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Theorem lgsdir 20622
Description: The Legendre symbol is completely multiplicative in its left argument. Together with lgsqr 20638 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 8840 . . . . . . 7  |-  1  e.  CC
2 0cn 8876 . . . . . . 7  |-  0  e.  CC
31, 2keepel 3656 . . . . . 6  |-  if ( ( B ^ 2 )  =  1 ,  1 ,  0 )  e.  CC
43mulid2i 8885 . . . . 5  |-  ( 1  x.  if ( ( B ^ 2 )  =  1 ,  1 ,  0 ) )  =  if ( ( B ^ 2 )  =  1 ,  1 ,  0 )
5 iftrue 3605 . . . . . . 7  |-  ( ( A ^ 2 )  =  1  ->  if ( ( A ^
2 )  =  1 ,  1 ,  0 )  =  1 )
65adantl 452 . . . . . 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 5915 . . . . 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 958 . . . . . . . . . . 11  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0
) )  ->  A  e.  ZZ )
98zcnd 10165 . . . . . . . . . 10  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0
) )  ->  A  e.  CC )
109ad2antrr 706 . . . . . . . . 9  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =  0 )  /\  ( A ^
2 )  =  1 )  ->  A  e.  CC )
11 simpl2 959 . . . . . . . . . . 11  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0
) )  ->  B  e.  ZZ )
1211zcnd 10165 . . . . . . . . . 10  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0
) )  ->  B  e.  CC )
1312ad2antrr 706 . . . . . . . . 9  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =  0 )  /\  ( A ^
2 )  =  1 )  ->  B  e.  CC )
1410, 13sqmuld 11304 . . . . . . . 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 447 . . . . . . . . 9  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =  0 )  /\  ( A ^
2 )  =  1 )  ->  ( A ^ 2 )  =  1 )
1615oveq1d 5915 . . . . . . . 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 11290 . . . . . . . . . 10  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0
) )  ->  ( B ^ 2 )  e.  CC )
1817ad2antrr 706 . . . . . . . . 9  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =  0 )  /\  ( A ^
2 )  =  1 )  ->  ( B ^ 2 )  e.  CC )
1918mulid2d 8898 . . . . . . . 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 2352 . . . . . . 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 2324 . . . . . 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 3617 . . . . 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 2374 . . . 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 9046 . . . . 5  |-  ( 0  x.  if ( ( B ^ 2 )  =  1 ,  1 ,  0 ) )  =  0
25 iffalse 3606 . . . . . . 7  |-  ( -.  ( A ^ 2 )  =  1  ->  if ( ( A ^
2 )  =  1 ,  1 ,  0 )  =  0 )
2625adantl 452 . . . . . 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 5915 . . . . 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 12597 . . . . . . . . . . . 12  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  A  ||  ( A  x.  B ) )
298, 11, 28syl2anc 642 . . . . . . . . . . 11  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0
) )  ->  A  ||  ( A  x.  B
) )
308, 11zmulcld 10170 . . . . . . . . . . . 12  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0
) )  ->  ( A  x.  B )  e.  ZZ )
31 dvdssq 12786 . . . . . . . . . . . 12  |-  ( ( A  e.  ZZ  /\  ( A  x.  B
)  e.  ZZ )  ->  ( A  ||  ( A  x.  B
)  <->  ( A ^
2 )  ||  (
( A  x.  B
) ^ 2 ) ) )
328, 30, 31syl2anc 642 . . . . . . . . . . 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 201 . . . . . . . . . 10  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0
) )  ->  ( A ^ 2 )  ||  ( ( A  x.  B ) ^ 2 ) )
3433adantr 451 . . . . . . . . 9  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =  0 )  ->  ( A ^
2 )  ||  (
( A  x.  B
) ^ 2 ) )
35 breq2 4064 . . . . . . . . 9  |-  ( ( ( A  x.  B
) ^ 2 )  =  1  ->  (
( A ^ 2 )  ||  ( ( A  x.  B ) ^ 2 )  <->  ( A ^ 2 )  ||  1 ) )
3634, 35syl5ibcom 211 . . . . . . . 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 732 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0
) )  ->  A  =/=  0 )
3837neneqd 2495 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0
) )  ->  -.  A  =  0 )
39 sqeq0 11215 . . . . . . . . . . . . . . . 16  |-  ( A  e.  CC  ->  (
( A ^ 2 )  =  0  <->  A  =  0 ) )
409, 39syl 15 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0
) )  ->  (
( A ^ 2 )  =  0  <->  A  =  0 ) )
4138, 40mtbird 292 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0
) )  ->  -.  ( A ^ 2 )  =  0 )
42 zsqcl2 11228 . . . . . . . . . . . . . . . . 17  |-  ( A  e.  ZZ  ->  ( A ^ 2 )  e. 
