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Theorem lgsquad3 21147
Description: Extend lgsquad2 21146 to integers which share a factor. (Contributed by Mario Carneiro, 19-Jun-2015.)
Assertion
Ref Expression
lgsquad3  |-  ( ( ( M  e.  NN  /\ 
-.  2  ||  M
)  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  -> 
( M  / L N )  =  ( ( -u 1 ^ ( ( ( M  -  1 )  / 
2 )  x.  (
( N  -  1 )  /  2 ) ) )  x.  ( N  / L M ) ) )

Proof of Theorem lgsquad3
StepHypRef Expression
1 simplrl 738 . . . . . . . . . 10  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  ( M  gcd  N )  =  1 )  ->  N  e.  NN )
2 nnz 10305 . . . . . . . . . 10  |-  ( N  e.  NN  ->  N  e.  ZZ )
31, 2syl 16 . . . . . . . . 9  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  ( M  gcd  N )  =  1 )  ->  N  e.  ZZ )
4 nnz 10305 . . . . . . . . . 10  |-  ( M  e.  NN  ->  M  e.  ZZ )
54ad3antrrr 712 . . . . . . . . 9  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  ( M  gcd  N )  =  1 )  ->  M  e.  ZZ )
6 lgscl 21096 . . . . . . . . 9  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ )  ->  ( N  / L M )  e.  ZZ )
73, 5, 6syl2anc 644 . . . . . . . 8  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  ( M  gcd  N )  =  1 )  ->  ( N  / L M )  e.  ZZ )
87zred 10377 . . . . . . 7  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  ( M  gcd  N )  =  1 )  ->  ( N  / L M )  e.  RR )
9 absresq 12109 . . . . . . 7  |-  ( ( N  / L M
)  e.  RR  ->  ( ( abs `  ( N  / L M ) ) ^ 2 )  =  ( ( N  / L M ) ^ 2 ) )
108, 9syl 16 . . . . . 6  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  ( M  gcd  N )  =  1 )  ->  ( ( abs `  ( N  / L M ) ) ^
2 )  =  ( ( N  / L M ) ^ 2 ) )
11 gcdcom 13022 . . . . . . . . . . 11  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ )  ->  ( N  gcd  M
)  =  ( M  gcd  N ) )
123, 5, 11syl2anc 644 . . . . . . . . . 10  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  ( M  gcd  N )  =  1 )  ->  ( N  gcd  M )  =  ( M  gcd  N ) )
13 simpr 449 . . . . . . . . . 10  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  ( M  gcd  N )  =  1 )  ->  ( M  gcd  N )  =  1 )
1412, 13eqtrd 2470 . . . . . . . . 9  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  ( M  gcd  N )  =  1 )  ->  ( N  gcd  M )  =  1 )
15 lgsabs1 21120 . . . . . . . . . 10  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ )  ->  ( ( abs `  ( N  / L M ) )  =  1  <->  ( N  gcd  M )  =  1 ) )
163, 5, 15syl2anc 644 . . . . . . . . 9  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  ( M  gcd  N )  =  1 )  ->  ( ( abs `  ( N  / L M ) )  =  1  <->  ( N  gcd  M )  =  1 ) )
1714, 16mpbird 225 . . . . . . . 8  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  ( M  gcd  N )  =  1 )  ->  ( abs `  ( N  / L M ) )  =  1 )
1817oveq1d 6098 . . . . . . 7  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  ( M  gcd  N )  =  1 )  ->  ( ( abs `  ( N  / L M ) ) ^
2 )  =  ( 1 ^ 2 ) )
19 sq1 11478 . . . . . . 7  |-  ( 1 ^ 2 )  =  1
2018, 19syl6eq 2486 . . . . . 6  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  ( M  gcd  N )  =  1 )  ->  ( ( abs `  ( N  / L M ) ) ^
