CC )
8281adantl 466 . . . . 5  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =/=  0 )  /\  ( x  e.  CC  /\  y  e.  CC ) )  ->  ( x  x.  y )  e.  CC )
83 mulcom 9581 . . . . . 6  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  x.  y
)  =  ( y  x.  x ) )
8483adantl 466 . . . . 5  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =/=  0 )  /\  ( x  e.  CC  /\  y  e.  CC ) )  ->  ( x  x.  y )  =  ( y  x.  x ) )
85 mulass 9583 . . . . . 6  |-  ( ( x  e.  CC  /\  y  e.  CC  /\  z  e.  CC )  ->  (
( x  x.  y
)  x.  z )  =  ( x  x.  ( y  x.  z
) ) )
8685adantl 466 . . . . 5  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =/=  0 )  /\  ( x  e.  CC  /\  y  e.  CC  /\  z  e.  CC )
)  ->  ( (
x  x.  y )  x.  z )  =  ( x  x.  (
y  x.  z ) ) )
87 simpl3 1002 . . . . . . 7  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0
) )  ->  N  e.  ZZ )
88 nnabscl 13140 . . . . . . 7  |-  ( ( N  e.  ZZ  /\  N  =/=  0 )  -> 
( abs `  N
)  e.  NN )
8987, 88sylan 471 . . . . . 6  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =/=  0 )  -> 
( abs `  N
)  e.  NN )
90 nnuz 11127 . . . . . 6  |-  NN  =  ( ZZ>= `  1 )
9189, 90syl6eleq 2541 . . . . 5  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =/=  0 )  -> 
( abs `  N
)  e.  ( ZZ>= ` 
1 ) )
92 simpll1 1036 . . . . . . . 8  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =/=  0 )  ->  A  e.  ZZ )
93 simpll3 1038 . . . . . . . 8  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =/=  0 )  ->  N  e.  ZZ )
94 simpr 461 . . . . . . . 8  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =/=  0 )  ->  N  =/=  0 )
95 eqid 2443 . . . . . . . . 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 ) )
9695lgsfcl3 23570 . . . . . . . 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 )
9792, 93, 94, 96syl3anc 1229 . . . . . . 7  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =/=  0 )  -> 
( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( A  /L n ) ^
( n  pCnt  N
) ) ,  1 ) ) : NN --> ZZ )
98 elfznn 11725 . . . . . . 7  |-  ( k  e.  ( 1 ... ( abs `  N
) )  ->  k  e.  NN )
99 ffvelrn 6014 . . . . . . 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 )
10097, 98, 99syl2an 477 . . . . . 6  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  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 )
101100zcnd 10977 . . . . 5  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  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 )
102 simpll2 1037 . . . . . . . 8  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =/=  0 )  ->  B  e.  ZZ )
103 eqid 2443 . . . . . . . . 9  |-  ( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( B  /L
n ) ^ (
n  pCnt  N )
) ,  1 ) )  =  ( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( B  /L
n ) ^ (
n  pCnt  N )
) ,  1 ) )
104103lgsfcl3 23570 . . . . . . . 8  |-  ( ( B  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  ->  (
n  e.  NN  |->  if ( n  e.  Prime ,  ( ( B  /L n ) ^
( n  pCnt  N
) ) ,  1 ) ) : NN --> ZZ )
105102, 93, 94, 104syl3anc 1229 . . . . . . 7  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =/=  0 )  -> 
( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( B  /L n ) ^
