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Theorem lgsdir 22668
Description: The Legendre symbol is completely multiplicative in its left argument. Together with lgsqr 22684 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 9339 . . . . . . 7  |-  1  e.  CC
2 0cn 9377 . . . . . . 7  |-  0  e.  CC
31, 2keepel 3856 . . . . . 6  |-  if ( ( B ^ 2 )  =  1 ,  1 ,  0 )  e.  CC
43mulid2i 9388 . . . . 5  |-  ( 1  x.  if ( ( B ^ 2 )  =  1 ,  1 ,  0 ) )  =  if ( ( B ^ 2 )  =  1 ,  1 ,  0 )
5 iftrue 3796 . . . . . . 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 6105 . . . . 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 991 . . . . . . . . . . 11  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0
) )  ->  A  e.  ZZ )
98zcnd 10747 . . . . . . . . . 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 992 . . . . . . . . . . 11  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0
) )  ->  B  e.  ZZ )
1211zcnd 10747 . . . . . . . . . 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 12019 . . . . . . . 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 6105 . . . . . . . 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 12005 . . . . . . . . . 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 9403 . . . . . . . 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 2478 . . . . . . 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 2450 . . . . . 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 3810 . . . . 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 2500 . . . 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 9557 . . . . 5  |-  ( 0  x.  if ( ( B ^ 2 )  =  1 ,  1 ,  0 ) )  =  0
25 iffalse 3798 . . . . . . 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 6105 . . . . 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 13553 . . . . . . . . . . . 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 10752 . . . . . . . . . . . 12  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0
) )  ->  ( A  x.  B )  e.  ZZ )
31 dvdssq 13743 . . . . . . . . . . . 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 4295 . . . . . . . . 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 755 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0
) )  ->  A  =/=  0 )
3837neneqd 2623 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0
) )  ->  -.  A  =  0 )
39 sqeq0 11929 . . . . . . . . . . . . . . . 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 11942 . . . . . . . . . . . . . . . . 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 10580 . . . . . . . . . . . . . . . 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 10745 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =  0 )  ->  ( A ^
2 )  e.  ZZ )
50 1nn 10332 . . . . . . . . . . 11  |-  1  e.  NN
51 dvdsle 13577 . . . . . . . . . . 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 10363 . . . . . . . . . 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 10336 . . . . . . . . . 10  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =  0 )  ->  ( A ^
2 )  e.  RR )
56 1re 9384 . . . . . . . . . 10  |-  1  e.  RR
57 letri3 9459 . . . . . . . . . 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 )
62 iffalse 3798 . . . . . 6  |-  ( -.  ( ( A  x.  B ) ^ 2 )  =  1  ->  if ( ( ( A  x.  B ) ^
2 )  =  1 ,  1 ,  0 )  =  0 )
6361, 62syl 16 . . . . 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 )
6424, 27, 633eqtr4a 2500 . . . 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 ) )
6523, 64pm2.61dan 789 . . 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 ) )
66 oveq2 6098 . . . . 5  |-  ( N  =  0  ->  ( A  /L N )  =  ( A  /L 0 ) )
67 lgs0 22647 . . . . . 6  |-  ( A  e.  ZZ  ->  ( A  /L 0 )  =  if ( ( A ^ 2 )  =  1 ,  1 ,  0 ) )
688, 67syl 16 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0
) )  ->  ( A  /L 0 )  =  if ( ( A ^ 2 )  =  1 ,  1 ,  0 ) )
6966, 68sylan9eqr 2496 . . . 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 ) )
70 oveq2 6098 . . . . 5  |-  ( N  =  0  ->  ( B  /L N )  =  ( B  /L 0 ) )
71 lgs0 22647 . . . . . 6  |-  ( B  e.  ZZ  ->  ( B  /L 0 )  =  if ( ( B ^ 2 )  =  1 ,  1 ,  0 ) )
7211, 71syl 16 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0
