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Theorem gcdneg 12721
Description: Negating one operand of the  gcd operator does not alter the result. (Contributed by Paul Chapman, 21-Mar-2011.)
Assertion
Ref Expression
gcdneg  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  gcd  -u N
)  =  ( M  gcd  N ) )

Proof of Theorem gcdneg
StepHypRef Expression
1 oveq12 5883 . . . . 5  |-  ( ( M  =  0  /\  N  =  0 )  ->  ( M  gcd  N )  =  ( 0  gcd  0 ) )
21adantl 452 . . . 4  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =  0  /\  N  =  0 ) )  -> 
( M  gcd  N
)  =  ( 0  gcd  0 ) )
3 zcn 10045 . . . . . . . . 9  |-  ( N  e.  ZZ  ->  N  e.  CC )
43negeq0d 9165 . . . . . . . 8  |-  ( N  e.  ZZ  ->  ( N  =  0  <->  -u N  =  0 ) )
54anbi2d 684 . . . . . . 7  |-  ( N  e.  ZZ  ->  (
( M  =  0  /\  N  =  0 )  <->  ( M  =  0  /\  -u N  =  0 ) ) )
65adantl 452 . . . . . 6  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( M  =  0  /\  N  =  0 )  <->  ( M  =  0  /\  -u N  =  0 ) ) )
7 oveq12 5883 . . . . . 6  |-  ( ( M  =  0  /\  -u N  =  0
)  ->  ( M  gcd  -u N )  =  ( 0  gcd  0
) )
86, 7syl6bi 219 . . . . 5  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( M  =  0  /\  N  =  0 )  ->  ( M  gcd  -u N )  =  ( 0  gcd  0
) ) )
98imp 418 . . . 4  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =  0  /\  N  =  0 ) )  -> 
( M  gcd  -u N
)  =  ( 0  gcd  0 ) )
102, 9eqtr4d 2331 . . 3  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =  0  /\  N  =  0 ) )  -> 
( M  gcd  N
)  =  ( M  gcd  -u N ) )
11 gcddvds 12710 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( M  gcd  N )  ||  M  /\  ( M  gcd  N ) 
||  N ) )
12 gcdcl 12712 . . . . . . . . . 10  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  gcd  N
)  e.  NN0 )
1312nn0zd 10131 . . . . . . . . 9  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  gcd  N
)  e.  ZZ )
14 dvdsnegb 12562 . . . . . . . . 9  |-  ( ( ( M  gcd  N
)  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( M  gcd  N )  ||  N  <->  ( M  gcd  N )  ||  -u N
) )
1513, 14sylancom 648 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( M  gcd  N )  ||  N  <->  ( M  gcd  N )  ||  -u N
) )
1615anbi2d 684 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( ( M  gcd  N )  ||  M  /\  ( M  gcd  N )  ||  N )  <-> 
( ( M  gcd  N )  ||  M  /\  ( M  gcd  N ) 
||  -u N ) ) )
1711, 16mpbid 201 . . . . . 6  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( M  gcd  N )  ||  M  /\  ( M  gcd  N ) 
||  -u N ) )
186notbid 285 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( -.  ( M  =  0  /\  N  =  0 )  <->  -.  ( M  =  0  /\  -u N  =  0 ) ) )
19 simpl 443 . . . . . . . . 9  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  M  e.  ZZ )
20 znegcl 10071 . . . . . . . . . 10  |-  ( N  e.  ZZ  ->  -u N  e.  ZZ )
2120adantl 452 . . . . . . . . 9  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  -> 
-u N  e.  ZZ )
22 dvdslegcd 12711 . . . . . . . . . 10  |-  ( ( ( ( M  gcd  N )  e.  ZZ  /\  M  e.  ZZ  /\  -u N  e.  ZZ )  /\  -.  ( M  =  0  /\  -u N  =  0 ) )  ->  (
( ( M  gcd  N )  ||  M  /\  ( M  gcd  N ) 
||  -u N )  -> 
( M  gcd  N
)  <_  ( M  gcd  -u N ) ) )
2322ex 423 . . . . . . . . 9  |-  ( ( ( M  gcd  N
)  e.  ZZ  /\  M  e.  ZZ  /\  -u N  e.  ZZ )  ->  ( -.  ( M  =  0  /\  -u N  =  0 )  ->  ( (
( M  gcd  N
)  ||  M  /\  ( M  gcd  N ) 
||  -u N )  -> 
( M  gcd  N
)  <_  ( M  gcd  -u N ) ) ) )
2413, 19, 21, 23syl3anc 1182 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( -.  ( M  =  0  /\  -u N  =  0 )  -> 
( ( ( M  gcd  N )  ||  M  /\  ( M  gcd  N )  ||  -u N
)  ->  ( M  gcd  N )  <_  ( M  gcd  -u N ) ) ) )
2518, 24sylbid 206 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( -.  ( M  =  0  /\  N  =  0 )  -> 
( ( ( M  gcd  N )  ||  M  /\  ( M  gcd  N )  ||  -u N
)  ->  ( M  gcd  N )  <_  ( M  gcd  -u N ) ) ) )
2625com12 27 . . . . . 6  |-  ( -.  ( M  =  0  /\  N  =  0 )  ->  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( ( M  gcd  N )  ||  M  /\  ( M  gcd  N ) 
||  -u N )  -> 
( M  gcd  N
)  <_  ( M  gcd  -u N ) ) ) )
2717, 26mpdi 38 . . . . 5  |-  ( -.  ( M  =  0  /\  N  =  0 )  ->  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  gcd  N )  <_ 
( M  gcd  -u N
) ) )
2827impcom 419 . . . 4  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  /\  N  =  0 ) )  ->  ( M  gcd  N )  <_  ( M  gcd  -u N ) )
29 gcddvds 12710 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  -u N  e.  ZZ )  ->  ( ( M  gcd  -u N )  ||  M  /\  ( M  gcd  -u N )  ||  -u N
