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Theorem lgsquad3 23502
Description: Extend lgsquad2 23501 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 759 . . . . . . . . . 10  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  ( M  gcd  N )  =  1 )  ->  N  e.  NN )
2 nnz 10898 . . . . . . . . . 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 10898 . . . . . . . . . 10  |-  ( M  e.  NN  ->  M  e.  ZZ )
54ad3antrrr 729 . . . . . . . . 9  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  ( M  gcd  N )  =  1 )  ->  M  e.  ZZ )
6 lgscl 23451 . . . . . . . . 9  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ )  ->  ( N  /L
M )  e.  ZZ )
73, 5, 6syl2anc 661 . . . . . . . 8  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  ( M  gcd  N )  =  1 )  ->  ( N  /L M )  e.  ZZ )
87zred 10978 . . . . . . 7  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  ( M  gcd  N )  =  1 )  ->  ( N  /L M )  e.  RR )
9 absresq 13115 . . . . . . 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 14034 . . . . . . . . . . 11  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ )  ->  ( N  gcd  M
)  =  ( M  gcd  N ) )
123, 5, 11syl2anc 661 . . . . . . . . . 10  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  ( M  gcd  N )  =  1 )  ->  ( N  gcd  M )  =  ( M  gcd  N ) )
13 simpr 461 . . . . . . . . . 10  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  ( M  gcd  N )  =  1 )  ->  ( M  gcd  N )  =  1 )
1412, 13eqtrd 2508 . . . . . . . . 9  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  ( M  gcd  N )  =  1 )  ->  ( N  gcd  M )  =  1 )
15 lgsabs1 23475 . . . . . . . . . 10  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ )  ->  ( ( abs `  ( N  /L M ) )  =  1  <->  ( N  gcd  M )  =  1 ) )
163, 5, 15syl2anc 661 . . . . . . . . 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 232 . . . . . . . 8  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  ( M  gcd  N )  =  1 )  ->  ( abs `  ( N  /L M ) )  =  1 )
1817oveq1d 6310 . . . . . . 7  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  ( M  gcd  N )  =  1 )  ->  ( ( abs `  ( N  /L
M ) ) ^
2 )  =  ( 1 ^ 2 ) )
19 sq1 12242 . . . . . . 7  |-  ( 1 ^ 2 )  =  1
2018, 19syl6eq 2524 . . . . . 6  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  ( M  gcd  N )  =  1 )  ->  ( ( abs `  ( N  /L
M ) ) ^
2 )  =  1 )
217zcnd 10979 . . . . . . 7  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  ( M  gcd  N )  =  1 )  ->  ( N  /L M )  e.  CC )
2221sqvald 12287 . . . . . 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 2516 . . . . 5  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  ( M  gcd  N )  =  1 )  ->  1  =  ( ( N  /L
M )  x.  ( N  /L M ) ) )
2423oveq2d 6311 . . . 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 23451 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  /L
N )  e.  ZZ )
265, 3, 25syl2anc 661 . . . . . 6  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  ( M  gcd  N )  =  1 )  ->  ( M  /L N )  e.  ZZ )
2726zcnd 10979 . . . . 5  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  ( M  gcd  N )  =  1 )  ->  ( M  /L N )  e.  CC )
2827, 21, 21mulassd 9631 . . . 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 2511 . . 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 9625 . . 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 757 . . . . 5  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  ( M  gcd  N )  =  1 )  ->  M  e.  NN )
32 simpllr 758 . . . . 5  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  ( M  gcd  N )  =  1 )  ->  -.  2  ||  M )
33 simplrr 760 . . . . 5  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  ( M  gcd  N )  =  1 )  ->  -.  2  ||  N )
3431, 32, 1, 33, 13lgsquad2 23501 . . . 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 6310 . . 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 2516 . 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 23474 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( M  /L N )  =/=  0  <->  ( M  gcd  N )  =  1 ) )
3837necon1bbid 2717 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( -.  ( M  gcd  N )  =  1  <->  ( M  /L N )  =  0 ) )
