MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  lgsdir Structured version   Unicode version

Theorem lgsdir 24200
Description: The Legendre symbol is completely multiplicative in its left argument. Together with lgsqr 24216 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 9548 . . . . . . 7  |-  1  e.  CC
2 0cn 9586 . . . . . . 7  |-  0  e.  CC
31, 2keepel 3921 . . . . . 6  |-  if ( ( B ^ 2 )  =  1 ,  1 ,  0 )  e.  CC
43mulid2i 9597 . . . . 5  |-  ( 1  x.  if ( ( B ^ 2 )  =  1 ,  1 ,  0 ) )  =  if ( ( B ^ 2 )  =  1 ,  1 ,  0 )
5 iftrue 3860 . . . . . . 7  |-  ( ( A ^ 2 )  =  1  ->  if ( ( A ^
2 )  =  1 ,  1 ,  0 )  =  1 )
65adantl 467 . . . . . 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 6264 . . . . 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 1008 . . . . . . . . . . 11  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0
) )  ->  A  e.  ZZ )
98zcnd 10992 . . . . . . . . . 10  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0
) )  ->  A  e.  CC )
109ad2antrr 730 . . . . . . . . 9  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =  0 )  /\  ( A ^
2 )  =  1 )  ->  A  e.  CC )
11 simpl2 1009 . . . . . . . . . . 11  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0
) )  ->  B  e.  ZZ )
1211zcnd 10992 . . . . . . . . . 10  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0
) )  ->  B  e.  CC )
1312ad2antrr 730 . . . . . . . . 9  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =  0 )  /\  ( A ^
2 )  =  1 )  ->  B  e.  CC )
1410, 13sqmuld 12378 . . . . . . . 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 462 . . . . . . . . 9  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =  0 )  /\  ( A ^
2 )  =  1 )  ->  ( A ^ 2 )  =  1 )
1615oveq1d 6264 . . . . . . . 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 12364 . . . . . . . . . 10  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0
) )  ->  ( B ^ 2 )  e.  CC )
1817ad2antrr 730 . . . . . . . . 9  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =  0 )  /\  ( A ^
2 )  =  1 )  ->  ( B ^ 2 )  e.  CC )
1918mulid2d 9612 . . . . . . . 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 2466 . . . . . . 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 2430 . . . . . 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 3876 . . . . 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 2488 . . . 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 9773 . . . . 5  |-  ( 0  x.  if ( ( B ^ 2 )  =  1 ,  1 ,  0 ) )  =  0
25 iffalse 3863 . . . . . . 7  |-  ( -.  ( A ^ 2 )  =  1  ->  if ( ( A ^
2 )  =  1 ,  1 ,  0 )  =  0 )
2625adantl 467 . . . . . 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 6264 . . . . 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 14267 . . . . . . . . . . . 12  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  A  ||  ( A  x.  B ) )
298, 11, 28syl2anc 665 . . . . . . . . . . 11  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0
) )  ->  A  ||  ( A  x.  B
) )
308, 11zmulcld 10997 . . . . . . . . . . . 12  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0
) )  ->  ( A  x.  B )  e.  ZZ )
31 dvdssq 14471 . . . . . . . . . . . 12  |-  ( ( A  e.  ZZ  /\  ( A  x.  B
)  e.  ZZ )  ->  ( A  ||  ( A  x.  B
)  <->  ( A ^
2 )  ||  (
( A  x.  B
) ^ 2 ) ) )
328, 30, 31syl2anc 665 . . . . . . . . . . 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 213 . . . . . . . . . 10  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0
) )  ->  ( A ^ 2 )  ||  ( ( A  x.  B ) ^ 2 ) )
3433adantr 466 . . . . . . . . 9  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =  0 )  ->  ( A ^
2 )  ||  (
( A  x.  B
) ^ 2 ) )
35 breq2 4370 . . . . . . . . 9  |-  ( ( ( A  x.  B
) ^ 2 )  =  1  ->  (
( A ^ 2 )  ||  ( ( A  x.  B ) ^ 2 )  <->  ( A ^ 2 )  ||  1 ) )
