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Theorem lgsquad3 20616
Description: Extend lgsquad2 20615 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 736 . . . . . . . . . 10  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  ( M  gcd  N )  =  1 )  ->  N  e.  NN )
2 nnz 10061 . . . . . . . . . 10  |-  ( N  e.  NN  ->  N  e.  ZZ )
31, 2syl 15 . . . . . . . . 9  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  ( M  gcd  N )  =  1 )  ->  N  e.  ZZ )
4 nnz 10061 . . . . . . . . . 10  |-  ( M  e.  NN  ->  M  e.  ZZ )
54ad3antrrr 710 . . . . . . . . 9  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  ( M  gcd  N )  =  1 )  ->  M  e.  ZZ )
6 lgscl 20565 . . . . . . . . 9  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ )  ->  ( N  / L M )  e.  ZZ )
73, 5, 6syl2anc 642 . . . . . . . 8  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  ( M  gcd  N )  =  1 )  ->  ( N  / L M )  e.  ZZ )
87zred 10133 . . . . . . 7  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  ( M  gcd  N )  =  1 )  ->  ( N  / L M )  e.  RR )
9 absresq 11803 . . . . . . 7  |-  ( ( N  / L M
)  e.  RR  ->  ( ( abs `  ( N  / L M ) ) ^ 2 )  =  ( ( N  / L M ) ^ 2 ) )
108, 9syl 15 . . . . . 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 12715 . . . . . . . . . . 11  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ )  ->  ( N  gcd  M
)  =  ( M  gcd  N ) )
123, 5, 11syl2anc 642 . . . . . . . . . 10  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  ( M  gcd  N )  =  1 )  ->  ( N  gcd  M )  =  ( M  gcd  N ) )
13 simpr 447 . . . . . . . . . 10  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  ( M  gcd  N )  =  1 )  ->  ( M  gcd  N )  =  1 )
1412, 13eqtrd 2328 . . . . . . . . 9  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  ( M  gcd  N )  =  1 )  ->  ( N  gcd  M )  =  1 )
15 lgsabs1 20589 . . . . . . . . . 10  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ )  ->  ( ( abs `  ( N  / L M ) )  =  1  <->  ( N  gcd  M )  =  1 ) )
163, 5, 15syl2anc 642 . . . . . . . . 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 223 . . . . . . . 8  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  ( M  gcd  N )  =  1 )  ->  ( abs `  ( N  / L M ) )  =  1 )
1817oveq1d 5889 . . . . . . 7  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  ( M  gcd  N )  =  1 )  ->  ( ( abs `  ( N  / L M ) ) ^
2 )  =  ( 1 ^ 2 ) )
19 sq1 11214 . . . . . . 7  |-  ( 1 ^ 2 )  =  1
2018, 19syl6eq 2344 . . . . . 6  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  ( M  gcd  N )  =  1 )  ->  ( ( abs `  ( N  / L M ) ) ^
2 )  =  1 )
217zcnd 10134 . . . . . . 7  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  ( M  gcd  N )  =  1 )  ->  ( N  / L M )  e.  CC )
2221sqvald 11258 . . . . . 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 2336 . . . . 5  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  ( M  gcd  N )  =  1 )  ->  1  =  ( ( N  / L M )  x.  ( N  / L M ) ) )
2423oveq2d 5890 . . . 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 20565 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  / L N )  e.  ZZ )
265, 3, 25syl2anc 642 . . . . . 6  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  ( M  gcd  N )  =  1 )  ->  ( M  / L N )  e.  ZZ )
2726zcnd 10134 . . . . 5  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  ( M  gcd  N )  =  1 )  ->  ( M  / L N )  e.  CC )
2827, 21, 21mulassd 8874 . . . 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 2331 . . 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 8868 . . 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 734 . . . . 5  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  ( M  gcd  N )  =  1 )  ->  M  e.  NN )
32 simpllr 735 . . . . 5  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  ( M  gcd  N )  =  1 )  ->  -.  2  ||  M )
33 simplrr 737 . . . . 5  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  ( M  gcd  N )  =  1 )  ->  -.  2  ||  N )
