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Theorem lgsdir 24337
Description: The Legendre symbol is completely multiplicative in its left argument. Together with lgsqr 24353 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 9615 . . . . . . 7  |-  1  e.  CC
2 0cn 9653 . . . . . . 7  |-  0  e.  CC
31, 2keepel 3939 . . . . . 6  |-  if ( ( B ^ 2 )  =  1 ,  1 ,  0 )  e.  CC
43mulid2i 9664 . . . . 5  |-  ( 1  x.  if ( ( B ^ 2 )  =  1 ,  1 ,  0 ) )  =  if ( ( B ^ 2 )  =  1 ,  1 ,  0 )
5 iftrue 3878 . . . . . . 7  |-  ( ( A ^ 2 )  =  1  ->  if ( ( A ^
2 )  =  1 ,  1 ,  0 )  =  1 )
65adantl 473 . . . . . 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 6323 . . . . 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 1033 . . . . . . . . . . 11  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0
) )  ->  A  e.  ZZ )
98zcnd 11064 . . . . . . . . . 10  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0
) )  ->  A  e.  CC )
109ad2antrr 740 . . . . . . . . 9  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =  0 )  /\  ( A ^
2 )  =  1 )  ->  A  e.  CC )
11 simpl2 1034 . . . . . . . . . . 11  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0
) )  ->  B  e.  ZZ )
1211zcnd 11064 . . . . . . . . . 10  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0
) )  ->  B  e.  CC )
1312ad2antrr 740 . . . . . . . . 9  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =  0 )  /\  ( A ^
2 )  =  1 )  ->  B  e.  CC )
1410, 13sqmuld 12466 . . . . . . . 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 468 . . . . . . . . 9  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =  0 )  /\  ( A ^
2 )  =  1 )  ->  ( A ^ 2 )  =  1 )
1615oveq1d 6323 . . . . . . . 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 12452 . . . . . . . . . 10  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0
) )  ->  ( B ^ 2 )  e.  CC )
1817ad2antrr 740 . . . . . . . . 9  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =  0 )  /\  ( A ^
2 )  =  1 )  ->  ( B ^ 2 )  e.  CC )
1918mulid2d 9679 . . . . . . . 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 2509 . . . . . . 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 2473 . . . . . 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 3894 . . . . 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 2531 . . . 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 9840 . . . . 5  |-  ( 0  x.  if ( ( B ^ 2 )  =  1 ,  1 ,  0 ) )  =  0
25 iffalse 3881 . . . . . . 7  |-  ( -.  ( A ^ 2 )  =  1  ->  if ( ( A ^
2 )  =  1 ,  1 ,  0 )  =  0 )
2625adantl 473 . . . . . 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 6323 . . . . 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 14401 . . . . . . . . . . . 12  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  A  ||  ( A  x.  B ) )
298, 11, 28syl2anc 673 . . . . . . . . . . 11  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0
) )  ->  A  ||  ( A  x.  B
) )
308, 11zmulcld 11069 . . . . . . . . . . . 12  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0
) )  ->  ( A  x.  B )  e.  ZZ )
31 dvdssq 14607 . . . . . . . . . . . 12  |-  ( ( A  e.  ZZ  /\  ( A  x.  B
)  e.  ZZ )  ->  ( A  ||  ( A  x.  B
)  <->  ( A ^
2 )  ||  (
( A  x.  B
) ^ 2 ) ) )
328, 30, 31syl2anc 673 . . . . . . . . . . 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 215 . . . . . . . . . 10  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0
) )  ->  ( A ^ 2 )  ||  ( ( A  x.  B ) ^ 2 ) )
3433adantr 472 . . . . . . . . 9  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =  0 )  ->  ( A ^
2 )  ||  (
( A  x.  B
) ^ 2 ) )
35 breq2 4399 . . . . . . . . 9  |-  ( ( ( A  x.  B
) ^ 2 )  =  1  ->  (
( A ^ 2 )  ||  ( ( A  x.  B ) ^ 2 )  <->  ( A ^ 2 )  ||  1 ) )
3634, 35syl5ibcom 228 . . . . . . . 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 772 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0
) )  ->  A  =/=  0 )
3837neneqd 2648 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0
) )  ->  -.  A  =  0 )
39 sqeq0 12377 . . . . . . . . . . . . . . . 16  |-  ( A  e.  CC  ->  (
( A ^ 2 )  =  0  <->  A  =  0 ) )
409, 39syl 17 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0
) )  ->  (
( A ^ 2 )  =  0  <->  A  =  0 ) )
4138, 40mtbird 308 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0
) )  ->  -.  ( A ^ 2 )  =  0 )
42 zsqcl2 12390 . . . . . . . . . . . . . . . . 17  |-  ( A  e.  ZZ  ->  ( A ^ 2 )  e. 