3634, 35syl5ibcom 223 . . . . . . . 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 762 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0
) )  ->  A  =/=  0 )
3837neneqd 2606 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0
) )  ->  -.  A  =  0 )
39 sqeq0 12289 . . . . . . . . . . . . . . . 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 302 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0
) )  ->  -.  ( A ^ 2 )  =  0 )
42 zsqcl2 12302 . . . . . . . . . . . . . . . . 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 10822 . . . . . . . . . . . . . . . 16  |-  ( ( A ^ 2 )  e.  NN0  <->  ( ( A ^ 2 )  e.  NN  \/  ( A ^ 2 )  =  0 ) )
4543, 44sylib 199 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0
) )  ->  (
( A ^ 2 )  e.  NN  \/  ( A ^ 2 )  =  0 ) )
4645ord 378 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0
) )  ->  ( -.  ( A ^ 2 )  e.  NN  ->  ( A ^ 2 )  =  0 ) )
4741, 46mt3d 128 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0
) )  ->  ( A ^ 2 )  e.  NN )
4847adantr 466 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =  0 )  ->  ( A ^
2 )  e.  NN )
4948nnzd 10990 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =  0 )  ->  ( A ^
2 )  e.  ZZ )
50 1nn 10571 . . . . . . . . . . 11  |-  1  e.  NN
51 dvdsle 14293 . . . . . . . . . . 11  |-  ( ( ( A ^ 2 )  e.  ZZ  /\  1  e.  NN )  ->  ( ( A ^
2 )  ||  1  ->  ( A ^ 2 )  <_  1 ) )
5249, 50, 51sylancl 666 . . . . . . . . . 10  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =  0 )  ->  ( ( A ^ 2 )  ||  1  ->  ( A ^
2 )  <_  1
) )
5348nnge1d 10603 . . . . . . . . . 10  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =  0 )  ->  1  <_  ( A ^ 2 ) )
5452, 53jctird 546 . . . . . . . . 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 10575 . . . . . . . . . 10  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =  0 )  ->  ( A ^
2 )  e.  RR )
56 1re 9593 . . . . . . . . . 10  |-  1  e.  RR
57 letri3 9670 . . . . . . . . . 10  |-  ( ( ( A ^ 2 )  e.  RR  /\  1  e.  RR )  ->  ( ( A ^
2 )  =  1  <-> 
( ( A ^
2 )  <_  1  /\  1  <_  ( A ^ 2 ) ) ) )
5855, 56, 57sylancl 666 . . . . . . . . 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 237 . . . . . . . 8  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =  0 )  ->  ( ( A ^ 2 )  ||  1  ->  ( A ^
2 )  =  1 ) )
6036, 59syld 45 . . . . . . 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 442 . . . . . 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 3865 . . . . 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 2488 . . . 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 798 . . 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 6257 . . . . 5  |-  ( N  =  0  ->  ( A  /L N )  =  ( A  /L 0 ) )
66 lgs0 24179 . . . . . 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 2484 . . . 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 6257 . . . . 5  |-  ( N  =  0  ->  ( B  /L N )  =  ( B  /L 0 ) )
70 lgs0 24179 . . . . . 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 2484 . . . 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 6267 . . 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 6257 . . . 4  |-  ( N  =  0  ->  (
( A  x.  B
)  /L N )  =  ( ( A  x.  B )  /L 0 ) )
75 lgs0 24179 . . . . 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 2484 . . 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 2473 . 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 24192 . . . . 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 466 . . . 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 9574 . . . . . 6  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  x.  y
)  e.  CC )
8281adantl 467 . . . . 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 9576 . . . . . 6  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  x.  y
)  =  ( y  x.  x ) )
8483adantl 467 . . . . 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 9578 . . . . . 6  |-  ( ( x  e.  CC  /\  y  e.  CC  /\  z  e.  CC )  ->  (