NN0 )
438, 42syl 15 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0
) )  ->  ( A ^ 2 )  e. 
NN0 )
44 elnn0 10014 . . . . . . . . . . . . . . . 16  |-  ( ( A ^ 2 )  e.  NN0  <->  ( ( A ^ 2 )  e.  NN  \/  ( A ^ 2 )  =  0 ) )
4543, 44sylib 188 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0
) )  ->  (
( A ^ 2 )  e.  NN  \/  ( A ^ 2 )  =  0 ) )
4645ord 366 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0
) )  ->  ( -.  ( A ^ 2 )  e.  NN  ->  ( A ^ 2 )  =  0 ) )
4741, 46mt3d 117 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0
) )  ->  ( A ^ 2 )  e.  NN )
4847adantr 451 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =  0 )  ->  ( A ^
2 )  e.  NN )
4948nnzd 10163 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =  0 )  ->  ( A ^
2 )  e.  ZZ )
50 1nn 9802 . . . . . . . . . . 11  |-  1  e.  NN
51 dvdsle 12621 . . . . . . . . . . 11  |-  ( ( ( A ^ 2 )  e.  ZZ  /\  1  e.  NN )  ->  ( ( A ^
2 )  ||  1  ->  ( A ^ 2 )  <_  1 ) )
5249, 50, 51sylancl 643 . . . . . . . . . 10  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =  0 )  ->  ( ( A ^ 2 )  ||  1  ->  ( A ^
2 )  <_  1
) )
5348nnge1d 9833 . . . . . . . . . 10  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =  0 )  ->  1  <_  ( A ^ 2 ) )
5452, 53jctird 528 . . . . . . . . 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 9806 . . . . . . . . . 10  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =  0 )  ->  ( A ^
2 )  e.  RR )
56 1re 8882 . . . . . . . . . 10  |-  1  e.  RR
57 letri3 8952 . . . . . . . . . 10  |-  ( ( ( A ^ 2 )  e.  RR  /\  1  e.  RR )  ->  ( ( A ^
2 )  =  1  <-> 
( ( A ^
2 )  <_  1  /\  1  <_  ( A ^ 2 ) ) ) )
5855, 56, 57sylancl 643 . . . . . . . . 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 225 . . . . . . . 8  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =  0 )  ->  ( ( A ^ 2 )  ||  1  ->  ( A ^
2 )  =  1 ) )
6036, 59syld 40 . . . . . . 7  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =  0 )  ->  ( ( ( A  x.  B ) ^ 2 )  =  1  ->  ( A ^ 2 )  =  1 ) )
6160con3and 428 . . . . . 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 3606 . . . . . 6  |-  ( -.  ( ( A  x.  B ) ^ 2 )  =  1  ->  if ( ( ( A  x.  B ) ^
2 )  =  1 ,  1 ,  0 )  =  0 )
6361, 62syl 15 . . . . 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 2374 . . . 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 766 . . 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 5908 . . . . 5  |-  ( N  =  0  ->  ( A  / L N )  =  ( A  / L 0 ) )
67 lgs0 20601 . . . . . 6  |-  ( A  e.  ZZ  ->  ( A  / L 0 )  =  if ( ( A ^ 2 )  =  1 ,  1 ,  0 ) )
688, 67syl 15 . . . . 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 2370 . . . 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 5908 . . . . 5  |-  ( N  =  0  ->  ( B  / L N )  =  ( B  / L 0 ) )
71 lgs0 20601 . . . . . 6  |-  ( B  e.  ZZ  ->  ( B  / L 0 )  =  if ( ( B ^ 2 )  =  1 ,  1 ,  0 ) )
7211, 71syl 15 . . . . 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 2370 . . . 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 5918 . . 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 5908 . . . 4  |-  ( N  =  0  ->  (
( A  x.  B
)  / L N
)  =  ( ( A  x.  B )  / L 0 ) )
76 lgs0 20601 . . . . 5  |-  ( ( A  x.  B )  e.  ZZ  ->  (
( A  x.  B
)  / L 0 )  =  if ( ( ( A  x.  B ) ^ 2 )  =  1 ,  1 ,  0 ) )