( n  pCnt  N
) ) ,  1 ) ) : NN --> ZZ )
106 ffvelrn 6014 . . . . . . 7  |-  ( ( ( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( B  /L n ) ^
( n  pCnt  N
) ) ,  1 ) ) : NN --> ZZ  /\  k  e.  NN )  ->  ( ( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( B  /L
n ) ^ (
n  pCnt  N )
) ,  1 ) ) `  k )  e.  ZZ )
107105, 98, 106syl2an 477 . . . . . 6  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =/=  0 )  /\  k  e.  ( 1 ... ( abs `  N
) ) )  -> 
( ( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( B  /L n ) ^ ( n 
pCnt  N ) ) ,  1 ) ) `  k )  e.  ZZ )
108107zcnd 10977 . . . . 5  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =/=  0 )  /\  k  e.  ( 1 ... ( abs `  N
) ) )  -> 
( ( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( B  /L n ) ^ ( n 
pCnt  N ) ) ,  1 ) ) `  k )  e.  CC )
10992adantr 465 . . . . . . . . . . . 12  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =/=  0 )  /\  k  e.  Prime )  ->  A  e.  ZZ )
110102adantr 465 . . . . . . . . . . . 12  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =/=  0 )  /\  k  e.  Prime )  ->  B  e.  ZZ )
111 simpr 461 . . . . . . . . . . . 12  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =/=  0 )  /\  k  e.  Prime )  -> 
k  e.  Prime )
112 lgsdirprm 23582 . . . . . . . . . . . 12  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  k  e.  Prime )  ->  (
( A  x.  B
)  /L k )  =  ( ( A  /L k )  x.  ( B  /L k ) ) )
113109, 110, 111, 112syl3anc 1229 . . . . . . . . . . 11  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =/=  0 )  /\  k  e.  Prime )  -> 
( ( A  x.  B )  /L
k )  =  ( ( A  /L
k )  x.  ( B  /L k ) ) )
114113oveq1d 6296 . . . . . . . . . 10  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =/=  0 )  /\  k  e.  Prime )  -> 
( ( ( A  x.  B )  /L k ) ^
( k  pCnt  N
) )  =  ( ( ( A  /L k )  x.  ( B  /L
k ) ) ^
( k  pCnt  N
) ) )
115 prmz 14203 . . . . . . . . . . . . 13  |-  ( k  e.  Prime  ->  k  e.  ZZ )
116 lgscl 23563 . . . . . . . . . . . . 13  |-  ( ( A  e.  ZZ  /\  k  e.  ZZ )  ->  ( A  /L
k )  e.  ZZ )
11792, 115, 116syl2an 477 . . . . . . . . . . . 12  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =/=  0 )  /\  k  e.  Prime )  -> 
( A  /L
k )  e.  ZZ )
118117zcnd 10977 . . . . . . . . . . 11  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =/=  0 )  /\  k  e.  Prime )  -> 
( A  /L
k )  e.  CC )
119 lgscl 23563 . . . . . . . . . . . . 13  |-  ( ( B  e.  ZZ  /\  k  e.  ZZ )  ->  ( B  /L
k )  e.  ZZ )
120102, 115, 119syl2an 477 . . . . . . . . . . . 12  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =/=  0 )  /\  k  e.  Prime )  -> 
( B  /L
k )  e.  ZZ )
121120zcnd 10977 . . . . . . . . . . 11  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =/=  0 )  /\  k  e.  Prime )  -> 
( B  /L
k )  e.  CC )
12293adantr 465 . . . . . . . . . . . 12  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =/=  0 )  /\  k  e.  Prime )  ->  N  e.  ZZ )
12394adantr 465 . . . . . . . . . . . 12  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =/=  0 )  /\  k  e.  Prime )  ->  N  =/=  0 )