( x  x.  y
)  x.  z )  =  ( x  x.  ( y  x.  z
) ) )
8685adantl 467 . . . . 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 1010 . . . . . . 7  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0
) )  ->  N  e.  ZZ )
88 nnabscl 13332 . . . . . . 7  |-  ( ( N  e.  ZZ  /\  N  =/=  0 )  -> 
( abs `  N
)  e.  NN )
8987, 88sylan 473 . . . . . 6  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =/=  0 )  -> 
( abs `  N
)  e.  NN )
90 nnuz 11145 . . . . . 6  |-  NN  =  ( ZZ>= `  1 )
9189, 90syl6eleq 2516 . . . . 5  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =/=  0 )  -> 
( abs `  N
)  e.  ( ZZ>= ` 
1 ) )
92 simpll1 1044 . . . . . . . 8  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =/=  0 )  ->  A  e.  ZZ )
93 simpll3 1046 . . . . . . . 8  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =/=  0 )  ->  N  e.  ZZ )
94 simpr 462 . . . . . . . 8  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =/=  0 )  ->  N  =/=  0 )
95 eqid 2428 . . . . . . . . 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 24187 . . . . . . . 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 1264 . . . . . . 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 11779 . . . . . . 7  |-  ( k  e.  ( 1 ... ( abs `  N
) )  ->  k  e.  NN )
99 ffvelrn 5979 . . . . . . 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 479 . . . . . 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 10992 . . . . 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 1045 . . . . . . . 8  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =/=  0 )  ->  B  e.  ZZ )
103 eqid 2428 . . . . . . . . 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 24187 . . . . . . . 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 1264 . . . . . . 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 5979 . . . . . . 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 479 . . . . . 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 10992 . . . . 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 466 . . . . . . . . . . . 12  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =/=  0 )  /\  k  e.  Prime )  ->  A  e.  ZZ )
110102adantr 466 . . . . . . . . . . . 12  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =/=  0 )  /\  k  e.  Prime )  ->  B  e.  ZZ )
111 simpr 462 . . . . . . . . . . . 12  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =/=  0 )  /\  k  e.  Prime )  -> 
k  e.  Prime )
112 lgsdirprm 24199 . . . . . . . . . . . 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 1264 . . . . . . . . . . 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 6264 . . . . . . . . . 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 14569 . . . . . . . . . . . . 13  |-  ( k  e.  Prime  ->  k  e.  ZZ )
116 lgscl 24180 . . . . . . . . . . . . 13  |-  ( ( A  e.  ZZ  /\  k  e.  ZZ )  ->  ( A  /L
k )  e.  ZZ )
11792, 115, 116syl2an 479 . . . . . . . . . . . 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 10992 . . . . . . . . . . 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 24180 . . . . . . . . . . . . 13  |-  ( ( B  e.  ZZ  /\  k  e.  ZZ )  ->  ( B  /L
k )  e.  ZZ )
120102, 115, 119syl2an 479 . . . . . . . . . . . 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 10992 . . . . . . . . . . 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 466 . . . . . . . . . . . 12  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =/=  0 )  /\  k  e.  Prime )  ->  N  e.  ZZ )
12394adantr 466 . . . . . . . . . . . 12  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =/=  0 )  /\  k  e.  Prime )  ->  N  =/=  0 )
124 pczcl 14741 . . . . . . . . . . . 12  |-  ( ( k  e.  Prime  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( k  pCnt  N
)  e.  NN0 )