7730, 76syl 15 . . . 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 2370 . . 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 2359 . 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 20614 . . . . 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 451 . . . 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 8866 . . . . . 6  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  x.  y
)  e.  CC )
8382adantl 452 . . . . 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 8868 . . . . . 6  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  x.  y
)  =  ( y  x.  x ) )
8584adantl 452 . . . . 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 8870 . . . . . 6  |-  ( ( x  e.  CC  /\  y  e.  CC  /\  z  e.  CC )  ->  (
( x  x.  y
)  x.  z )  =  ( x  x.  ( y  x.  z
) ) )
8786adantl 452 . . . . 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 960 . . . . . . 7  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0
) )  ->  N  e.  ZZ )
89 nnabscl 11856 . . . . . . 7  |-  ( ( N  e.  ZZ  /\  N  =/=  0 )  -> 
( abs `  N
)  e.  NN )
9088, 89sylan 457 . . . . . 6  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =/=  0 )  -> 
( abs `  N
)  e.  NN )
91 nnuz 10310 . . . . . 6  |-  NN  =  ( ZZ>= `  1 )
9290, 91syl6eleq 2406 . . . . 5  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =/=  0 )  -> 
( abs `  N
)  e.  ( ZZ>= ` 
1 ) )
93 simpll1 994 . . . . . . . 8  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =/=  0 )  ->  A  e.  ZZ )
94 simpll3 996 . . . . . . . 8  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =/=  0 )  ->  N  e.  ZZ )
95 simpr 447 . . . . . . . 8  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =/=  0 )  ->  N  =/=  0 )
96 eqid 2316 . . . . . . . . 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 20609 . . . . . . . 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 1182 . . . . . . 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 10866 . . . . . . 7  |-  ( k  e.  ( 1 ... ( abs `  N
) )  ->  k  e.  NN )
100 ffvelrn 5701 . . . . . . 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 463 . . . . . 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 10165 . . . . 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 995 . . . . . . . 8  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =/=  0 )  ->  B  e.  ZZ )
104 eqid 2316 . . . . . . . . 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 20609 . . . . . . . 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 1182 . . . . . . 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 5701 . . . . . . 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 463 . . . . . 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 10165 . . . . 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 451 . . . . . . . . . . . 12  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =/=  0 )  /\  k  e.  Prime )  ->  A  e.  ZZ )
111103adantr 451 . . . . . . . . . . . 12  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =/=  0 )  /\  k  e.  Prime )  ->  B  e.  ZZ )
112 simpr 447 . . . . . . . . . . . 12  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =/=  0 )  /\  k  e.  Prime )  -> 
k  e.  Prime )
113 lgsdirprm 20621 . . . . . . . . . . . 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 1182 . . . . . . . . . . 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 5915 . . . . . . . . . 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 12809 . . . . . . . . . . . . 13  |-  ( k  e.  Prime  ->  k  e.  ZZ )
117 lgscl 20602 . . . . . . . . . . . . 13  |-  ( ( A  e.  ZZ  /\  k  e.  ZZ )  ->  ( A  / L
k )  e.  ZZ )