) )  ->  ( B  /L 0 )  =  if ( ( B ^ 2 )  =  1 ,  1 ,  0 ) )
7370, 72sylan9eqr 2496 . . . 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 ) )
7469, 73oveq12d 6108 . . 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 ) ) )
75 oveq2 6098 . . . 4  |-  ( N  =  0  ->  (
( A  x.  B
)  /L N )  =  ( ( A  x.  B )  /L 0 ) )
76 lgs0 22647 . . . . 5  |-  ( ( A  x.  B )  e.  ZZ  ->  (
( A  x.  B
)  /L 0 )  =  if ( ( ( A  x.  B ) ^ 2 )  =  1 ,  1 ,  0 ) )
7730, 76syl 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 ) )
7875, 77sylan9eqr 2496 . . 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 ) )
7965, 74, 783eqtr4rd 2485 . 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 ) ) )
80 lgsdilem 22660 . . . . 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 ) ) )
8180adantr 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 ) ) )
82 mulcl 9365 . . . . . 6  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  x.  y
)  e.  CC )
8382adantl 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 )
84 mulcom 9367 . . . . . 6  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  x.  y
)  =  ( y  x.  x ) )
8584adantl 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 ) )
86 mulass 9369 . . . . . 6  |-  ( ( x  e.  CC  /\  y  e.  CC  /\  z  e.  CC )  ->  (
( x  x.  y
)  x.  z )  =  ( x  x.  ( y  x.  z
) ) )
8786adantl 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 ) ) )
88 simpl3 993 . . . . . . 7  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0
) )  ->  N  e.  ZZ )
89 nnabscl 12812 . . . . . . 7  |-  ( ( N  e.  ZZ  /\  N  =/=  0 )  -> 
( abs `  N
)  e.  NN )
9088, 89sylan 471 . . . . . 6  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =/=  0 )  -> 
( abs `  N
)  e.  NN )
91 nnuz 10895 . . . . . 6  |-  NN  =  ( ZZ>= `  1 )
9290, 91syl6eleq 2532 . . . . 5  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =/=  0 )  -> 
( abs `  N
)  e.  ( ZZ>= ` 
1 ) )
93 simpll1 1027 . . . . . . . 8  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =/=  0 )  ->  A  e.  ZZ )
94 simpll3 1029 . . . . . . . 8  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =/=  0 )  ->  N  e.  ZZ )
95 simpr 461 . . . . . . . 8  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =/=  0 )  ->  N  =/=  0 )
96 eqid 2442 . . . . . . . . 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 ) )
9796lgsfcl3 22655 . . . . . . . 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 )
9893, 94, 95, 97syl3anc 1218 . . . . . . 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 )
99 elfznn 11477 . . . . . . 7  |-  ( k  e.  ( 1 ... ( abs `  N
) )  ->  k  e.  NN )
100 ffvelrn 5840 . . . . . . 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 )
10198, 99, 100syl2an 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 )
102101zcnd 10747 . . . . 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 )
103 simpll2 1028 . . . . . . . 8  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =/=  0 )  ->  B  e.  ZZ )
104 eqid 2442 . . . . . . . . 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 ) )
105104lgsfcl3 22655 . . . . . . . 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 )
106103, 94, 95, 105syl3anc 1218 . . . . . . 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 )
107 ffvelrn 5840 . . . . . . 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 )
108106, 99, 107syl2an 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 )
109108zcnd 10747 . . . . 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 )
11093adantr 465 . . . . . . . . . . . 12  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =/=  0 )  /\  k  e.  Prime )  ->  A  e.  ZZ )
111103adantr 465 . . . . . . . . . . . 12  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =/=  0 )  /\  k  e.  Prime )  ->  B  e.  ZZ )
112 simpr 461 . . . . . . . . . . . 12  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =/=  0 )  /\  k  e.  Prime )  -> 
k  e.  Prime )
113 lgsdirprm 22667 . . . . . . . . . . . 12  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  k  e.  Prime )  ->  (
( A  x.  B
)  /L k )  =  ( ( A  /L k )  x.  ( B  /L k ) ) )
114110, 111, 112, 113syl3anc 1218 . . . . . . . . . . 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 ) ) )
