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Theorem gcdabs 11712
Description: The gcd of two integers is the same as that of their absolute values. (Contributed by Paul Chapman, 31-Mar-2011.)
Assertion
Ref Expression
gcdabs  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( abs `  M
)  gcd  ( abs `  N ) )  =  ( M  gcd  N
) )

Proof of Theorem gcdabs
StepHypRef Expression
1 zq 9445 . . 3  |-  ( M  e.  ZZ  ->  M  e.  QQ )
2 zq 9445 . . 3  |-  ( N  e.  ZZ  ->  N  e.  QQ )
3 qabsor 10879 . . . 4  |-  ( M  e.  QQ  ->  (
( abs `  M
)  =  M  \/  ( abs `  M )  =  -u M ) )
4 qabsor 10879 . . . 4  |-  ( N  e.  QQ  ->  (
( abs `  N
)  =  N  \/  ( abs `  N )  =  -u N ) )
53, 4anim12i 336 . . 3  |-  ( ( M  e.  QQ  /\  N  e.  QQ )  ->  ( ( ( abs `  M )  =  M  \/  ( abs `  M
)  =  -u M
)  /\  ( ( abs `  N )  =  N  \/  ( abs `  N )  =  -u N ) ) )
61, 2, 5syl2an 287 . 2  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( ( abs `  M )  =  M  \/  ( abs `  M
)  =  -u M
)  /\  ( ( abs `  N )  =  N  \/  ( abs `  N )  =  -u N ) ) )
7 oveq12 5791 . . . 4  |-  ( ( ( abs `  M
)  =  M  /\  ( abs `  N )  =  N )  -> 
( ( abs `  M
)  gcd  ( abs `  N ) )  =  ( M  gcd  N
) )
87a1i 9 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( ( abs `  M )  =  M  /\  ( abs `  N
)  =  N )  ->  ( ( abs `  M )  gcd  ( abs `  N ) )  =  ( M  gcd  N ) ) )
9 oveq12 5791 . . . . 5  |-  ( ( ( abs `  M
)  =  -u M  /\  ( abs `  N
)  =  N )  ->  ( ( abs `  M )  gcd  ( abs `  N ) )  =  ( -u M  gcd  N ) )
10 neggcd 11707 . . . . 5  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( -u M  gcd  N )  =  ( M  gcd  N ) )
119, 10sylan9eqr 2195 . . . 4  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( ( abs `  M )  =  -u M  /\  ( abs `  N
)  =  N ) )  ->  ( ( abs `  M )  gcd  ( abs `  N
) )  =  ( M  gcd  N ) )
1211ex 114 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( ( abs `  M )  =  -u M  /\  ( abs `  N
)  =  N )  ->  ( ( abs `  M )  gcd  ( abs `  N ) )  =  ( M  gcd  N ) ) )
13 oveq12 5791 . . . . 5  |-  ( ( ( abs `  M
)  =  M  /\  ( abs `  N )  =  -u N )  -> 
( ( abs `  M
)  gcd  ( abs `  N ) )  =  ( M  gcd  -u N
) )
14 gcdneg 11706 . . . . 5  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  gcd  -u N
)  =  ( M  gcd  N ) )
1513, 14sylan9eqr 2195 . . . 4  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( ( abs `  M )  =  M  /\  ( abs `  N
)  =  -u N
) )  ->  (
( abs `  M
)  gcd  ( abs `  N ) )  =  ( M  gcd  N
) )
1615ex 114 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( ( abs `  M )  =  M  /\  ( abs `  N
)  =  -u N
)  ->  ( ( abs `  M )  gcd  ( abs `  N
) )  =  ( M  gcd  N ) ) )
17 oveq12 5791 . . . . 5  |-  ( ( ( abs `  M
)  =  -u M  /\  ( abs `  N
)  =  -u N
)  ->  ( ( abs `  M )  gcd  ( abs `  N
) )  =  (
-u M  gcd  -u N
) )
18 znegcl 9109 . . . . . . 7  |-  ( M  e.  ZZ  ->  -u M  e.  ZZ )
19 gcdneg 11706 . . . . . . 7  |-  ( (
-u M  e.  ZZ  /\  N  e.  ZZ )  ->  ( -u M  gcd  -u N )  =  ( -u M  gcd  N ) )