11893, 116, 117syl2an 463 . . . . . . . . . . . 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 10165 . . . . . . . . . . 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 20602 . . . . . . . . . . . . 13  |-  ( ( B  e.  ZZ  /\  k  e.  ZZ )  ->  ( B  / L
k )  e.  ZZ )
121103, 116, 120syl2an 463 . . . . . . . . . . . 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 10165 . . . . . . . . . . 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 451 . . . . . . . . . . . 12  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =/=  0 )  /\  k  e.  Prime )  ->  N  e.  ZZ )
12495adantr 451 . . . . . . . . . . . 12  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =/=  0 )  /\  k  e.  Prime )  ->  N  =/=  0 )
125 pczcl 12948 . . . . . . . . . . . 12  |-  ( ( k  e.  Prime  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( k  pCnt  N
)  e.  NN0 )
126112, 123, 124, 125syl12anc 1180 . . . . . . . . . . 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 11307 . . . . . . . . . 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 2348 . . . . . . . . 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 3605 . . . . . . . . . 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 452 . . . . . . . . 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 3605 . . . . . . . . . . 11  |-  ( k  e.  Prime  ->  if ( k  e.  Prime ,  ( ( A  / L
k ) ^ (
k  pCnt  N )
) ,  1 )  =  ( ( A  / L k ) ^ ( k  pCnt  N ) ) )
132 iftrue 3605 . . . . . . . . . . 11  |-  ( k  e.  Prime  ->  if ( k  e.  Prime ,  ( ( B  / L
k ) ^ (
k  pCnt  N )
) ,  1 )  =  ( ( B  / L k ) ^ ( k  pCnt  N ) ) )
133131, 132oveq12d 5918 . . . . . . . . . 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 452 . . . . . . . . 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 2358 . . . . . . . 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 9917 . . . . . . . . . . 11  |-  ( 1  x.  1 )  =  1
137136eqcomi 2320 . . . . . . . . . 10  |-  1  =  ( 1  x.  1 )
138 iffalse 3606 . . . . . . . . . 10  |-  ( -.  k  e.  Prime  ->  if ( k  e.  Prime ,  ( ( ( A  x.  B )  / L k ) ^
( k  pCnt  N
) ) ,  1 )  =  1 )
139 iffalse 3606 . . . . . . . . . . 11  |-  ( -.  k  e.  Prime  ->  if ( k  e.  Prime ,  ( ( A  / L k ) ^
( k  pCnt  N
) ) ,  1 )  =  1 )
140 iffalse 3606 . . . . . . . . . . 11  |-  ( -.  k  e.  Prime  ->  if ( k  e.  Prime ,  ( ( B  / L k ) ^
( k  pCnt  N
) ) ,  1 )  =  1 )
141139, 140oveq12d 5918 . . . . . . . . . 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 2374 . . . . . . . . 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 452 . . . . . . . 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 766 . . . . . . 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 451 . . . . . 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 452 . . . . . . 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 2376 . . . . . . . . 9  |-  ( n  =  k  ->  (
n  e.  Prime  <->  k  e.  Prime ) )
148 oveq2 5908 . . . . . . . . . 10  |-  ( n  =  k  ->  (
( A  x.  B
)  / L n )  =  ( ( A  x.  B )  / L k ) )
149 oveq1 5907 . . . . . . . . . 10  |-  ( n  =  k  ->  (
n  pCnt  N )  =  ( k  pCnt  N ) )
150148, 149oveq12d 5918 . . . . . . . . 9  |-  ( n  =  k  ->  (
( ( A  x.  B )  / L
n ) ^ (
n  pCnt  N )
)  =  ( ( ( A  x.  B
)  / L k ) ^ ( k 
pCnt  N ) ) )
151 eqidd 2317 . . . . . . . . 9  |-  ( n  =  k  ->  1  =  1 )
152147, 150, 151ifbieq12d 3621 . . . . . . . 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 ) )
153 eqid 2316 . . . . . . . 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 ) )
154 ovex 5925 . . . . . . . . 9  |-  ( ( ( A  x.  B
)  / L k ) ^ ( k 
pCnt  N ) )  e. 