115114oveq1d 6105 . . . . . . . . . 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
) ) )
116 prmz 13766 . . . . . . . . . . . . 13  |-  ( k  e.  Prime  ->  k  e.  ZZ )
117 lgscl 22648 . . . . . . . . . . . . 13  |-  ( ( A  e.  ZZ  /\  k  e.  ZZ )  ->  ( A  /L
k )  e.  ZZ )
11893, 116, 117syl2an 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 )
119118zcnd 10747 . . . . . . . . . . 11  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =/=  0 )  /\  k  e.  Prime )  -> 
( A  /L
k )  e.  CC )
120 lgscl 22648 . . . . . . . . . . . . 13  |-  ( ( B  e.  ZZ  /\  k  e.  ZZ )  ->  ( B  /L
k )  e.  ZZ )
121103, 116, 120syl2an 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 )
122121zcnd 10747 . . . . . . . . . . 11  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =/=  0 )  /\  k  e.  Prime )  -> 
( B  /L
k )  e.  CC )
12394adantr 465 . . . . . . . . . . . 12  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =/=  0 )  /\  k  e.  Prime )  ->  N  e.  ZZ )
12495adantr 465 . . . . . . . . . . . 12  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =/=  0 )  /\  k  e.  Prime )  ->  N  =/=  0 )
125 pczcl 13914 . . . . . . . . . . . 12  |-  ( ( k  e.  Prime  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( k  pCnt  N
)  e.  NN0 )
126112, 123, 124, 125syl12anc 1216 . . . . . . . . . . 11  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =/=  0 )  /\  k  e.  Prime )  -> 
( k  pCnt  N
)  e.  NN0 )
127119, 122, 126mulexpd 12022 . . . . . . . . . 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 )
) ) )
128115, 127eqtrd 2474 . . . . . . . . 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 )
) ) )
129 iftrue 3796 . . . . . . . . . 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 ) ) )
130129adantl 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 ) ) )
131 iftrue 3796 . . . . . . . . . . 11  |-  ( k  e.  Prime  ->  if ( k  e.  Prime ,  ( ( A  /L
k ) ^ (
k  pCnt  N )
) ,  1 )  =  ( ( A  /L k ) ^ ( k  pCnt  N ) ) )
132 iftrue 3796 . . . . . . . . . . 11  |-  ( k  e.  Prime  ->  if ( k  e.  Prime ,  ( ( B  /L
k ) ^ (
k  pCnt  N )
) ,  1 )  =  ( ( B  /L k ) ^ ( k  pCnt  N ) ) )
133131, 132oveq12d 6108 . . . . . . . . . 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 ) ) ) )
134133adantl 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 )
) ) )
135128, 130, 1343eqtr4d 2484 . . . . . . . 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 ) ) )
136 1t1e1 10468 . . . . . . . . . . 11  |-  ( 1  x.  1 )  =  1
137136eqcomi 2446 . . . . . . . . . 10  |-  1  =  ( 1  x.  1 )
138 iffalse 3798 . . . . . . . . . 10  |-  ( -.  k  e.  Prime  ->  if ( k  e.  Prime ,  ( ( ( A  x.  B )  /L k ) ^
( k  pCnt  N
) ) ,  1 )  =  1 )
139 iffalse 3798 . . . . . . . . . . 11  |-  ( -.  k  e.  Prime  ->  if ( k  e.  Prime ,  ( ( A  /L k ) ^
( k  pCnt  N
) ) ,  1 )  =  1 )
140 iffalse 3798 . . . . . . . . . . 11  |-  ( -.  k  e.  Prime  ->  if ( k  e.  Prime ,  ( ( B  /L k ) ^
( k  pCnt  N
) ) ,  1 )  =  1 )
141139, 140oveq12d 6108 . . . . . . . . . 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 ) )
142137, 138, 1413eqtr4a 2500 . . . . . . . . 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 ) ) )
143142adantl 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 ) ) )
144135, 143pm2.61dan 789 . . . . . . 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 ) ) )
145144adantr 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 ) ) )
14699adantl 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 )
147 eleq1 2502 . . . . . . . . 9  |-  ( n  =  k  ->  (
n  e.  Prime  <->  k  e.  Prime ) )
148 oveq2 6098 . . . . . . . . . 10  |-  ( n  =  k  ->  (
( A  x.  B
)  /L n )  =  ( ( A  x.  B )  /L k ) )
149 oveq1 6097 . . . . . . . . . 10  |-  ( n  =  k  ->  (
n  pCnt  N )  =  ( k  pCnt  N ) )
150148, 149oveq12d 6108 . . . . . . . . 9  |-  ( n  =  k  ->  (
( ( A  x.  B )  /L
n ) ^ (
n  pCnt  N )
)  =  ( ( ( A  x.  B
)  /L k ) ^ ( k 
pCnt  N ) ) )
151147, 150ifbieq1d 3811 . . . . . . . 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 ) )
152 eqid 2442 . . . . . . . 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 ) )
153 ovex 6115 . . . . . . . . 9  |-  ( ( ( A  x.  B
)  /L k ) ^ ( k 
pCnt  N ) )  e. 