124 pczcl 14354 . . . . . . . . . . . 12  |-  ( ( k  e.  Prime  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( k  pCnt  N
)  e.  NN0 )
125111, 122, 123, 124syl12anc 1227 . . . . . . . . . . 11  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =/=  0 )  /\  k  e.  Prime )  -> 
( k  pCnt  N
)  e.  NN0 )
126118, 121, 125mulexpd 12307 . . . . . . . . . 10  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =/=  0 )  /\  k  e.  Prime )  -> 
( ( ( A  /L k )  x.  ( B  /L k ) ) ^ ( k  pCnt  N ) )  =  ( ( ( A  /L k ) ^
( k  pCnt  N
) )  x.  (
( B  /L
k ) ^ (
k  pCnt  N )
) ) )
127114, 126eqtrd 2484 . . . . . . . . 9  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =/=  0 )  /\  k  e.  Prime )  -> 
( ( ( A  x.  B )  /L k ) ^
( k  pCnt  N
) )  =  ( ( ( A  /L k ) ^
( k  pCnt  N
) )  x.  (
( B  /L
k ) ^ (
k  pCnt  N )
) ) )
128 iftrue 3932 . . . . . . . . . 10  |-  ( k  e.  Prime  ->  if ( k  e.  Prime ,  ( ( ( A  x.  B )  /L
k ) ^ (
k  pCnt  N )
) ,  1 )  =  ( ( ( A  x.  B )  /L k ) ^ ( k  pCnt  N ) ) )
129128adantl 466 . . . . . . . . 9  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =/=  0 )  /\  k  e.  Prime )  ->  if ( k  e.  Prime ,  ( ( ( A  x.  B )  /L k ) ^
( k  pCnt  N
) ) ,  1 )  =  ( ( ( A  x.  B
)  /L k ) ^ ( k 
pCnt  N ) ) )
130 iftrue 3932 . . . . . . . . . . 11  |-  ( k  e.  Prime  ->  if ( k  e.  Prime ,  ( ( A  /L
k ) ^ (
k  pCnt  N )
) ,  1 )  =  ( ( A  /L k ) ^ ( k  pCnt  N ) ) )
131 iftrue 3932 . . . . . . . . . . 11  |-  ( k  e.  Prime  ->  if ( k  e.  Prime ,  ( ( B  /L
k ) ^ (
k  pCnt  N )
) ,  1 )  =  ( ( B  /L k ) ^ ( k  pCnt  N ) ) )
132130, 131oveq12d 6299 . . . . . . . . . 10  |-  ( k  e.  Prime  ->  ( if ( k  e.  Prime ,  ( ( A  /L k ) ^
( k  pCnt  N
) ) ,  1 )  x.  if ( k  e.  Prime ,  ( ( B  /L
k ) ^ (
k  pCnt  N )
) ,  1 ) )  =  ( ( ( A  /L
k ) ^ (
k  pCnt  N )
)  x.  ( ( B  /L k ) ^ ( k 
pCnt  N ) ) ) )
133132adantl 466 . . . . . . . . 9  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =/=  0 )  /\  k  e.  Prime )  -> 
( if ( k  e.  Prime ,  ( ( A  /L k ) ^ ( k 
pCnt  N ) ) ,  1 )  x.  if ( k  e.  Prime ,  ( ( B  /L k ) ^
( k  pCnt  N
) ) ,  1 ) )  =  ( ( ( A  /L k ) ^
( k  pCnt  N
) )  x.  (
( B  /L
k ) ^ (
k  pCnt  N )
) ) )
134127, 129, 1333eqtr4d 2494 . . . . . . . 8  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =/=  0 )  /\  k  e.  Prime )  ->  if ( k  e.  Prime ,  ( ( ( A  x.  B )  /L k ) ^
( k  pCnt  N
) ) ,  1 )  =  ( if ( k  e.  Prime ,  ( ( A  /L k ) ^
( k  pCnt  N
) ) ,  1 )  x.  if ( k  e.  Prime ,  ( ( B  /L
k ) ^ (
k  pCnt  N )
) ,  1 ) ) )
135 1t1e1 10690 . . . . . . . . . . 11  |-  ( 1  x.  1 )  =  1
136135eqcomi 2456 . . . . . . . . . 10  |-  1  =  ( 1  x.  1 )
137 iffalse 3935 . . . . . . . . . 10  |-  ( -.  k  e.  Prime  ->  if ( k  e.  Prime ,  ( ( ( A  x.  B )  /L k ) ^
( k  pCnt  N
) ) ,  1 )  =  1 )
138 iffalse 3935 . . . . . . . . . . 11  |-  ( -.  k  e.  Prime  ->  if ( k  e.  Prime ,  ( ( A  /L k ) ^
( k  pCnt  N
) ) ,  1 )  =  1 )
139 iffalse 3935 . . . . . . . . . . 11  |-  ( -.  k  e.  Prime  ->  if ( k  e.  Prime ,  ( ( B  /L k ) ^
( k  pCnt  N
) ) ,  1 )  =  1 )
140138, 139oveq12d 6299 . . . . . . . . . 10  |-  ( -.  k  e.  Prime  ->  ( if ( k  e. 