2 )  =  1 )
217zcnd 10378 . . . . . . 7  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  ( M  gcd  N )  =  1 )  ->  ( N  / L M )  e.  CC )
2221sqvald 11522 . . . . . 6  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  ( M  gcd  N )  =  1 )  ->  ( ( N  / L M ) ^ 2 )  =  ( ( N  / L M )  x.  ( N  / L M ) ) )
2310, 20, 223eqtr3d 2478 . . . . 5  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  ( M  gcd  N )  =  1 )  ->  1  =  ( ( N  / L M )  x.  ( N  / L M ) ) )
2423oveq2d 6099 . . . 4  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  ( M  gcd  N )  =  1 )  ->  ( ( M  / L N )  x.  1 )  =  ( ( M  / L N )  x.  (
( N  / L M )  x.  ( N  / L M ) ) ) )
25 lgscl 21096 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  / L N )  e.  ZZ )
265, 3, 25syl2anc 644 . . . . . 6  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  ( M  gcd  N )  =  1 )  ->  ( M  / L N )  e.  ZZ )
2726zcnd 10378 . . . . 5  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  ( M  gcd  N )  =  1 )  ->  ( M  / L N )  e.  CC )
2827, 21, 21mulassd 9113 . . . 4  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  ( M  gcd  N )  =  1 )  ->  ( ( ( M  / L N
)  x.  ( N  / L M ) )  x.  ( N  / L M ) )  =  ( ( M  / L N
)  x.  ( ( N  / L M
)  x.  ( N  / L M ) ) ) )
2924, 28eqtr4d 2473 . . 3  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  ( M  gcd  N )  =  1 )  ->  ( ( M  / L N )  x.  1 )  =  ( ( ( M  / L N )  x.  ( N  / L M ) )  x.  ( N  / L M ) ) )
3027mulid1d 9107 . . 3  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  ( M  gcd  N )  =  1 )  ->  ( ( M  / L N )  x.  1 )  =  ( M  / L N ) )
31 simplll 736 . . . . 5  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  ( M  gcd  N )  =  1 )  ->  M  e.  NN )
32 simpllr 737 . . . . 5  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  ( M  gcd  N )  =  1 )  ->  -.  2  ||  M )
33 simplrr 739 . . . . 5  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  ( M  gcd  N )  =  1 )  ->  -.  2  ||  N )
3431, 32, 1, 33, 13lgsquad2 21146 . . . 4  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  ( M  gcd  N )  =  1 )  ->  ( ( M  / L N )  x.  ( N  / L M ) )  =  ( -u 1 ^ ( ( ( M  -  1 )  / 
2 )  x.  (
( N  -  1 )  /  2 ) ) ) )
3534oveq1d 6098 . . 3  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  ( M  gcd  N )  =  1 )  ->  ( ( ( M  / L N
)  x.  ( N  / L M ) )  x.  ( N  / L M ) )  =  ( (
-u 1 ^ (
( ( M  - 
1 )  /  2
)  x.  ( ( N  -  1 )  /  2 ) ) )  x.  ( N  / L M ) ) )
3629, 30, 353eqtr3d 2478 . 2  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  ( M  gcd  N )  =  1 )  ->  ( M  / L N )  =  ( ( -u 1 ^ ( ( ( M  -  1 )  / 
2 )  x.  (
( N  -  1 )  /  2 ) ) )  x.  ( N  / L M ) ) )
37 lgsne0 21119 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( M  / L N )  =/=  0  <->  ( M  gcd  N )  =  1 ) )
3837necon1bbid 2660 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( -.  ( M  gcd  N )  =  1  <->  ( M  / L N )  =  0 ) )
394, 2, 38syl2an 465 . . . . . 6  |-  ( ( M  e.  NN  /\  N  e.  NN )  ->  ( -.  ( M  gcd  N )  =  1  <->  ( M  / L N )  =  0 ) )
4039ad2ant2r 729 . . . . 5  |-  ( ( ( M  e.  NN  /\ 