_V
154 1ex 9380 . . . . . . . . 9  |-  1  e.  _V
155153, 154ifex 3857 . . . . . . . 8  |-  if ( k  e.  Prime ,  ( ( ( A  x.  B )  /L
k ) ^ (
k  pCnt  N )
) ,  1 )  e.  _V
156151, 152, 155fvmpt 5773 . . . . . . 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 ) )
157146, 156syl 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 ) )
158 oveq2 6098 . . . . . . . . . . 11  |-  ( n  =  k  ->  ( A  /L n )  =  ( A  /L k ) )
159158, 149oveq12d 6108 . . . . . . . . . 10  |-  ( n  =  k  ->  (
( A  /L
n ) ^ (
n  pCnt  N )
)  =  ( ( A  /L k ) ^ ( k 
pCnt  N ) ) )
160147, 159ifbieq1d 3811 . . . . . . . . 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 ) )
161 ovex 6115 . . . . . . . . . 10  |-  ( ( A  /L k ) ^ ( k 
pCnt  N ) )  e. 
_V
162161, 154ifex 3857 . . . . . . . . 9  |-  if ( k  e.  Prime ,  ( ( A  /L
k ) ^ (
k  pCnt  N )
) ,  1 )  e.  _V
163160, 96, 162fvmpt 5773 . . . . . . . 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 ) )
164146, 163syl 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 ) )
165 oveq2 6098 . . . . . . . . . . 11  |-  ( n  =  k  ->  ( B  /L n )  =  ( B  /L k ) )
166165, 149oveq12d 6108 . . . . . . . . . 10  |-  ( n  =  k  ->  (
( B  /L
n ) ^ (
n  pCnt  N )
)  =  ( ( B  /L k ) ^ ( k 
pCnt  N ) ) )
167147, 166ifbieq1d 3811 . . . . . . . . 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 ) )
168 ovex 6115 . . . . . . . . . 10  |-  ( ( B  /L k ) ^ ( k 
pCnt  N ) )  e. 