Prime ,  ( ( A  /L k ) ^ ( k  pCnt  N ) ) ,  1 )  x.  if ( k  e.  Prime ,  ( ( B  /L
k ) ^ (
k  pCnt  N )
) ,  1 ) )  =  ( 1  x.  1 ) )
141136, 137, 1403eqtr4a 2510 . . . . . . . . 9  |-  ( -.  k  e.  Prime  ->  if ( k  e.  Prime ,  ( ( ( A  x.  B )  /L k ) ^
( k  pCnt  N
) ) ,  1 )  =  ( if ( k  e.  Prime ,  ( ( A  /L k ) ^
( k  pCnt  N
) ) ,  1 )  x.  if ( k  e.  Prime ,  ( ( B  /L
k ) ^ (
k  pCnt  N )
) ,  1 ) ) )
142141adantl 466 . . . . . . . 8  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =/=  0 )  /\  -.  k  e.  Prime )  ->  if ( k  e.  Prime ,  ( ( ( A  x.  B
)  /L k ) ^ ( k 
pCnt  N ) ) ,  1 )  =  ( if ( k  e. 
Prime ,  ( ( A  /L k ) ^ ( k  pCnt  N ) ) ,  1 )  x.  if ( k  e.  Prime ,  ( ( B  /L
k ) ^ (
k  pCnt  N )
) ,  1 ) ) )
143134, 142pm2.61dan 791 . . . . . . 7  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =/=  0 )  ->  if ( k  e.  Prime ,  ( ( ( A  x.  B )  /L k ) ^
( k  pCnt  N
) ) ,  1 )  =  ( if ( k  e.  Prime ,  ( ( A  /L k ) ^
( k  pCnt  N
) ) ,  1 )  x.  if ( k  e.  Prime ,  ( ( B  /L
k ) ^ (
k  pCnt  N )
) ,  1 ) ) )
144143adantr 465 . . . . . 6  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =/=  0 )  /\  k  e.  ( 1 ... ( abs `  N
) ) )  ->  if ( k  e.  Prime ,  ( ( ( A  x.  B )  /L k ) ^
( k  pCnt  N
) ) ,  1 )  =  ( if ( k  e.  Prime ,  ( ( A  /L k ) ^
( k  pCnt  N
) ) ,  1 )  x.  if ( k  e.  Prime ,  ( ( B  /L
k ) ^ (
k  pCnt  N )
) ,  1 ) ) )
14598adantl 466 . . . . . . 7  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =/=  0 )  /\  k  e.  ( 1 ... ( abs `  N
) ) )  -> 
k  e.  NN )
146 eleq1 2515 . . . . . . . . 9  |-  ( n  =  k  ->  (
n  e.  Prime  <->  k  e.  Prime ) )
147 oveq2 6289 . . . . . . . . . 10  |-  ( n  =  k  ->  (
( A  x.  B
)  /L n )  =  ( ( A  x.  B )  /L k ) )
148 oveq1 6288 . . . . . . . . . 10  |-  ( n  =  k  ->  (
n  pCnt  N )  =  ( k  pCnt  N ) )
149147, 148oveq12d 6299 . . . . . . . . 9  |-  ( n  =  k  ->  (
( ( A  x.  B )  /L
n ) ^ (
n  pCnt  N )
)  =  ( ( ( A  x.  B
)  /L k ) ^ ( k 
pCnt  N ) ) )
150146, 149ifbieq1d 3949 . . . . . . . 8  |-  ( n  =  k  ->  if ( n  e.  Prime ,  ( ( ( A  x.  B )  /L n ) ^
( n  pCnt  N
) ) ,  1 )  =  if ( k  e.  Prime ,  ( ( ( A  x.  B )  /L
k ) ^ (
k  pCnt  N )
) ,  1 ) )
151 eqid 2443 . . . . . . . 8  |-  ( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( ( A  x.  B )  /L
n ) ^ (
n  pCnt  N )
) ,  1 ) )  =  ( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( ( A  x.  B )  /L
n ) ^ (
n  pCnt  N )
) ,  1 ) )
152 ovex 6309 . . . . . . . . 9  |-  ( ( ( A  x.  B
)  /L k ) ^ ( k 
pCnt  N ) )  e. 