-.  2  ||  M
)  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  -> 
( -.  ( M  gcd  N )  =  1  <->  ( M  / L N )  =  0 ) )
4140biimpa 472 . . . 4  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  -.  ( M  gcd  N )  =  1 )  ->  ( M  / L N )  =  0 )
42 neg1cn 10069 . . . . . . 7  |-  -u 1  e.  CC
4342a1i 11 . . . . . 6  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  -.  ( M  gcd  N )  =  1 )  ->  -u 1  e.  CC )
44 ax-1cn 9050 . . . . . . . 8  |-  1  e.  CC
45 ax-1ne0 9061 . . . . . . . 8  |-  1  =/=  0
4644, 45negne0i 9377 . . . . . . 7  |-  -u 1  =/=  0
4746a1i 11 . . . . . 6  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  -.  ( M  gcd  N )  =  1 )  ->  -u 1  =/=  0 )
484ad3antrrr 712 . . . . . . . . 9  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  -.  ( M  gcd  N )  =  1 )  ->  M  e.  ZZ )
49 simpllr 737 . . . . . . . . 9  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  -.  ( M  gcd  N )  =  1 )  ->  -.  2  ||  M )
50 1z 10313 . . . . . . . . . 10  |-  1  e.  ZZ
5150a1i 11 . . . . . . . . 9  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  -.  ( M  gcd  N )  =  1 )  ->  1  e.  ZZ )
52 2prm 13097 . . . . . . . . . 10  |-  2  e.  Prime
53 nprmdvds1 13113 . . . . . . . . . 10  |-  ( 2  e.  Prime  ->  -.  2  ||  1 )
5452, 53mp1i 12 . . . . . . . . 9  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  -.  ( M  gcd  N )  =  1 )  ->  -.  2  ||  1 )
55 omoe 13188 . . . . . . . . 9  |-  ( ( ( M  e.  ZZ  /\ 
-.  2  ||  M
)  /\  ( 1  e.  ZZ  /\  -.  2  ||  1 ) )  ->  2  ||  ( M  -  1 ) )
5648, 49, 51, 54, 55syl22anc 1186 . . . . . . . 8  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  -.  ( M  gcd  N )  =  1 )  ->  2  ||  ( M  -  1 ) )
57 2z 10314 . . . . . . . . . 10  |-  2  e.  ZZ
5857a1i 11 . . . . . . . . 9  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  -.  ( M  gcd  N )  =  1 )  ->  2  e.  ZZ )
59 2ne0 10085 . . . . . . . . . 10  |-  2  =/=  0
6059a1i 11 . . . . . . . . 9  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  -.  ( M  gcd  N )  =  1 )  ->  2  =/=  0 )
61 peano2zm 10322 . . . . . . . . . 10  |-  ( M  e.  ZZ  ->  ( M  -  1 )  e.  ZZ )
6248, 61syl 16 . . . . . . . . 9  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  -.  ( M  gcd  N )  =  1 )  ->  ( M  -  1 )  e.  ZZ )
63 dvdsval2 12857 . . . . . . . . 9  |-  ( ( 2  e.  ZZ  /\  2  =/=  0  /\  ( M  -  1 )  e.  ZZ )  -> 
( 2  ||  ( M  -  1 )  <-> 
( ( M  - 
1 )  /  2
)  e.  ZZ ) )
6458, 60, 62, 63syl3anc 1185 . . . . . . . 8  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  -.  ( M  gcd  N )  =  1 )  ->  (
2  ||  ( M  -  1 )  <->  ( ( M  -  1 )  /  2 )  e.  ZZ ) )
6556, 64mpbid 203 . . . . . . 7  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  -.  ( M  gcd  N )  =  1 )  ->  (
( M  -  1 )  /  2 )  e.  ZZ )
662adantr 453 . . . . . . . . . 10  |-  ( ( N  e.  NN  /\  -.  2  ||  N )  ->  N  e.  ZZ )
6766ad2antlr 709 . . . . . . . . 9  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  -.  ( M  gcd  N )  =  1 )  ->  N  e.  ZZ )