394, 2, 38syl2an 477 . . . . . 6  |-  ( ( M  e.  NN  /\  N  e.  NN )  ->  ( -.  ( M  gcd  N )  =  1  <->  ( M  /L N )  =  0 ) )
4039ad2ant2r 746 . . . . 5  |-  ( ( ( M  e.  NN  /\ 
-.  2  ||  M
)  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  -> 
( -.  ( M  gcd  N )  =  1  <->  ( M  /L N )  =  0 ) )
4140biimpa 484 . . . 4  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  -.  ( M  gcd  N )  =  1 )  ->  ( M  /L N )  =  0 )
42 neg1cn 10651 . . . . . . 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 neg1ne0 10653 . . . . . . 7  |-  -u 1  =/=  0
4544a1i 11 . . . . . 6  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  -.  ( M  gcd  N )  =  1 )  ->  -u 1  =/=  0 )
464ad3antrrr 729 . . . . . . . . 9  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  -.  ( M  gcd  N )  =  1 )  ->  M  e.  ZZ )
47 simpllr 758 . . . . . . . . 9  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  -.  ( M  gcd  N )  =  1 )  ->  -.  2  ||  M )
48 1zzd 10907 . . . . . . . . 9  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  -.  ( M  gcd  N )  =  1 )  ->  1  e.  ZZ )
49 2prm 14109 . . . . . . . . . 10  |-  2  e.  Prime
50 nprmdvds1 14128 . . . . . . . . . 10  |-  ( 2  e.  Prime  ->  -.  2  ||  1 )
5149, 50mp1i 12 . . . . . . . . 9  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  -.  ( M  gcd  N )  =  1 )  ->  -.  2  ||  1 )
52 omoe 14212 . . . . . . . . 9  |-  ( ( ( M  e.  ZZ  /\ 
-.  2  ||  M
)  /\  ( 1  e.  ZZ  /\  -.  2  ||  1 ) )  ->  2  ||  ( M  -  1 ) )
5346, 47, 48, 51, 52syl22anc 1229 . . . . . . . 8  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  -.  ( M  gcd  N )  =  1 )  ->  2  ||  ( M  -  1 ) )
54 2z 10908 . . . . . . . . . 10  |-  2  e.  ZZ
5554a1i 11 . . . . . . . . 9  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  -.  ( M  gcd  N )  =  1 )  ->  2  e.  ZZ )
56 2ne0 10640 . . . . . . . . . 10  |-  2  =/=  0
5756a1i 11 . . . . . . . . 9  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  -.  ( M  gcd  N )  =  1 )  ->  2  =/=  0 )
58 peano2zm 10918 . . . . . . . . . 10  |-  ( M  e.  ZZ  ->  ( M  -  1 )  e.  ZZ )
5946, 58syl 16 . . . . . . . . 9  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  -.  ( M  gcd  N )  =  1 )  ->  ( M  -  1 )  e.  ZZ )
60 dvdsval2 13867 . . . . . . . . 9  |-  ( ( 2  e.  ZZ  /\  2  =/=  0  /\  ( M  -  1 )  e.  ZZ )  -> 
( 2  ||  ( M  -  1 )  <-> 
( ( M  - 
1 )  /  2
)  e.  ZZ ) )
6155, 57, 59, 60syl3anc 1228 . . . . . . . 8  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  -.  ( M  gcd  N )  =  1 )  ->  (
2  ||  ( M  -  1 )  <->  ( ( M  -  1 )  /  2 )  e.  ZZ ) )
6253, 61mpbid 210 . . . . . . 7  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  -.  ( M  gcd  N )  =  1 )  ->  (
( M  -  1 )  /  2 )  e.  ZZ )
632adantr 465 . . . . . . . . . 10  |-  ( ( N  e.  NN  /\  -.  2  ||  N )  ->  N  e.  ZZ )
6463ad2antlr 726 . . . . . . . . 9  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  -.  ( M  gcd  N )  =  1 )  ->  N  e.  ZZ )
65 simplrr 760 . . . . . . . . 9  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  -.  ( M  gcd  N )  =  1 )  ->  -.  2  ||  N )
66 omoe 14212 . . . . . . . . 9  |-  ( ( ( N  e.  ZZ  /\ 
-.  2  ||  N
)  /\  ( 1  e.  ZZ  /\  -.  2  ||  1 ) )  ->  2  ||  ( N  -  1 ) )
6764, 65, 48, 51, 66syl22anc 1229 . . . . . . . 8  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  -.  ( M  gcd  N )  =  1 )  ->  2  ||  ( N  -  1 ) )
68 peano2zm 10918 . . . . . . . . . 10  |-  ( N  e.  ZZ  ->  ( N  -  1 )  e.  ZZ )
6964, 68syl 16 . . . . . . . . 9  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  -.  ( M  gcd  N )  =  1 )  ->  ( N  -  1 )  e.  ZZ )
70 dvdsval2 13867 . . . . . . . . 9  |-  ( ( 2  e.  ZZ  /\  2  =/=  0  /\  ( N  -  1 )  e.  ZZ )  -> 
( 2  ||  ( N  -  1 )  <-> 
( ( N  - 
1 )  /  2
)  e.  ZZ ) )
7155, 57, 69, 70syl3anc 1228 . . . . . . . 8  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  -.  ( M  gcd  N )  =  1 )  ->  (
2  ||  ( N  -  1 )  <->  ( ( N  -  1 )  /  2 )  e.  ZZ ) )
7267, 71mpbid 210 . . . . . . 7  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  -.  ( M  gcd  N )  =  1 )  ->  (
( N  -  1 )  /  2 )  e.  ZZ )
7362, 72zmulcld 10984 . . . . . 6  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  -.  ( M  gcd  N )  =  1 )  ->  (