_V
153 1ex 9594 . . . . . . . . 9  |-  1  e.  _V
154152, 153ifex 3995 . . . . . . . 8  |-  if ( k  e.  Prime ,  ( ( ( A  x.  B )  /L
k ) ^ (
k  pCnt  N )
) ,  1 )  e.  _V
155150, 151, 154fvmpt 5941 . . . . . . 7  |-  ( k  e.  NN  ->  (
( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( ( A  x.  B )  /L n ) ^
( n  pCnt  N
) ) ,  1 ) ) `  k
)  =  if ( k  e.  Prime ,  ( ( ( A  x.  B )  /L
k ) ^ (
k  pCnt  N )
) ,  1 ) )
156145, 155syl 16 . . . . . 6  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =/=  0 )  /\  k  e.  ( 1 ... ( abs `  N
) ) )  -> 
( ( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( ( A  x.  B
)  /L n ) ^ ( n 
pCnt  N ) ) ,  1 ) ) `  k )  =  if ( k  e.  Prime ,  ( ( ( A  x.  B )  /L k ) ^
( k  pCnt  N
) ) ,  1 ) )
157 oveq2 6289 . . . . . . . . . . 11  |-  ( n  =  k  ->  ( A  /L n )  =  ( A  /L k ) )
158157, 148oveq12d 6299 . . . . . . . . . 10  |-  ( n  =  k  ->  (
( A  /L
n ) ^ (
n  pCnt  N )
)  =  ( ( A  /L k ) ^ ( k 
pCnt  N ) ) )
159146, 158ifbieq1d 3949 . . . . . . . . 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 ) )
160 ovex 6309 . . . . . . . . . 10  |-  ( ( A  /L k ) ^ ( k 
pCnt  N ) )  e. 
_V
161160, 153ifex 3995 . . . . . . . . 9  |-  if ( k  e.  Prime ,  ( ( A  /L
k ) ^ (
k  pCnt  N )
) ,  1 )  e.  _V
162159, 95, 161fvmpt 5941 . . . . . . . 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 ) )
163145, 162syl 16 . . . . . . 7  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =/=  0 )  /\  k  e.  ( 1 ... ( abs `  N
) ) )  -> 
( ( 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 ) )
164 oveq2 6289 . . . . . . . . . . 11  |-  ( n  =  k  ->  ( B  /L n )  =  ( B  /L k ) )
165164, 148oveq12d 6299 . . . . . . . . . 10  |-  ( n  =  k  ->  (
( B  /L
n ) ^ (
n  pCnt  N )
)  =  ( ( B  /L k ) ^ ( k 
pCnt  N ) ) )
166146, 165ifbieq1d 3949 . . . . . . . . 9  |-  ( n  =  k  ->  if ( n  e.  Prime ,  ( ( B  /L n ) ^
( n  pCnt  N
) ) ,  1 )  =  if ( k  e.  Prime ,  ( ( B  /L
k ) ^ (
k  pCnt  N )
) ,  1 ) )
167 ovex 6309 . . . . . . . . . 10  |-  ( ( B  /L k ) ^ ( k 
pCnt  N ) )  e. 