NN0 )
438, 42syl 17 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0
) )  ->  ( A ^ 2 )  e. 
NN0 )
44 elnn0 10895 . . . . . . . . . . . . . . . 16  |-  ( ( A ^ 2 )  e.  NN0  <->  ( ( A ^ 2 )  e.  NN  \/  ( A ^ 2 )  =  0 ) )
4543, 44sylib 201 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0
) )  ->  (
( A ^ 2 )  e.  NN  \/  ( A ^ 2 )  =  0 ) )
4645ord 384 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0
) )  ->  ( -.  ( A ^ 2 )  e.  NN  ->  ( A ^ 2 )  =  0 ) )
4741, 46mt3d 130 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0
) )  ->  ( A ^ 2 )  e.  NN )
4847adantr 472 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =  0 )  ->  ( A ^
2 )  e.  NN )
4948nnzd 11062 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =  0 )  ->  ( A ^
2 )  e.  ZZ )
50 1nn 10642 . . . . . . . . . . 11  |-  1  e.  NN
51 dvdsle 14427 . . . . . . . . . . 11  |-  ( ( ( A ^ 2 )  e.  ZZ  /\  1  e.  NN )  ->  ( ( A ^
2 )  ||  1  ->  ( A ^ 2 )  <_  1 ) )
5249, 50, 51sylancl 675 . . . . . . . . . 10  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =  0 )  ->  ( ( A ^ 2 )  ||  1  ->  ( A ^
2 )  <_  1
) )
5348nnge1d 10674 . . . . . . . . . 10  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =  0 )  ->  1  <_  ( A ^ 2 ) )
5452, 53jctird 553 . . . . . . . . 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 10646 . . . . . . . . . 10  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =  0 )  ->  ( A ^
2 )  e.  RR )
56 1re 9660 . . . . . . . . . 10  |-  1  e.  RR
57 letri3 9737 . . . . . . . . . 10  |-  ( ( ( A ^ 2 )  e.  RR  /\  1  e.  RR )  ->  ( ( A ^
2 )  =  1  <-> 
( ( A ^
2 )  <_  1  /\  1  <_  ( A ^ 2 ) ) ) )
5855, 56, 57sylancl 675 . . . . . . . . 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 242 . . . . . . . 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 448 . . . . . 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 3883 . . . . 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 2531 . . . 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 808 . . 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 6316 . . . . 5  |-  ( N  =  0  ->  ( A  /L N )  =  ( A  /L 0 ) )
66 lgs0 24316 . . . . . 6  |-  ( A  e.  ZZ  ->  ( A  /L 0 )  =  if ( ( A ^ 2 )  =  1 ,  1 ,  0 ) )
678, 66syl 17 . . . . 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 2527 . . . 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 6316 . . . . 5  |-  ( N  =  0  ->  ( B  /L N )  =  ( B  /L 0 ) )
70 lgs0 24316 . . . . . 6  |-  ( B  e.  ZZ  ->  ( B  /L 0 )  =  if ( ( B ^ 2 )  =  1 ,  1 ,  0 ) )
7111, 70syl 17 . . . . 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 2527 . . . 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 6326 . . 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 6316 . . . 4  |-  ( N  =  0  ->  (
( A  x.  B
)  /L N )  =  ( ( A  x.  B )  /L 0 ) )
75 lgs0 24316 . . . . 5  |-  ( ( A  x.  B )  e.  ZZ  ->  (
( A  x.  B
)  /L 0 )  =  if ( ( ( A  x.  B ) ^ 2 )  =  1 ,  1 ,  0 ) )
7630, 75syl 17 . . . 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 2527 . . 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 2516 . 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 24329 . . . . 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 472 . . . 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 9641 . . . . . 6  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  x.  y