) )
3020, 29sylan2 460 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( M  gcd  -u N )  ||  M  /\  ( M  gcd  -u N
)  ||  -u N ) )
31 gcdcl 12712 . . . . . . . . . . 11  |-  ( ( M  e.  ZZ  /\  -u N  e.  ZZ )  ->  ( M  gcd  -u N )  e.  NN0 )
3231nn0zd 10131 . . . . . . . . . 10  |-  ( ( M  e.  ZZ  /\  -u N  e.  ZZ )  ->  ( M  gcd  -u N )  e.  ZZ )
3320, 32sylan2 460 . . . . . . . . 9  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  gcd  -u N
)  e.  ZZ )
34 dvdsnegb 12562 . . . . . . . . 9  |-  ( ( ( M  gcd  -u N
)  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( M  gcd  -u N )  ||  N  <->  ( M  gcd  -u N
)  ||  -u N ) )
3533, 34sylancom 648 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( M  gcd  -u N )  ||  N  <->  ( M  gcd  -u N
)  ||  -u N ) )
3635anbi2d 684 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( ( M  gcd  -u N )  ||  M  /\  ( M  gcd  -u N )  ||  N
)  <->  ( ( M  gcd  -u N )  ||  M  /\  ( M  gcd  -u N )  ||  -u N
) ) )
3730, 36mpbird 223 . . . . . 6  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( M  gcd  -u N )  ||  M  /\  ( M  gcd  -u N
)  ||  N )
)
38 simpr 447 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  N  e.  ZZ )
39 dvdslegcd 12711 . . . . . . . . 9  |-  ( ( ( ( M  gcd  -u N )  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  /\  N  =  0 ) )  -> 
( ( ( M  gcd  -u N )  ||  M  /\  ( M  gcd  -u N )  ||  N
)  ->  ( M  gcd  -u N )  <_ 
( M  gcd  N
) ) )
4039ex 423 . . . . . . . 8  |-  ( ( ( M  gcd  -u N
)  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( -.  ( M  =  0  /\  N  =  0 )  ->  ( (
( M  gcd  -u N
)  ||  M  /\  ( M  gcd  -u N
)  ||  N )  ->  ( M  gcd  -u N
)  <_  ( M  gcd  N ) ) ) )
4133, 19, 38, 40syl3anc 1182 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( -.  ( M  =  0  /\  N  =  0 )  -> 
( ( ( M  gcd  -u N )  ||  M  /\  ( M  gcd  -u N )  ||  N
)  ->  ( M  gcd  -u N )  <_ 
( M  gcd  N
) ) ) )
4241com12 27 . . . . . 6  |-  ( -.  ( M  =  0  /\  N  =  0 )  ->  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( ( M  gcd  -u N )  ||  M  /\  ( M  gcd  -u N
)  ||  N )  ->  ( M  gcd  -u N
)  <_  ( M  gcd  N ) ) ) )
4337, 42mpdi 38 . . . . 5  |-  ( -.  ( M  =  0  /\  N  =  0 )  ->  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  gcd  -u N )  <_ 
( M  gcd  N
) ) )
4443impcom 419 . . . 4  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  /\  N  =  0 ) )  ->  ( M  gcd  -u N )  <_  ( M  gcd  N ) )
4513zred 10133 . . . . . 6  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  gcd  N
)  e.  RR )
4633zred 10133 . . . . . 6  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  gcd  -u N
)  e.  RR )
4745, 46letri3d 8977 . . . . 5  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( M  gcd  N )  =  ( M  gcd  -u N )  <->  ( ( M  gcd  N )  <_ 
( M  gcd  -u N
)  /\  ( M  gcd  -u N )  <_ 
( M  gcd  N
) ) ) )
4847adantr 451 . . . 4  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  /\  N  =  0 ) )  ->  ( ( M  gcd  N )  =  ( M  gcd  -u N
)  <->  ( ( M  gcd  N )  <_ 
( M  gcd  -u N
)  /\  ( M  gcd  -u N )  <_ 
( M  gcd  N
) ) ) )
4928, 44, 48mpbir2and 888 . . 3  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  /\  N  =  0 ) )  ->  ( M  gcd  N )  =  ( M  gcd  -u N ) )
5010, 49pm2.61dan 766 . 2  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  gcd  N
)  =  ( M  gcd  -u N ) )
5150eqcomd 2301 1  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  gcd  -u N
)  =  ( M  gcd  N ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   class class class wbr 4039  (class class class)co 5874   0cc0 8753    <_ cle 8884   -ucneg 9054   ZZcz 10040    || cdivides 12547    gcd cgcd 12701
This theorem is referenced by:  neggcd  12722  gcdabs  12728  odinv  14890
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-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  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
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-iun 3923  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-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-ov 5877  df-oprab 5878  df-mpt2 5879  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-sup 7210  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-n0 9982  df-z 10041  df-uz 10247  df-rp 10371  df-seq 11063  df-exp 11121  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-dvds 12548  df-gcd 12702
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