_V
168167, 153ifex 3995 . . . . . . . . 9  |-  if ( k  e.  Prime ,  ( ( B  /L
k ) ^ (
k  pCnt  N )
) ,  1 )  e.  _V
169166, 103, 168fvmpt 5941 . . . . . . . 8  |-  ( k  e.  NN  ->  (
( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( B  /L n ) ^
( n  pCnt  N
) ) ,  1 ) ) `  k
)  =  if ( k  e.  Prime ,  ( ( B  /L
k ) ^ (
k  pCnt  N )
) ,  1 ) )
170145, 169syl 16 . . . . . . 7  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =/=  0 )  /\  k  e.  ( 1 ... ( abs `  N
) ) )  -> 
( ( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( B  /L n ) ^ ( n 
pCnt  N ) ) ,  1 ) ) `  k )  =  if ( k  e.  Prime ,  ( ( B  /L k ) ^
( k  pCnt  N
) ) ,  1 ) )
171163, 170oveq12d 6299 . . . . . 6  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =/=  0 )  /\  k  e.  ( 1 ... ( abs `  N
) ) )  -> 
( ( ( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( A  /L
n ) ^ (
n  pCnt  N )
) ,  1 ) ) `  k )  x.  ( ( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( B  /L
n ) ^ (
n  pCnt  N )
) ,  1 ) ) `  k ) )  =  ( if ( k  e.  Prime ,  ( ( A  /L k ) ^
( k  pCnt  N
) ) ,  1 )  x.  if ( k  e.  Prime ,  ( ( B  /L
k ) ^ (
k  pCnt  N )
) ,  1 ) ) )
172144, 156, 1713eqtr4d 2494 . . . . 5  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =/=  0 )  /\  k  e.  ( 1 ... ( abs `  N
) ) )  -> 
( ( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( ( A  x.  B
)  /L n ) ^ ( n 
pCnt  N ) ) ,  1 ) ) `  k )  =  ( ( ( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( A  /L n ) ^ ( n 
pCnt  N ) ) ,  1 ) ) `  k )  x.  (
( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( B  /L n ) ^
( n  pCnt  N
) ) ,  1 ) ) `  k
) ) )
17382, 84, 86, 91, 101, 108, 172seqcaopr 12126 . . . 4  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =/=  0 )  -> 
(  seq 1 (  x.  ,  ( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( ( A  x.  B
)  /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.  (  seq 1
(  x.  ,  ( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( B  /L n ) ^
( n  pCnt  N
) ) ,  1 ) ) ) `  ( abs `  N ) ) ) )
17480, 173oveq12d 6299 . . 3  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =/=  0 )  -> 
( if ( ( N  <  0  /\  ( A  x.  B
)  <  0 ) ,  -u 1 ,  1 )  x.  (  seq 1 (  x.  , 
( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( ( A  x.  B )  /L n ) ^
( n  pCnt  N
) ) ,  1 ) ) ) `  ( abs `  N ) ) )  =  ( ( if ( ( N  <  0  /\  A  <  0 ) ,  -u 1 ,  1 )  x.  if ( ( N  <  0  /\  B  <  0
) ,  -u 1 ,  1 ) )  x.  ( (  seq 1 (  x.  , 
( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( A  /L n ) ^
( n  pCnt  N
) ) ,  1 ) ) ) `  ( abs `  N ) )  x.  (  seq 1 (  x.  , 
( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( B  /L n ) ^
( n  pCnt  N
) ) ,  1 ) ) ) `  ( abs `  N ) ) ) ) )
17530adantr 465 . . . 4  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =/=  0 )  -> 
( A  x.  B
)  e.  ZZ )
176151lgsval4 23569 . . . 4  |-  ( ( ( A  x.  B
)  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  ->  (
( A  x.  B
)  /L N )  =  ( if ( ( N  <  0  /\  ( A  x.  B )  <  0 ) ,  -u