)  e.  CC )
8281adantl 473 . . . . 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 9643 . . . . . 6  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  x.  y
)  =  ( y  x.  x ) )
8483adantl 473 . . . . 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 9645 . . . . . 6  |-  ( ( x  e.  CC  /\  y  e.  CC  /\  z  e.  CC )  ->  (
( x  x.  y
)  x.  z )  =  ( x  x.  ( y  x.  z
) ) )
8685adantl 473 . . . . 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 1035 . . . . . . 7  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0
) )  ->  N  e.  ZZ )
88 nnabscl 13465 . . . . . . 7  |-  ( ( N  e.  ZZ  /\  N  =/=  0 )  -> 
( abs `  N
)  e.  NN )
8987, 88sylan 479 . . . . . 6  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =/=  0 )  -> 
( abs `  N
)  e.  NN )
90 nnuz 11218 . . . . . 6  |-  NN  =  ( ZZ>= `  1 )
9189, 90syl6eleq 2559 . . . . 5  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =/=  0 )  -> 
( abs `  N
)  e.  ( ZZ>= ` 
1 ) )
92 simpll1 1069 . . . . . . . 8  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =/=  0 )  ->  A  e.  ZZ )
93 simpll3 1071 . . . . . . . 8  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =/=  0 )  ->  N  e.  ZZ )
94 simpr 468 . . . . . . . 8  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =/=  0 )  ->  N  =/=  0 )
95 eqid 2471 . . . . . . . . 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 24324 . . . . . . . 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 1292 . . . . . . 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 11854 . . . . . . 7  |-  ( k  e.  ( 1 ... ( abs `  N
) )  ->  k  e.  NN )
99 ffvelrn 6035 . . . . . . 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 485 . . . . . 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 11064 . . . . 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 1070 . . . . . . . 8  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =/=  0 )  ->  B  e.  ZZ )
103 eqid 2471 . . . . . . . . 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 24324 . . . . . . . 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 1292 . . . . . . 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 6035 . . . . . . 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 485 . . . . . 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 11064 . . . . 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 472 . . . . . . . . . . . 12  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =/=  0 )  /\  k  e.  Prime )  ->  A  e.  ZZ )
110102adantr 472 . . . . . . . . . . . 12  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =/=  0 )  /\  k  e.  Prime )  ->  B  e.  ZZ )
111 simpr 468 . . . . . . . . . . . 12  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =/=  0 )  /\  k  e.  Prime )  -> 
k  e.  Prime )
112 lgsdirprm 24336 . . . . . . . . . . . 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 1292 . . . . . . . . . . 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 6323 . . . . . . . . . 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 14705 . . . . . . . . . . . . 13  |-  ( k  e.  Prime  ->  k  e.  ZZ )
116 lgscl 24317 . . . . . . . . . . . . 13  |-  ( ( A  e.  ZZ  /\  k  e.  ZZ )  ->  ( A  /L
k )  e.  ZZ )
11792, 115, 116syl2an 485 . . . . . . . . . . . 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 11064 . . . . . . . . . . 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 24317 . . . . . . . . . . . . 13  |-  ( ( B  e.  ZZ  /\  k  e.  ZZ )  ->  ( B  /L