3431, 32, 1, 33, 13lgsquad2 20615 . . . 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 5889 . . 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 2336 . 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 20588 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( M  / L N )  =/=  0  <->  ( M  gcd  N )  =  1 ) )
3837necon1bbid 2513 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( -.  ( M  gcd  N )  =  1  <->  ( M  / L N )  =  0 ) )
394, 2, 38syl2an 463 . . . . . 6  |-  ( ( M  e.  NN  /\  N  e.  NN )  ->  ( -.  ( M  gcd  N )  =  1  <->  ( M  / L N )  =  0 ) )
4039ad2ant2r 727 . . . . 5  |-  ( ( ( M  e.  NN  /\ 
-.  2  ||  M
)  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  -> 
( -.  ( M  gcd  N )  =  1  <->  ( M  / L N )  =  0 ) )
4140biimpa 470 . . . 4  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  -.  ( M  gcd  N )  =  1 )  ->  ( M  / L N )  =  0 )
42 neg1cn 9829 . . . . . . 7  |-  -u 1  e.  CC
4342a1i 10 . . . . . 6  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  -.  ( M  gcd  N )  =  1 )  ->  -u 1  e.  CC )
44 ax-1cn 8811 . . . . . . . 8  |-  1  e.  CC
45 ax-1ne0 8822 . . . . . . . 8  |-  1  =/=  0
4644, 45negne0i 9137 . . . . . . 7  |-  -u 1  =/=  0
4746a1i 10 . . . . . 6  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  -.  ( M  gcd  N )  =  1 )  ->  -u 1  =/=  0 )
484ad3antrrr 710 . . . . . . . . 9  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  -.  ( M  gcd  N )  =  1 )  ->  M  e.  ZZ )
49 simpllr 735 . . . . . . . . 9  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  -.  ( M  gcd  N )  =  1 )  ->  -.  2  ||  M )
50 1z 10069 . . . . . . . . . 10  |-  1  e.  ZZ
5150a1i 10 . . . . . . . . 9  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  -.  ( M  gcd  N )  =  1 )  ->  1  e.  ZZ )
52 2prm 12790 . . . . . . . . . 10  |-  2  e.  Prime
53 nprmdvds1 12806 . . . . . . . . . 10  |-  ( 2  e.  Prime  ->  -.  2  ||  1 )
5452, 53mp1i 11 . . . . . . . . 9  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  -.  ( M  gcd  N )  =  1 )  ->  -.  2  ||  1 )
55 omoe 12881 . . . . . . . . 9  |-  ( ( ( M  e.  ZZ  /\ 
-.  2  ||  M
)  /\  ( 1  e.  ZZ  /\  -.  2  ||  1 ) )  ->  2  ||  ( M  -  1 ) )
5648, 49, 51, 54, 55syl22anc 1183 . . . . . . . 8  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  -.  ( M  gcd  N )  =  1 )  ->  2  ||  ( M  -  1 ) )
57 2z 10070 . . . . . . . . . 10  |-  2  e.  ZZ
5857a1i 10 . . . . . . . . 9  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  -.  ( M  gcd  N )  =  1 )  ->  2  e.  ZZ )
59 2ne0 9845 . . . . . . . . . 10  |-  2  =/=  0
6059a1i 10 . . . . . . . . 9  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  -.  ( M  gcd  N )  =  1 )  ->  2  =/=  0 )
61 peano2zm 10078 . . . . . . . . . 10  |-  ( M  e.  ZZ  ->  ( M  -  1 )  e.  ZZ )
6248, 61syl 15 . . . . . . . . 9  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  -.  ( M  gcd  N )  =  1 )  ->  ( M  -  1 )  e.  ZZ )
63 dvdsval2 12550 . . . . . . . . 9  |-  ( ( 2  e.  ZZ  /\  2  =/=  0  /\  ( M  -  1 )  e.  ZZ )  -> 
( 2  ||  ( M  -  1 )  <-> 
( ( M  - 
1 )  /  2
)  e.  ZZ ) )
6458, 60, 62, 63syl3anc 1182 . . . . . . . 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 201 . . . . . . 7  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  -.  ( M  gcd  N )  =  1 )  ->  (
( M  -  1 )  /  2 )  e.  ZZ )
662adantr 451 . . . . . . . . . 10  |-  ( ( N  e.  NN  /\  -.  2  ||  N )  ->  N  e.  ZZ )
6766ad2antlr 707 . . . . . . . . 9  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  -.  ( M  gcd  N )  =  1 )  ->  N  e.  ZZ )
68 simplrr 737 . . . . . . . . 9  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  -.  ( M  gcd  N )  =  1 )  ->  -.  2  ||  N )
69 omoe 12881 . . . . . . . . 9  |-  ( ( ( N  e.  ZZ  /\ 
-.  2  ||  N
)  /\  ( 1  e.  ZZ  /\  -.  2  ||  1 ) )  ->  2  ||  ( N  -  1 ) )
7067, 68, 51, 54, 69syl22anc 1183 . . . . . . . 8  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  -.  ( M  gcd  N )  =  1 )  ->  2  ||  ( N  -  1 ) )