1 ,  1 )  x.  (  seq 1
(  x.  ,  ( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( ( A  x.  B )  /L n ) ^
( n  pCnt  N
) ) ,  1 ) ) ) `  ( abs `  N ) ) ) )
177175, 93, 94, 176syl3anc 1229 . . 3  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =/=  0 )  -> 
( ( A  x.  B )  /L
N )  =  ( if ( ( N  <  0  /\  ( A  x.  B )  <  0 ) ,  -u
1 ,  1 )  x.  (  seq 1
(  x.  ,  ( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( ( A  x.  B )  /L n ) ^
( n  pCnt  N
) ) ,  1 ) ) ) `  ( abs `  N ) ) ) )
17895lgsval4 23569 . . . . . 6  |-  ( ( 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 ) ) ) )
17992, 93, 94, 178syl3anc 1229 . . . . 5  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  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 ) ) ) )
180103lgsval4 23569 . . . . . 6  |-  ( ( B  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  ->  ( B  /L N )  =  ( if ( ( N  <  0  /\  B  <  0
) ,  -u 1 ,  1 )  x.  (  seq 1 (  x.  ,  ( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( B  /L
n ) ^ (
n  pCnt  N )
) ,  1 ) ) ) `  ( abs `  N ) ) ) )
181102, 93, 94, 180syl3anc 1229 . . . . 5  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =/=  0 )  -> 
( B  /L
N )  =  ( if ( ( N  <  0  /\  B  <  0 ) ,  -u
1 ,  1 )  x.  (  seq 1
(  x.  ,  ( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( B  /L n ) ^
( n  pCnt  N
) ) ,  1 ) ) ) `  ( abs `  N ) ) ) )
182179, 181oveq12d 6299 . . . 4  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =/=  0 )  -> 
( ( A  /L N )  x.  ( B  /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 ) ) )  x.  ( if ( ( N  <  0  /\  B  <  0 ) ,  -u
1 ,  1 )  x.  (  seq 1
(  x.  ,  ( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( B  /L n ) ^
( n  pCnt  N
) ) ,  1 ) ) ) `  ( abs `  N ) ) ) ) )
183 neg1cn 10646 . . . . . . 7  |-  -u 1  e.  CC
184183, 1keepel 3994 . . . . . 6  |-  if ( ( N  <  0  /\  A  <  0
) ,  -u 1 ,  1 )  e.  CC
185184a1i 11 . . . . 5  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =/=  0 )  ->  if ( ( N  <  0  /\  A  <  0 ) ,  -u
1 ,  1 )  e.  CC )
186 mulcl 9579 . . . . . . 7  |-  ( ( k  e.  CC  /\  x  e.  CC )  ->  ( k  x.  x
)  e.  CC )
187186adantl 466 . . . . . 6  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =/=  0 )  /\  ( k  e.  CC  /\  x  e.  CC ) )  ->  ( k  x.  x )  e.  CC )
18891, 101, 187seqcl 12109 . . . . 5  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =/=  0 )  -> 
(  seq 1 (  x.  ,  ( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( A  /L n ) ^ ( n 
pCnt  N ) ) ,  1 ) ) ) `
 ( abs `  N
) )  e.  CC )
189183, 1keepel 3994 . . . . . 6  |-  if ( ( N  <  0  /\  B  <  0
) ,  -u 1 ,  1 )  e.  CC
190189a1i 11 . . . . 5  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =/=  0 )  ->  if ( ( N  <  0  /\  B  <  0 ) ,  -u
1 ,  1 )  e.  CC )
19191, 108, 187seqcl 12109 . . . . 5  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =/=  0 )  -> 
(  seq 1 (  x.  ,  ( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( B  /L n ) ^ ( n 
pCnt  N ) ) ,  1 ) ) ) `
 ( abs `  N