2018, 19sylan 281 . . . . . 6  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( -u M  gcd  -u N )  =  (
-u M  gcd  N
) )
2120, 10eqtrd 2173 . . . . 5  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( -u M  gcd  -u N )  =  ( M  gcd  N ) )
2217, 21sylan9eqr 2195 . . . 4  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( ( abs `  M )  =  -u M  /\  ( abs `  N
)  =  -u N
) )  ->  (
( abs `  M
)  gcd  ( abs `  N ) )  =  ( M  gcd  N
) )
2322ex 114 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( ( abs `  M )  =  -u M  /\  ( abs `  N
)  =  -u N
)  ->  ( ( abs `  M )  gcd  ( abs `  N
) )  =  ( M  gcd  N ) ) )
248, 12, 16, 23ccased 950 . 2  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( ( ( abs `  M )  =  M  \/  ( abs `  M )  = 
-u M )  /\  ( ( abs `  N
)  =  N  \/  ( abs `  N )  =  -u N ) )  ->  ( ( abs `  M )  gcd  ( abs `  N ) )  =  ( M  gcd  N ) ) )
256, 24mpd 13 1  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( abs `  M
)  gcd  ( abs `  N ) )  =  ( M  gcd  N
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    \/ wo 698    = wceq 1332    e. wcel 1481   ` cfv 5131  (class class class)co 5782   -ucneg 7958   ZZcz 9078   QQcq 9438   abscabs 10801    gcd cgcd 11671
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4051  ax-sep 4054  ax-nul 4062  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-iinf 4510  ax-cnex 7735  ax-resscn 7736  ax-1cn 7737  ax-1re 7738  ax-icn 7739  ax-addcl 7740  ax-addrcl 7741  ax-mulcl 7742  ax-mulrcl 7743  ax-addcom 7744  ax-mulcom 7745  ax-addass 7746  ax-mulass 7747  ax-distr 7748  ax-i2m1 7749  ax-0lt1 7750  ax-1rid 7751  ax-0id 7752  ax-rnegex 7753  ax-precex 7754  ax-cnre 7755  ax-pre-ltirr 7756  ax-pre-ltwlin 7757  ax-pre-lttrn 7758  ax-pre-apti 7759  ax-pre-ltadd 7760  ax-pre-mulgt0 7761  ax-pre-mulext 7762  ax-arch 7763  ax-caucvg 7764
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-reu 2424  df-rmo 2425  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-if 3480  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-int 3780  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-tr 4035  df-id 4223  df-po 4226  df-iso 4227  df-iord 4296  df-on 4298  df-ilim 4299  df-suc 4301  df-iom 4513  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-riota 5738  df-ov 5785  df-oprab 5786  df-mpo 5787  df-1st 6046  df-2nd 6047  df-recs 6210  df-frec 6296  df-sup 6879  df-pnf 7826  df-mnf 7827  df-xr 7828  df-ltxr 7829  df-le 7830  df-sub 7959  df-neg 7960  df-reap 8361  df-ap 8368  df-div 8457  df-inn 8745  df-2 8803  df-3 8804  df-4 8805  df-n0 9002  df-z 9079  df-uz 9351  df-q 9439  df-rp 9471  df-fz 9822  df-fzo 9951  df-fl 10074  df-mod 10127  df-seqfrec 10250  df-exp 10324  df-cj 10646  df-re 10647  df-im 10648  df-rsqrt 10802  df-abs 10803  df-dvds 11530  df-gcd 11672
This theorem is referenced by:  absmulgcd  11741  lcmgcd  11795  lcmgcdeq  11800  zgcdsq  11915
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