( ( M  - 
1 )  /  2
)  x.  ( ( N  -  1 )  /  2 ) )  e.  ZZ )
7443, 45, 73expclzd 12295 . . . . 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 )
7574mul01d 9790 . . . 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 )
7641, 75eqtr4d 2511 . . 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 ) )
77 lgsne0 23474 . . . . . . . . 9  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ )  ->  ( ( N  /L M )  =/=  0  <->  ( N  gcd  M )  =  1 ) )
7811eqeq1d 2469 . . . . . . . . 9  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ )  ->  ( ( N  gcd  M )  =  1  <->  ( M  gcd  N )  =  1 ) )
7977, 78bitrd 253 . . . . . . . 8  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ )  ->  ( ( N  /L M )  =/=  0  <->  ( M  gcd  N )  =  1 ) )
802, 4, 79syl2anr 478 . . . . . . 7  |-  ( ( M  e.  NN  /\  N  e.  NN )  ->  ( ( N  /L M )  =/=  0  <->  ( M  gcd  N )  =  1 ) )
8180necon1bbid 2717 . . . . . 6  |-  ( ( M  e.  NN  /\  N  e.  NN )  ->  ( -.  ( M  gcd  N )  =  1  <->  ( N  /L M )  =  0 ) )
8281ad2ant2r 746 . . . . 5  |-  ( ( ( M  e.  NN  /\ 
-.  2  ||  M
)  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  -> 
( -.  ( M  gcd  N )  =  1  <->  ( N  /L M )  =  0 ) )
8382biimpa 484 . . . 4  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  -.  ( M  gcd  N )  =  1 )  ->  ( N  /L M )  =  0 )
8483oveq2d 6311 . . 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 ) )
8576, 84eqtr4d 2511 . 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 ) ) )
8636, 85pm2.61dan 789 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   class class class wbr 4453   ` cfv 5594  (class class class)co 6295   CCcc 9502   RRcr 9503   0cc0 9504   1c1 9505    x. cmul 9509    - cmin 9817   -ucneg 9818    / cdiv 10218   NNcn 10548   2c2 10597   ZZcz 10876   ^cexp 12146   abscabs 13047    || cdivides 13864    gcd cgcd 14020   Primecprime 14093    /Lclgs 23435
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-inf2 8070  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-pre-sup 9582  ax-addf 9583  ax-mulf 9584
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-disj 4424  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-se 4845  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-of 6535  df-om 6696  df-1st 6795  df-2nd 6796  df-supp 6914  df-tpos 6967  df-recs 7054  df-rdg 7088  df-1o 7142  df-2o 7143  df-oadd 7146  df-er 7323  df-ec 7325  df-qs 7329  df-map 7434  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-fsupp 7842  df-sup 7913  df-oi 7947  df-card 8332  df-cda 8560  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-nn 10549  df-2 10606  df-3 10607  df-4 10608  df-5 10609  df-6 10610  df-7 10611  df-8 10612  df-9 10613  df-10 10614  df-n0 10808  df-z 10877  df-dec 10989  df-uz 11095  df-q 11195  df-rp 11233  df-fz 11685  df-fzo 11805  df-fl 11909  df-mod 11977  df-seq 12088  df-exp 12147  df-hash 12386  df-cj 12912  df-re 12913  df-im 12914  df-sqrt 13048  df-abs 13049  df-clim 13291  df-sum 13489  df-dvds 13865  df-gcd 14021  df-prm 14094  df-phi 14172  df-pc 14237  df-struct 14509  df-ndx 14510  df-slot 14511  df-base 14512  df-sets 14513  df-ress 14514  df-plusg 14585  df-mulr 14586  df-starv 14587  df-sca 14588  df-vsca 14589  df-ip 14590  df-tset 14591  df-ple 14592  df-ds 14594  df-unif 14595  df-0g 14714  df-gsum 14715  df-imas 14780  df-qus 14781  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-mhm 15839  df-submnd 15840  df-grp 15929  df-minusg 15930  df-sbg 15931  df-mulg 15932  df-subg 16070  df-nsg 16071  df-eqg 16072  df-ghm 16137  df-cntz 16227  df-cmn 16673  df-abl 16674  df-mgp 17014  df-ur 17026  df-ring 17072  df-cring 17073  df-oppr 17144  df-dvdsr 17162  df-unit 17163  df-invr 17193  df-dvr 17204  df-rnghom 17236  df-drng 17269  df-field 17270  df-subrg 17298  df-lmod 17385  df-lss 17450  df-lsp 17489  df-sra 17689  df-rgmod 17690  df-lidl 17691  df-rsp 17692  df-2idl 17750  df-nzr 17776  df-rlreg 17801  df-domn 17802  df-idom 17803  df-cnfld 18291  df-zring 18359  df-zrh 18410  df-zn 18413  df-lgs 23436
This theorem is referenced by: (None)
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