k )  e.  ZZ )
120102, 115, 119syl2an 485 . . . . . . . . . . . 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 11064 . . . . . . . . . . 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 472 . . . . . . . . . . . 12  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =/=  0 )  /\  k  e.  Prime )  ->  N  e.  ZZ )
12394adantr 472 . . . . . . . . . . . 12  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =/=  0 )  /\  k  e.  Prime )  ->  N  =/=  0 )
124 pczcl 14877 . . . . . . . . . . . 12  |-  ( ( k  e.  Prime  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( k  pCnt  N
)  e.  NN0 )
125111, 122, 123, 124syl12anc 1290 . . . . . . . . . . 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 12469 . . . . . . . . . 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 2505 . . . . . . . . 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 3878 . . . . . . . . . 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 473 . . . . . . . . 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 3878 . . . . . . . . . . 11  |-  ( k  e.  Prime  ->  if ( k  e.  Prime ,  ( ( A  /L
k ) ^ (
k  pCnt  N )
) ,  1 )  =  ( ( A  /L k ) ^ ( k  pCnt  N ) ) )
131 iftrue 3878 . . . . . . . . . . 11  |-  ( k  e.  Prime  ->  if ( k  e.  Prime ,  ( ( B  /L
k ) ^ (
k  pCnt  N )
) ,  1 )  =  ( ( B  /L k ) ^ ( k  pCnt  N ) ) )
132130, 131oveq12d 6326 . . . . . . . . . 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 473 . . . . . . . . 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 2515 . . . . . . . 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 10780 . . . . . . . . . . 11  |-  ( 1  x.  1 )  =  1
136135eqcomi 2480 . . . . . . . . . 10  |-  1  =  ( 1  x.  1 )
137 iffalse 3881 . . . . . . . . . 10  |-  ( -.  k  e.  Prime  ->  if ( k  e.  Prime ,  ( ( ( A  x.  B )  /L k ) ^
( k  pCnt  N
) ) ,  1 )  =  1 )
138 iffalse 3881 . . . . . . . . . . 11  |-  ( -.  k  e.  Prime  ->  if ( k  e.  Prime ,  ( ( A  /L k ) ^
( k  pCnt  N
) ) ,  1 )  =  1 )
139 iffalse 3881 . . . . . . . . . . 11  |-  ( -.  k  e.  Prime  ->  if ( k  e.  Prime ,  ( ( B  /L k ) ^
( k  pCnt  N
) ) ,  1 )  =  1 )
140138, 139oveq12d 6326 . . . . . . . . . 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 2531 . . . . . . . . 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 473 . . . . . . . 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 808 . . . . . . 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 472 . . . . . 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 473 . . . . . . 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 2537 . . . . . . . . 9  |-  ( n  =  k  ->  (
n  e.  Prime  <->  k  e.  Prime ) )
147 oveq2 6316 . . . . . . . . . 10  |-  ( n  =  k  ->  (
( A  x.  B
)  /L n )  =  ( ( A  x.  B )  /L k ) )
148 oveq1 6315 . . . . . . . . . 10  |-  ( n  =  k  ->  (
n  pCnt  N )  =  ( k  pCnt  N ) )
149147, 148oveq12d 6326 . . . . . . . . 9  |-  ( n  =  k  ->  (
( ( A  x.  B )  /L
n ) ^ (
n  pCnt  N )
)  =  ( ( ( A  x.  B
)  /L k ) ^ ( k 
pCnt  N ) ) )
150146, 149ifbieq1d 3895 . . . . . . . 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 2471 . . . . . . . 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 6336 . . . . . . . . 9  |-  ( ( ( A  x.  B
)  /L k ) ^ ( k 
pCnt  N ) )  e. 