_V
169168, 154ifex 3857 . . . . . . . . 9  |-  if ( k  e.  Prime ,  ( ( B  /L
k ) ^ (
k  pCnt  N )
) ,  1 )  e.  _V
170167, 104, 169fvmpt 5773 . . . . . . . 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 ) )
171146, 170syl 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 ) )
172164, 171oveq12d 6108 . . . . . 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 ) ) )
173145, 157, 1723eqtr4d 2484 . . . . 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
) ) )
17483, 85, 87, 92, 102, 109, 173seqcaopr 11842 . . . 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 ) ) ) )
17581, 174oveq12d 6108 . . 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 ) ) ) ) )
17630adantr 465 . . . 4  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =/=  0 )  -> 
( A  x.  B
)  e.  ZZ )
177152lgsval4 22654 . . . 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 ) ) ) )
178176, 94, 95, 177syl3anc 1218 . . 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 ) ) ) )
17996lgsval4 22654 . . . . . 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 ) ) ) )
18093, 94, 95, 179syl3anc 1218 . . . . 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 ) ) ) )
181104lgsval4 22654 . . . . . 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 ) ) ) )
182103, 94, 95, 181syl3anc 1218 . . . . 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 ) ) ) )
183180, 182oveq12d 6108 . . . 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 ) ) ) ) )
184 neg1cn 10424 . . . . . . 7  |-  -u 1  e.  CC
185184, 1keepel 3856 . . . . . 6  |-  if ( ( N  <  0  /\  A  <  0
) ,  -u 1 ,  1 )  e.  CC
186185a1i 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 )
187 mulcl 9365 . . . . . . 7  |-  ( ( k  e.  CC  /\  x  e.  CC )  ->  ( k  x.  x
)  e.  CC )
188187adantl 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 )
18992, 102, 188seqcl 11825 . . . . 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 )
190184, 1keepel 3856 . . . . . 6  |-  if ( ( N  <  0  /\  B  <  0
) ,  -u 1 ,  1 )  e.  CC
191190a1i 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 )
19292, 109, 188seqcl 11825 . . . . 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 )
193186, 189, 191, 192mul4d 9580 . . . 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 ) ) ) ) )
194183, 193eqtrd 2474 . . 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 ) ) ) ) )
195175, 178, 1943eqtr4d 2484 . 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 ) ) )
19679, 195pm2.61dane 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 184    \/ wo 368    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2605   ifcif 3790   class class class wbr 4291    e. cmpt 4349   -->wf 5413   ` cfv 5417  (class class class)co 6090   CCcc 9279   RRcr 9280   0cc0 9281   1c1 9282    x. cmul 9286    < clt 9417    <_ cle 9418   -ucneg 9595   NNcn 10321   2c2 10370   NN0cn0 10578   ZZcz 10645   ZZ>=cuz 10860   ...cfz 11436    seqcseq 11805   ^cexp 11864   abscabs 12722    || cdivides 13534   Primecprime 13762    pCnt cpc 13902    /Lclgs 22632
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4402  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371  ax-cnex 9337  ax-resscn 9338  ax-1cn 9339  ax-icn 9340  ax-addcl 9341  ax-addrcl 9342  ax-mulcl 9343  ax-mulrcl 9344  ax-mulcom 9345  ax-addass 9346  ax-mulass 9347  ax-distr 9348  ax-i2m1 9349  ax-1ne0 9350  ax-1rid 9351  ax-rnegex 9352  ax-rrecex 9353  ax-cnre 9354  ax-pre-lttri 9355  ax-pre-lttrn 9356  ax-pre-ltadd 9357  ax-pre-mulgt0 9358  ax-pre-sup 9359
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-pss 3343  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-tp 3881  df-op 3883  df-uni 4091  df-int 4128  df-iun 4172  df-br 4292  df-opab 4350  df-mpt 4351  df-tr 4385  df-eprel 4631  df-id 4635  df-po 4640  df-so 4641  df-fr 4678  df-we 4680  df-ord 4721  df-on 4722  df-lim 4723  df-suc 4724  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-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-riota 6051  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6831  df-rdg 6865  df-1o 6919  df-2o 6920  df-oadd 6923  df-er 7100  df-map 7215  df-en 7310  df-dom 7311  df-sdom 7312  df-fin 7313  df-sup 7690  df-card 8108  df-cda 8336  df-pnf 9419  df-mnf 9420  df-xr 9421  df-ltxr 9422  df-le 9423  df-sub 9596  df-neg 9597  df-div 9993  df-nn 10322  df-2 10379  df-3 10380  df-4 10381  df-5 10382  df-6 10383  df-7 10384  df-8 10385  df-9 10386  df-n0 10579  df-z 10646  df-uz 10861  df-q 10953  df-rp 10991  df-fz 11437  df-fzo 11548  df-fl 11641  df-mod 11708  df-seq 11806  df-exp 11865  df-hash 12103  df-cj 12587  df-re 12588  df-im 12589  df-sqr 12723  df-abs 12724  df-dvds 13535  df-gcd 13690  df-prm 13763  df-phi 13840  df-pc 13903  df-lgs 22633
This theorem is referenced by:  lgssq  22673  lgsdirnn0  22677  lgsquad2lem1  22696
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