_V
155 1ex 8878 . . . . . . . . 9  |-  1  e.  _V
156154, 155ifex 3657 . . . . . . . 8  |-  if ( k  e.  Prime ,  ( ( ( A  x.  B )  / L
k ) ^ (
k  pCnt  N )
) ,  1 )  e.  _V
157152, 153, 156fvmpt 5640 . . . . . . 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 ) )
158146, 157syl 15 . . . . . 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 ) )
159 oveq2 5908 . . . . . . . . . . 11  |-  ( n  =  k  ->  ( A  / L n )  =  ( A  / L k ) )
160159, 149oveq12d 5918 . . . . . . . . . 10  |-  ( n  =  k  ->  (
( A  / L
n ) ^ (
n  pCnt  N )
)  =  ( ( A  / L k ) ^ ( k 
pCnt  N ) ) )
161147, 160, 151ifbieq12d 3621 . . . . . . . . 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 ) )
162 ovex 5925 . . . . . . . . . 10  |-  ( ( A  / L k ) ^ ( k 
pCnt  N ) )  e. 
_V
163162, 155ifex 3657 . . . . . . . . 9  |-  if ( k  e.  Prime ,  ( ( A  / L
k ) ^ (
k  pCnt  N )
) ,  1 )  e.  _V
164161, 96, 163fvmpt 5640 . . . . . . . 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 ) )
165146, 164syl 15 . . . . . . 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 ) )
166 oveq2 5908 . . . . . . . . . . 11  |-  ( n  =  k  ->  ( B  / L n )  =  ( B  / L k ) )
167166, 149oveq12d 5918 . . . . . . . . . 10  |-  ( n  =  k  ->  (
( B  / L
n ) ^ (
n  pCnt  N )
)  =  ( ( B  / L k ) ^ ( k 
pCnt  N ) ) )
168147, 167, 151ifbieq12d 3621 . . . . . . . . 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 ) )
169 ovex 5925 . . . . . . . . . 10  |-  ( ( B  / L k ) ^ ( k 
pCnt  N ) )  e. 
_V
170169, 155ifex 3657 . . . . . . . . 9  |-  if ( k  e.  Prime ,  ( ( B  / L
k ) ^ (
k  pCnt  N )
) ,  1 )  e.  _V
171168, 104, 170fvmpt 5640 . . . . . . . 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 ) )
172146, 171syl 15 . . . . . . 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 ) )
173165, 172oveq12d 5918 . . . . . 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 ) ) )
174145, 158, 1733eqtr4d 2358 . . . . 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
) ) )
17583, 85, 87, 92, 102, 109, 174seqcaopr 11130 . . . 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 ) ) ) )
17681, 175oveq12d 5918 . . 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 ) ) ) ) )
17730adantr 451 . . . 4  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  =/=  0 )  -> 
( A  x.  B
)  e.  ZZ )
178153lgsval4 20608 . . . 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 ) ) ) )
179177, 94, 95, 178syl3anc 1182 . . 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 ) ) ) )
18096lgsval4 20608 . . . . . 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 ) ) ) )
18193, 94, 95, 180syl3anc 1182 . . . . 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 ) ) ) )
182104lgsval4 20608 . . . . . 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 ) ) ) )
183103, 94, 95, 182syl3anc 1182 . . . . 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 ) ) ) )
184181, 183oveq12d 5918 . . . 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 ) ) ) ) )
185 neg1cn 9858 . . . . . . 7  |-  -u 1  e.  CC
186185, 1keepel 3656 . . . . . 6  |-  if ( ( N  <  0  /\  A  <  0
) ,  -u 1 ,  1 )  e.  CC