_V
153 1ex 9656 . . . . . . . . 9  |-  1  e.  _V
154152, 153ifex 3940 . . . . . . . 8  |-  if ( k  e.  Prime ,  ( ( ( A  x.  B )  /L
k ) ^ (
k  pCnt  N )
) ,  1 )  e.  _V
155150, 151, 154fvmpt 5963 . . . . . . 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 17 . . . . . 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 6316 . . . . . . . . . . 11  |-  ( n  =  k  ->  ( A  /L n )  =  ( A  /L k ) )
158157, 148oveq12d 6326 . . . . . . . . . 10  |-  ( n  =  k  ->  (
( A  /L
n ) ^ (
n  pCnt  N )
)  =  ( ( A  /L k ) ^ ( k 
pCnt  N ) ) )
159146, 158ifbieq1d 3895 . . . . . . . . 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 6336 . . . . . . . . . 10  |-  ( ( A  /L k ) ^ ( k 
pCnt  N ) )  e. 
_V
161160, 153ifex 3940 . . . . . . . . 9  |-  if ( k  e.  Prime ,  ( ( A  /L
k ) ^ (
k  pCnt  N )
) ,  1 )  e.  _V
162159, 95, 161fvmpt 5963 . . . . . . . 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 17 . . . . . . 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 6316 . . . . . . . . . . 11  |-  ( n  =  k  ->  ( B  /L n )  =  ( B  /L k ) )
165164, 148oveq12d 6326 . . . . . . . . . 10  |-  ( n  =  k  ->  (
( B  /L
n ) ^ (
n  pCnt  N )
)  =  ( ( B  /L k ) ^ ( k 
pCnt  N ) ) )
166146, 165ifbieq1d 3895 . . . . . . . . 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 6336 . . . . . . . . . 10  |-  ( ( B  /L k ) ^ ( k 
pCnt  N ) )  e. 
_V
168167, 153ifex 3940 . . . . . . . . 9  |-  if ( k  e.  Prime ,  ( ( B  /L
k ) ^ (
k  pCnt  N )
) ,  1 )  e.  _V
169166, 103, 168fvmpt 5963 . . . . . . . 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 17 . . . . . . 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 6326 . . . . . 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 2515 . . . . 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 12288 . . . 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 6326 . . 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 472 . . . 4  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =/=  0 )  -> 
( A  x.  B
)  e.  ZZ )
176151lgsval4 24323 . . . 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 1292 . . 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 24323 . . . . . 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 1292 . . . . 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 24323 . . . . . 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 1292 . . . . 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 6326 . . . 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 10735 . . . . . . 7  |-  -u 1  e.  CC
184183, 1keepel 3939 . . . . . 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 9641 . . . . . . 7  |-  ( ( k  e.  CC  /\  x  e.  CC )  ->  ( k  x.  x
)  e.  CC )
187186adantl 473 . . . . . 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 12271 . . . . 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 3939 . . . . . 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 12271 . . . . 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 9863 . . . 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 2505 . . 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 2515 . 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 2730 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 189    \/ wo 375    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904    =/= wne 2641   ifcif 3872   class class class wbr 4395    |-> cmpt 4454   -->wf 5585   ` cfv 5589  (class class class)co 6308   CCcc 9555   RRcr 9556   0cc0 9557   1c1 9558    x. cmul 9562    < clt 9693    <_ cle 9694   -ucneg 9881   NNcn 10631   2c2 10681   NN0cn0 10893   ZZcz 10961   ZZ>=cuz 11182   ...cfz 11810    seqcseq 12251   ^cexp 12310   abscabs 13374    || cdvds 14382   Primecprime 14701    pCnt cpc 14865    /Lclgs 24301
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-2o 7201  df-oadd 7204  df-er 7381  df-map 7492  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-sup 7974  df-inf 7975  df-card 8391  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-4 10692  df-5 10693  df-6 10694  df-7 10695  df-8 10696  df-9 10697  df-n0 10894  df-z 10962  df-uz 11183  df-q 11288  df-rp 11326  df-fz 11811  df-fzo 11943  df-fl 12061  df-mod 12130  df-seq 12252  df-exp 12311  df-hash 12554  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-dvds 14383  df-gcd 14548  df-prm 14702  df-phi 14793  df-pc 14866  df-lgs 24302
This theorem is referenced by:  lgssq  24342  lgsdirnn0  24346  lgsquad2lem1  24365
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