) )  e.  CC )
192185, 188, 190, 191mul4d 9795 . . . 4  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =/=  0 )  -> 
( ( 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 ) ) )  x.  ( if ( ( N  <  0  /\  B  <  0 ) ,  -u
1 ,  1 )  x.  (  seq 1
(  x.  ,  ( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( B  /L n ) ^
( n  pCnt  N
) ) ,  1 ) ) ) `  ( abs `  N ) ) ) )  =  ( ( if ( ( N  <  0  /\  A  <  0
) ,  -u 1 ,  1 )  x.  if ( ( N  <  0  /\  B  <  0 ) ,  -u
1 ,  1 ) )  x.  ( (  seq 1 (  x.  ,  ( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( A  /L n ) ^ ( n 
pCnt  N ) ) ,  1 ) ) ) `
 ( abs `  N
) )  x.  (  seq 1 (  x.  , 
( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( B  /L n ) ^
( n  pCnt  N
) ) ,  1 ) ) ) `  ( abs `  N ) ) ) ) )
193182, 192eqtrd 2484 . . 3  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =/=  0 )  -> 
( ( A  /L N )  x.  ( B  /L
N ) )  =  ( ( if ( ( N  <  0  /\  A  <  0
) ,  -u 1 ,  1 )  x.  if ( ( N  <  0  /\  B  <  0 ) ,  -u
1 ,  1 ) )  x.  ( (  seq 1 (  x.  ,  ( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( A  /L n ) ^ ( n 
pCnt  N ) ) ,  1 ) ) ) `
 ( abs `  N
) )  x.  (  seq 1 (  x.  , 
( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( B  /L n ) ^
( n  pCnt  N
) ) ,  1 ) ) ) `  ( abs `  N ) ) ) ) )
194174, 177, 1933eqtr4d 2494 . 2  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =/=  0 )  -> 
( ( A  x.  B )  /L
N )  =  ( ( A  /L
N )  x.  ( B  /L N ) ) )
19578, 194pm2.61dane 2761 1  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0
) )  ->  (
( A  x.  B
)  /L N )  =  ( ( A  /L N )  x.  ( B  /L N ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    /\ w3a 974    = wceq 1383    e. wcel 1804    =/= wne 2638   ifcif 3926   class class class wbr 4437    |-> cmpt 4495   -->wf 5574   ` cfv 5578  (class class class)co 6281   CCcc 9493   RRcr 9494   0cc0 9495   1c1 9496    x. cmul 9500    < clt 9631    <_ cle 9632   -ucneg 9811   NNcn 10543   2c2 10592   NN0cn0 10802   ZZcz 10871   ZZ>=cuz 11092   ...cfz 11683    seqcseq 12089   ^cexp 12148   abscabs 13049    || cdvds 13968   Primecprime 14199    pCnt cpc 14342    /Lclgs 23547
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572  ax-pre-sup 9573
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-1o 7132  df-2o 7133  df-oadd 7136  df-er 7313  df-map 7424  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-sup 7903  df-card 8323  df-cda 8551  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-div 10214  df-nn 10544  df-2 10601  df-3 10602  df-4 10603  df-5 10604  df-6 10605  df-7 10606  df-8 10607  df-9 10608  df-n0 10803  df-z 10872  df-uz 11093  df-q 11194  df-rp 11232  df-fz 11684  df-fzo 11807  df-fl 11911  df-mod 11979  df-seq 12090  df-exp 12149  df-hash 12388  df-cj 12914  df-re 12915  df-im 12916  df-sqrt 13050  df-abs 13051  df-dvds 13969  df-gcd 14127  df-prm 14200  df-phi 14278  df-pc 14343  df-lgs 23548
This theorem is referenced by:  lgssq  23588  lgsdirnn0  23592  lgsquad2lem1  23611
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