187186a1i 10 . . . . 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 )
188 mulcl 8866 . . . . . . 7  |-  ( ( k  e.  CC  /\  x  e.  CC )  ->  ( k  x.  x
)  e.  CC )
189188adantl 452 . . . . . 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 )
19092, 102, 189seqcl 11113 . . . . 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 )
191185, 1keepel 3656 . . . . . 6  |-  if ( ( N  <  0  /\  B  <  0
) ,  -u 1 ,  1 )  e.  CC
192191a1i 10 . . . . 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 )
19392, 109, 189seqcl 11113 . . . . 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 )
194187, 190, 192, 193mul4d 9069 . . . 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 ) ) ) ) )
195184, 194eqtrd 2348 . . 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 ) ) ) ) )
196176, 179, 1953eqtr4d 2358 . 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 ) ) )
19779, 196pm2.61dane 2557 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 set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1633    e. wcel 1701    =/= wne 2479   ifcif 3599   class class class wbr 4060    e. cmpt 4114   -->wf 5288   ` cfv 5292  (class class class)co 5900   CCcc 8780   RRcr 8781   0cc0 8782   1c1 8783    x. cmul 8787    < clt 8912    <_ cle 8913   -ucneg 9083   NNcn 9791   2c2 9840   NN0cn0 10012   ZZcz 10071   ZZ>=cuz 10277   ...cfz 10829    seq cseq 11093   ^cexp 11151   abscabs 11766    || cdivides 12578   Primecprime 12805    pCnt cpc 12936    / Lclgs 20586
This theorem is referenced by:  lgssq  20627  lgsdirnn0  20631  lgsquad2lem1  20650
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549  ax-cnex 8838  ax-resscn 8839  ax-1cn 8840  ax-icn 8841  ax-addcl 8842  ax-addrcl 8843  ax-mulcl 8844  ax-mulrcl 8845  ax-mulcom 8846  ax-addass 8847  ax-mulass 8848  ax-distr 8849  ax-i2m1 8850  ax-1ne0 8851  ax-1rid 8852  ax-rnegex 8853  ax-rrecex 8854  ax-cnre 8855  ax-pre-lttri 8856  ax-pre-lttrn 8857  ax-pre-ltadd 8858  ax-pre-mulgt0 8859  ax-pre-sup 8860
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-reu 2584  df-rmo 2585  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-int 3900  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-tr 4151  df-eprel 4342  df-id 4346  df-po 4351  df-so 4352  df-fr 4389  df-we 4391  df-ord 4432  df-on 4433  df-lim 4434  df-suc 4435  df-om 4694  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-1st 6164  df-2nd 6165  df-riota 6346  df-recs 6430  df-rdg 6465  df-1o 6521  df-2o 6522  df-oadd 6525  df-er 6702  df-map 6817  df-en 6907  df-dom 6908  df-sdom 6909  df-fin 6910  df-sup 7239  df-card 7617  df-cda 7839  df-pnf 8914  df-mnf 8915  df-xr 8916  df-ltxr 8917  df-le 8918  df-sub 9084  df-neg 9085  df-div 9469  df-nn 9792  df-2 9849  df-3 9850  df-4 9851  df-5 9852  df-6 9853  df-7 9854  df-8 9855  df-9 9856  df-n0 10013  df-z 10072  df-uz 10278  df-q 10364  df-rp 10402  df-fz 10830  df-fzo 10918  df-fl 10972  df-mod 11021  df-seq 11094  df-exp 11152  df-hash 11385  df-cj 11631  df-re 11632  df-im 11633  df-sqr 11767  df-abs 11768  df-dvds 12579  df-gcd 12733  df-prm 12806  df-phi 12881  df-pc 12937  df-lgs 20587
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