125111, 122, 123, 124syl12anc 1262 . . . . . . . . . . 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 12381 . . . . . . . . . 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 2462 . . . . . . . . 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 3860 . . . . . . . . . 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 467 . . . . . . . . 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 3860 . . . . . . . . . . 11  |-  ( k  e.  Prime  ->  if ( k  e.  Prime ,  ( ( A  /L
k ) ^ (
k  pCnt  N )
) ,  1 )  =  ( ( A  /L k ) ^ ( k  pCnt  N ) ) )
131 iftrue 3860 . . . . . . . . . . 11  |-  ( k  e.  Prime  ->  if ( k  e.  Prime ,  ( ( B  /L
k ) ^ (
k  pCnt  N )
) ,  1 )  =  ( ( B  /L k ) ^ ( k  pCnt  N ) ) )
132130, 131oveq12d 6267 . . . . . . . . . 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 467 . . . . . . . . 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 2472 . . . . . . . 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 10708 . . . . . . . . . . 11  |-  ( 1  x.  1 )  =  1
136135eqcomi 2437 . . . . . . . . . 10  |-  1  =  ( 1  x.  1 )
137 iffalse 3863 . . . . . . . . . 10  |-  ( -.  k  e.  Prime  ->  if ( k  e.  Prime ,  ( ( ( A  x.  B )  /L k ) ^
( k  pCnt  N
) ) ,  1 )  =  1 )
138 iffalse 3863 . . . . . . . . . . 11  |-  ( -.  k  e.  Prime  ->  if ( k  e.  Prime ,  ( ( A  /L k ) ^
( k  pCnt  N
) ) ,  1 )  =  1 )
139 iffalse 3863 . . . . . . . . . . 11  |-  ( -.  k  e.  Prime  ->  if ( k  e.  Prime ,  ( ( B  /L k ) ^
( k  pCnt  N
) ) ,  1 )  =  1 )
140138, 139oveq12d 6267 . . . . . . . . . 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 2488 . . . . . . . . 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 467 . . . . . . . 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 798 . . . . . . 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 466 . . . . . 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 467 . . . . . . 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 2494 . . . . . . . . 9  |-  ( n  =  k  ->  (
n  e.  Prime  <->  k  e.  Prime ) )
147 oveq2 6257 . . . . . . . . . 10  |-  ( n  =  k  ->  (
( A  x.  B
)  /L n )  =  ( ( A  x.  B )  /L k ) )
148 oveq1 6256 . . . . . . . . . 10  |-  ( n  =  k  ->  (
n  pCnt  N )  =  ( k  pCnt  N ) )
149147, 148oveq12d 6267 . . . . . . . . 9  |-  ( n  =  k  ->  (
( ( A  x.  B )  /L
n ) ^ (
n  pCnt  N )
)  =  ( ( ( A  x.  B
)  /L k ) ^ ( k 
pCnt  N ) ) )
150146, 149ifbieq1d 3877 . . . . . . . 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 2428 . . . . . . . 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 6277 . . . . . . . . 9  |-  ( ( ( A  x.  B
)  /L k ) ^ ( k 
pCnt  N ) )  e. 
_V
153 1ex 9589 . . . . . . . . 9  |-  1  e.  _V
154152, 153ifex 3922 . . . . . . . 8  |-  if ( k  e.  Prime ,  ( ( ( A  x.  B )  /L
k ) ^ (
k  pCnt  N )
) ,  1 )  e.  _V
155150, 151, 154fvmpt 5908 . . . . . . 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 6257 . . . . . . . . . . 11  |-  ( n  =  k  ->  ( A  /L n )  =  ( A  /L k ) )
158157, 148oveq12d 6267 . . . . . . . . . 10  |-  ( n  =  k  ->  (
( A  /L
n ) ^ (
n  pCnt  N )
)  =  ( ( A  /L k ) ^ ( k 
pCnt  N ) ) )
159146, 158ifbieq1d 3877 . . . . . . . . 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 6277 . . . . . . . . . 10  |-  ( ( A  /L k ) ^ ( k 
pCnt  N ) )  e. 
_V
161160, 153ifex 3922 . . . . . . . . 9  |-  if ( k  e.  Prime ,  ( ( A  /L
k ) ^ (
k  pCnt  N )
) ,  1 )  e.  _V
162159, 95, 161fvmpt 5908 . . . . . . . 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 6257 . . . . . . . . . . 11  |-  ( n  =  k  ->  ( B  /L n )  =  ( B  /L k ) )
165164, 148oveq12d 6267 . . . . . . . . . 10  |-  ( n  =  k  ->  (
( B  /L
n ) ^ (
n  pCnt  N )
)  =  ( ( B  /L k ) ^ ( k 
pCnt  N ) ) )
166146, 165ifbieq1d 3877 . . . . . . . . 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 6277 . . . . . . . . . 10  |-  ( ( B  /L k ) ^ ( k 
pCnt  N ) )  e. 
