MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  lgsquad3 Unicode version

Theorem lgsquad3 20600
Description: Extend lgsquad2 20599 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 10045 . . . . . . . . . 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 10045 . . . . . . . . . 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 20549 . . . . . . . . 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 10117 . . . . . . 7  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  ( M  gcd  N )  =  1 )  ->  ( N  / L M )  e.  RR )
9 absresq 11787 . . . . . . 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 12699 . . . . . . . . . . 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 2315 . . . . . . . . 9  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  ( M  gcd  N )  =  1 )  ->  ( N  gcd  M )  =  1 )
15 lgsabs1 20573 . . . . . . . . . 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 5873 . . . . . . 7  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  ( M  gcd  N )  =  1 )  ->  ( ( abs `  ( N  / L M ) ) ^
2 )  =  ( 1 ^ 2 ) )
19 sq1 11198 . . . . . . 7  |-  ( 1 ^ 2 )  =  1
2018, 19syl6eq 2331 . . . . . 6  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  ( M  gcd  N )  =  1 )  ->  ( ( abs `  ( N  / L M ) ) ^
2 )  =  1 )
217zcnd 10118 . . . . . . 7  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  ( M  gcd  N )  =  1 )  ->  ( N  / L M )  e.  CC )
2221sqvald 11242 . . . . . 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 2323 . . . . 5  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  ( M  gcd  N )  =  1 )  ->  1  =  ( ( N  / L M )  x.  ( N  / L M ) ) )
2423oveq2d 5874 . . . 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 20549 . . . . . . 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 10118 . . . . 5  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  ( M  gcd  N )  =  1 )  ->  ( M  / L N )  e.  CC )
2827, 21, 21mulassd 8858 . . . 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 2318 . . 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 8852 . . 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 20599 . . . 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 5873 . . 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 2323 . 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 20572 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( M  / L N )  =/=  0  <->  ( M  gcd  N )  =  1 ) )
3837necon1bbid 2500 . . . . . . 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 9813 . . . . . . 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 8795 . . . . . . . 8  |-  1  e.  CC
45 ax-1ne0 8806 . . . . . . . 8  |-  1  =/=  0
4644, 45negne0i 9121 . . . . . . 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 10053 . . . . . . . . . 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 12774 . . . . . . . . . 10  |-  2  e.  Prime
53 nprmdvds1 12790 . . . . . . . . . 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 12865 . . . . . . . . 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 10054 . . . . . . . . . 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 9829 . . . . . . . . . 10  |-  2  =/=  0
6059a1i 10 . . . . . . . . 9  |-  ( ( ( ( M  e.  NN  /\  -.  2  ||  M )  /\  ( N  e.  NN  /\  -.  2  ||  N ) )  /\  -.  ( M  gcd  N )  =  1 )  ->  2  =/=  0 )
61 peano2zm 10062 . . . . . . . . . 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 12534 . . . . . . . . 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 12865 . . . . . . . . 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 10062 . . . . . . . . . 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 12534 . . . . . . . . 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 10123 . . . . . 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 11250 . . . . 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 9011 . . . 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 2318 . . 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 20572 . . . . . . . . 9  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ )  ->  ( ( N  / L M )  =/=  0  <->  ( N  gcd  M )  =  1 ) )
8111eqeq1d 2291 . . . . . . . . 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 2500 . . . . . 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 5874 . . 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 2318 . 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 1623    e. wcel 1684    =/= wne 2446   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738    x. cmul 8742    - cmin 9037   -ucneg 9038    / cdiv 9423   NNcn 9746   2c2 9795   ZZcz 10024   ^cexp 11104   abscabs 11719    || cdivides 12531    gcd cgcd 12685   Primecprime 12758    / Lclgs 20533
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816  ax-mulf 8817
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-disj 3994  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-tpos 6234  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-ec 6662  df-qs 6666  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-q 10317  df-rp 10355  df-fz 10783  df-fzo 10871  df-fl 10925  df-mod 10974  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962  df-sum 12159  df-dvds 12532  df-gcd 12686  df-prm 12759  df-phi 12834  df-pc 12890  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-starv 13223  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-0g 13404  df-gsum 13405  df-imas 13411  df-divs 13412  df-mnd 14367  df-mhm 14415  df-submnd 14416  df-grp 14489  df-minusg 14490  df-sbg 14491  df-mulg 14492  df-subg 14618  df-nsg 14619  df-eqg 14620  df-ghm 14681  df-cntz 14793  df-cmn 15091  df-abl 15092  df-mgp 15326  df-rng 15340  df-cring 15341  df-ur 15342  df-oppr 15405  df-dvdsr 15423  df-unit 15424  df-invr 15454  df-dvr 15465  df-rnghom 15496  df-drng 15514  df-field 15515  df-subrg 15543  df-lmod 15629  df-lss 15690  df-lsp 15729  df-sra 15925  df-rgmod 15926  df-lidl 15927  df-rsp 15928  df-2idl 15984  df-nzr 16010  df-rlreg 16024  df-domn 16025  df-idom 16026  df-cnfld 16378  df-zrh 16455  df-zn 16458  df-lgs 20534
  Copyright terms: Public domain W3C validator