71 peano2zm 10078 . . . . . . . . . 10  |-  ( N  e.  ZZ  ->  ( N  -  1 )  e.  ZZ )
7267, 71syl 15 . . . . . . . . 9  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  -.  ( M  gcd  N )  =  1 )  ->  ( N  -  1 )  e.  ZZ )
73 dvdsval2 12550 . . . . . . . . 9  |-  ( ( 2  e.  ZZ  /\  2  =/=  0  /\  ( N  -  1 )  e.  ZZ )  -> 
( 2  ||  ( N  -  1 )  <-> 
( ( N  - 
1 )  /  2
)  e.  ZZ ) )
7458, 60, 72, 73syl3anc 1182 . . . . . . . 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 201 . . . . . . 7  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  -.  ( M  gcd  N )  =  1 )  ->  (
( N  -  1 )  /  2 )  e.  ZZ )
7665, 75zmulcld 10139 . . . . . 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 11266 . . . . 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 9027 . . . 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 2331 . . 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 20588 . . . . . . . . 9  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ )  ->  ( ( N  / L M )  =/=  0  <->  ( N  gcd  M )  =  1 ) )
8111eqeq1d 2304 . . . . . . . . 9  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ )  ->  ( ( N  gcd  M )  =  1  <->  ( M  gcd  N )  =  1 ) )
8280, 81bitrd 244 . . . . . . . 8  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ )  ->  ( ( N  / L M )  =/=  0  <->  ( M  gcd  N )  =  1 ) )
832, 4, 82syl2anr 464 . . . . . . 7  |-  ( ( M  e.  NN  /\  N  e.  NN )  ->  ( ( N  / L M )  =/=  0  <->  ( M  gcd  N )  =  1 ) )
8483necon1bbid 2513 . . . . . 6  |-  ( ( M  e.  NN  /\  N  e.  NN )  ->  ( -.  ( M  gcd  N )  =  1  <->  ( N  / L M )  =  0 ) )
8584ad2ant2r 727 . . . . 5  |-  ( ( ( M  e.  NN  /\ 
-.  2  ||  M
)  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  -> 
( -.  ( M  gcd  N )  =  1  <->  ( N  / L M )  =  0 ) )
8685biimpa 470 . . . 4  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  -.  ( M  gcd  N )  =  1 )  ->  ( N  / L M )  =  0 )
8786oveq2d 5890 . . 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 2331 . 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 766 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 176    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   CCcc 8751   RRcr 8752   0cc0 8753   1c1 8754    x. cmul 8758    - cmin 9053   -ucneg 9054    / cdiv 9439   NNcn 9762   2c2 9811   ZZcz 10040   ^cexp 11120   abscabs 11735    || cdivides 12547    gcd cgcd 12701   Primecprime 12774    / Lclgs 20549
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831  ax-addf 8832  ax-mulf 8833
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-disj 4010  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-tpos 6250  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-ec 6678  df-qs 6682  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-oi 7241  df-card 7588  df-cda 7810  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-q 10333  df-rp 10371  df-fz 10799  df-fzo 10887  df-fl 10941  df-mod 10990  df-seq 11063  df-exp 11121  df-hash 11354  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-clim 11978  df-sum 12175  df-dvds 12548  df-gcd 12702  df-prm 12775  df-phi 12850  df-pc 12906  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-starv 13239  df-sca 13240  df-vsca 13241  df-tset 13243  df-ple 13244  df-ds 13246  df-0g 13420  df-gsum 13421  df-imas 13427  df-divs 13428  df-mnd 14383  df-mhm 14431  df-submnd 14432  df-grp 14505  df-minusg 14506  df-sbg 14507  df-mulg 14508  df-subg 14634  df-nsg 14635  df-eqg 14636  df-ghm 14697  df-cntz 14809  df-cmn 15107  df-abl 15108  df-mgp 15342  df-rng 15356  df-cring 15357  df-ur 15358  df-oppr 15421  df-dvdsr 15439  df-unit 15440  df-invr 15470  df-dvr 15481  df-rnghom 15512  df-drng 15530  df-field 15531  df-subrg 15559  df-lmod 15645  df-lss 15706  df-lsp 15745  df-sra 15941  df-rgmod 15942  df-lidl 15943  df-rsp 15944  df-2idl 16000  df-nzr 16026  df-rlreg 16040  df-domn 16041  df-idom 16042  df-cnfld 16394  df-zrh 16471  df-zn 16474  df-lgs 20550
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