_V
168167, 153ifex 3922 . . . . . . . . 9  |-  if ( k  e.  Prime ,  ( ( B  /L
k ) ^ (
k  pCnt  N )
) ,  1 )  e.  _V
169166, 103, 168fvmpt 5908 . . . . . . . 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 6267 . . . . . 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 2472 . . . . 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 12200 . . . 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 6267 . . 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 466 . . . 4  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =/=  0 )  -> 
( A  x.  B
)  e.  ZZ )
176151lgsval4 24186 . . . 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 1264 . . 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 24186 . . . . . 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 1264 . . . . 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 24186 . . . . . 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 1264 . . . . 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 6267 . . . 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 10664 . . . . . . 7  |-  -u 1  e.  CC
184183, 1keepel 3921 . . . . . 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 9574 . . . . . . 7  |-  ( ( k  e.  CC  /\  x  e.  CC )  ->  ( k  x.  x
)  e.  CC )
187186adantl 467 . . . . . 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 12183 . . . . 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 3921 . . . . . 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 12183 . . . . 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 9796 . . . 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 2462 . . 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 2472 . 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 2688 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 187    \/ wo 369    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1872    =/= wne 2599   ifcif 3854   class class class wbr 4366    |-> cmpt 4425   -->wf 5540   ` cfv 5544  (class class class)co 6249   CCcc 9488   RRcr 9489   0cc0 9490   1c1 9491    x. cmul 9495    < clt 9626    <_ cle 9627   -ucneg 9812   NNcn 10560   2c2 10610   NN0cn0 10820   ZZcz 10888   ZZ>=cuz 11110   ...cfz 11735    seqcseq 12163   ^cexp 12222   abscabs 13241    || cdvds 14248   Primecprime 14565    pCnt cpc 14729    /Lclgs 24164
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-rep 4479  ax-sep 4489  ax-nul 4498  ax-pow 4545  ax-pr 4603  ax-un 6541  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567  ax-pre-sup 9568
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 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-nel 2602  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 3024  df-sbc 3243  df-csb 3339  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-pss 3395  df-nul 3705  df-if 3855  df-pw 3926  df-sn 3942  df-pr 3944  df-tp 3946  df-op 3948  df-uni 4163  df-int 4199  df-iun 4244  df-br 4367  df-opab 4426  df-mpt 4427  df-tr 4462  df-eprel 4707  df-id 4711  df-po 4717  df-so 4718  df-fr 4755  df-we 4757  df-xp 4802  df-rel 4803  df-cnv 4804  df-co 4805  df-dm 4806  df-rn 4807  df-res 4808  df-ima 4809  df-pred 5342  df-ord 5388  df-on 5389  df-lim 5390  df-suc 5391  df-iota 5508  df-fun 5546  df-fn 5547  df-f 5548  df-f1 5549  df-fo 5550  df-f1o 5551  df-fv 5552  df-riota 6211  df-ov 6252  df-oprab 6253  df-mpt2 6254  df-om 6651  df-1st 6751  df-2nd 6752  df-wrecs 6983  df-recs 7045  df-rdg 7083  df-1o 7137  df-2o 7138  df-oadd 7141  df-er 7318  df-map 7429  df-en 7525  df-dom 7526  df-sdom 7527  df-fin 7528  df-sup 7909  df-inf 7910  df-card 8325  df-cda 8549  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9813  df-neg 9814  df-div 10221  df-nn 10561  df-2 10619  df-3 10620  df-4 10621  df-5 10622  df-6 10623  df-7 10624  df-8 10625  df-9 10626  df-n0 10821  df-z 10889  df-uz 11111  df-q 11216  df-rp 11254  df-fz 11736  df-fzo 11867  df-fl 11978  df-mod 12047  df-seq 12164  df-exp 12223  df-hash 12466  df-cj 13106  df-re 13107  df-im 13108  df-sqrt 13242  df-abs 13243  df-dvds 14249  df-gcd 14412  df-prm 14566  df-phi 14657  df-pc 14730  df-lgs 24165
This theorem is referenced by:  lgssq  24205  lgsdirnn0  24209  lgsquad2lem1  24228
  Copyright terms: Public domain W3C validator