68 simplrr 739 . . . . . . . . 9  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  -.  ( M  gcd  N )  =  1 )  ->  -.  2  ||  N )
69 omoe 13188 . . . . . . . . 9  |-  ( ( ( N  e.  ZZ  /\ 
-.  2  ||  N
)  /\  ( 1  e.  ZZ  /\  -.  2  ||  1 ) )  ->  2  ||  ( N  -  1 ) )
7067, 68, 51, 54, 69syl22anc 1186 . . . . . . . 8  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  -.  ( M  gcd  N )  =  1 )  ->  2  ||  ( N  -  1 ) )
71 peano2zm 10322 . . . . . . . . . 10  |-  ( N  e.  ZZ  ->  ( N  -  1 )  e.  ZZ )
7267, 71syl 16 . . . . . . . . 9  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  -.  ( M  gcd  N )  =  1 )  ->  ( N  -  1 )  e.  ZZ )
73 dvdsval2 12857 . . . . . . . . 9  |-  ( ( 2  e.  ZZ  /\  2  =/=  0  /\  ( N  -  1 )  e.  ZZ )  -> 
( 2  ||  ( N  -  1 )  <-> 
( ( N  - 
1 )  /  2
)  e.  ZZ ) )
7458, 60, 72, 73syl3anc 1185 . . . . . . . 8  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  -.  ( M  gcd  N )  =  1 )  ->  (
2  ||  ( N  -  1 )  <->  ( ( N  -  1 )  /  2 )  e.  ZZ ) )
7570, 74mpbid 203 . . . . . . 7  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  -.  ( M  gcd  N )  =  1 )  ->  (
( N  -  1 )  /  2 )  e.  ZZ )
7665, 75zmulcld 10383 . . . . . 6  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  -.  ( M  gcd  N )  =  1 )  ->  (
( ( M  - 
1 )  /  2
)  x.  ( ( N  -  1 )  /  2 ) )  e.  ZZ )
7743, 47, 76expclzd 11530 . . . . 5  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  -.  ( M  gcd  N )  =  1 )  ->  ( -u 1 ^ ( ( ( M  -  1 )  /  2 )  x.  ( ( N  -  1 )  / 
2 ) ) )  e.  CC )
7877mul01d 9267 . . . 4  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  -.  ( M  gcd  N )  =  1 )  ->  (
( -u 1 ^ (
( ( M  - 
1 )  /  2
)  x.  ( ( N  -  1 )  /  2 ) ) )  x.  0 )  =  0 )
7941, 78eqtr4d 2473 . . 3  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  -.  ( M  gcd  N )  =  1 )  ->  ( M  / L N )  =  ( ( -u
1 ^ ( ( ( M  -  1 )  /  2 )  x.  ( ( N  -  1 )  / 
2 ) ) )  x.  0 ) )
80 lgsne0 21119 . . . . . . . . 9  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ )  ->  ( ( N  / L M )  =/=  0  <->  ( N  gcd  M )  =  1 ) )
8111eqeq1d 2446 . . . . . . . . 9  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ )  ->  ( ( N  gcd  M )  =  1  <->  ( M  gcd  N )  =  1 ) )
8280, 81bitrd 246 . . . . . . . 8  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ )  ->  ( ( N  / L M )  =/=  0  <->  ( M  gcd  N )  =  1 ) )
832, 4, 82syl2anr 466 . . . . . . 7  |-  ( ( M  e.  NN  /\  N  e.  NN )  ->  ( ( N  / L M )  =/=  0  <->  ( M  gcd  N )  =  1 ) )
8483necon1bbid 2660 . . . . . 6  |-  ( ( M  e.  NN  /\  N  e.  NN )  ->  ( -.  ( M  gcd  N )  =  1  <->  ( N  / L M )  =  0 ) )
8584ad2ant2r 729 . . . . 5  |-  ( ( ( M  e.  NN  /\ 
-.  2  ||  M
)  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  -> 
( -.  ( M  gcd  N )  =  1  <->  ( N  / L M )  =  0 ) )
8685biimpa 472 . . . 4  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  -.  ( M  gcd  N )  =  1 )  ->  ( N  / L M )  =  0 )
8786oveq2d 6099 . . 3  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  -.  ( M  gcd  N )  =  1 )  ->  (
( -u 1 ^ (
( ( M  - 
1 )  /  2
)  x.  ( ( N  -  1 )  /  2 ) ) )  x.  ( N  / L M ) )  =  ( (
-u 1 ^ (
( ( M  - 
1 )  /  2
)  x.  ( ( N  -  1 )  /  2 ) ) )  x.  0 ) )
8879, 87eqtr4d 2473 . 2  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  -.  ( M  gcd  N )  =  1 )  ->  ( M  / L N )  =  ( ( -u
1 ^ ( ( ( M  -  1 )  /  2 )  x.  ( ( N  -  1 )  / 
2 ) ) )  x.  ( N  / L M ) ) )
8936, 88pm2.61dan 768 1  |-  ( ( ( M  e.  NN  /\ 
-.  2  ||  M
)  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  -> 
( M  / L N )  =  ( ( -u 1 ^ ( ( ( M  -  1 )  / 
2 )  x.  (
( N  -  1 )  /  2 ) ) )  x.  ( N  / L M ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726    =/= wne 2601   class class class wbr 4214   ` cfv 5456  (class class class)co 6083   CCcc 8990   RRcr 8991   0cc0 8992   1c1 8993    x. cmul 8997    - cmin 9293   -ucneg 9294    / cdiv 9679   NNcn 10002   2c2 10051   ZZcz 10284   ^cexp 11384   abscabs 12041    || cdivides 12854    gcd cgcd 13008   Primecprime 13081    / Lclgs 21080
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-inf2 7598  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069  ax-pre-sup 9070  ax-addf 9071  ax-mulf 9072
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-disj 4185  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-se 4544  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-isom 5465  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-of 6307  df-1st 6351  df-2nd 6352  df-tpos 6481  df-riota 6551  df-recs 6635  df-rdg 6670  df-1o 6726  df-2o 6727  df-oadd 6730  df-er 6907  df-ec 6909  df-qs 6913  df-map 7022  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-sup 7448  df-oi 7481  df-card 7828  df-cda 8050  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-div 9680  df-nn 10003  df-2 10060  df-3 10061  df-4 10062  df-5 10063  df-6 10064  df-7 10065  df-8 10066  df-9 10067  df-10 10068  df-n0 10224  df-z 10285  df-dec 10385  df-uz 10491  df-q 10577  df-rp 10615  df-fz 11046  df-fzo 11138  df-fl 11204  df-mod 11253  df-seq 11326  df-exp 11385  df-hash 11621  df-cj 11906  df-re 11907  df-im 11908  df-sqr 12042  df-abs 12043  df-clim 12284  df-sum 12482  df-dvds 12855  df-gcd 13009  df-prm 13082  df-phi 13157  df-pc 13213  df-struct 13473  df-ndx 13474  df-slot 13475  df-base 13476  df-sets 13477  df-ress 13478  df-plusg 13544  df-mulr 13545  df-starv 13546  df-sca 13547  df-vsca 13548  df-tset 13550  df-ple 13551  df-ds 13553  df-unif 13554  df-0g 13729  df-gsum 13730  df-imas 13736  df-divs 13737  df-mnd 14692  df-mhm 14740  df-submnd 14741  df-grp 14814  df-minusg 14815  df-sbg 14816  df-mulg 14817  df-subg 14943  df-nsg 14944  df-eqg 14945  df-ghm 15006  df-cntz 15118  df-cmn 15416  df-abl 15417  df-mgp 15651  df-rng 15665  df-cring 15666  df-ur 15667  df-oppr 15730  df-dvdsr 15748  df-unit 15749  df-invr 15779  df-dvr 15790  df-rnghom 15821  df-drng 15839  df-field 15840  df-subrg 15868  df-lmod 15954  df-lss 16011  df-lsp 16050  df-sra 16246  df-rgmod 16247  df-lidl 16248  df-rsp 16249  df-2idl 16305  df-nzr 16331  df-rlreg 16345  df-domn 16346  df-idom 16347  df-cnfld 16706  df-zrh 16784  df-zn 16787  df-lgs 21081
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