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Theorem gcdabs 12809
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 zre 10120 . . 3  |-  ( M  e.  ZZ  ->  M  e.  RR )
2 zre 10120 . . 3  |-  ( N  e.  ZZ  ->  N  e.  RR )
3 absor 11881 . . . 4  |-  ( M  e.  RR  ->  (
( abs `  M
)  =  M  \/  ( abs `  M )  =  -u M ) )
4 absor 11881 . . . 4  |-  ( N  e.  RR  ->  (
( abs `  N
)  =  N  \/  ( abs `  N )  =  -u N ) )
53, 4anim12i 549 . . 3  |-  ( ( M  e.  RR  /\  N  e.  RR )  ->  ( ( ( abs `  M )  =  M  \/  ( abs `  M
)  =  -u M
)  /\  ( ( abs `  N )  =  N  \/  ( abs `  N )  =  -u N ) ) )
61, 2, 5syl2an 463 . 2  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( ( abs `  M )  =  M  \/  ( abs `  M
)  =  -u M
)  /\  ( ( abs `  N )  =  N  \/  ( abs `  N )  =  -u N ) ) )
7 oveq12 5954 . . . 4  |-  ( ( ( abs `  M
)  =  M  /\  ( abs `  N )  =  N )  -> 
( ( abs `  M
)  gcd  ( abs `  N ) )  =  ( M  gcd  N
) )
87a1i 10 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( ( abs `  M )  =  M  /\  ( abs `  N
)  =  N )  ->  ( ( abs `  M )  gcd  ( abs `  N ) )  =  ( M  gcd  N ) ) )
9 oveq12 5954 . . . . 5  |-  ( ( ( abs `  M
)  =  -u M  /\  ( abs `  N
)  =  N )  ->  ( ( abs `  M )  gcd  ( abs `  N ) )  =  ( -u M  gcd  N ) )
10 neggcd 12803 . . . . 5  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( -u M  gcd  N )  =  ( M  gcd  N ) )
119, 10sylan9eqr 2412 . . . 4  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( ( abs `  M )  =  -u M  /\  ( abs `  N
)  =  N ) )  ->  ( ( abs `  M )  gcd  ( abs `  N
) )  =  ( M  gcd  N ) )
1211ex 423 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( ( abs `  M )  =  -u M  /\  ( abs `  N
)  =  N )  ->  ( ( abs `  M )  gcd  ( abs `  N ) )  =  ( M  gcd  N ) ) )
13 oveq12 5954 . . . . 5  |-  ( ( ( abs `  M
)  =  M  /\  ( abs `  N )  =  -u N )  -> 
( ( abs `  M
)  gcd  ( abs `  N ) )  =  ( M  gcd  -u N
) )
14 gcdneg 12802 . . . . 5  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  gcd  -u N
)  =  ( M  gcd  N ) )
1513, 14sylan9eqr 2412 . . . 4  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( ( abs `  M )  =  M  /\  ( abs `  N
)  =  -u N
) )  ->  (
( abs `  M
)  gcd  ( abs `  N ) )  =  ( M  gcd  N
) )
1615ex 423 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( ( abs `  M )  =  M  /\  ( abs `  N
)  =  -u N
)  ->  ( ( abs `  M )  gcd  ( abs `  N
) )  =  ( M  gcd  N ) ) )
17 oveq12 5954 . . . . 5  |-  ( ( ( abs `  M
)  =  -u M  /\  ( abs `  N
)  =  -u N
)  ->  ( ( abs `  M )  gcd  ( abs `  N
) )  =  (
-u M  gcd  -u N
) )
18 znegcl 10147 . . . . . . 7  |-  ( M  e.  ZZ  ->  -u M  e.  ZZ )
19 gcdneg 12802 . . . . . . 7  |-  ( (
-u M  e.  ZZ  /\  N  e.  ZZ )  ->  ( -u M  gcd  -u N )  =  ( -u M  gcd  N ) )
2018, 19sylan 457 . . . . . 6  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( -u M  gcd  -u N )  =  (
-u M  gcd  N
) )
2120, 10eqtrd 2390 . . . . 5  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( -u M  gcd  -u N )  =  ( M  gcd  N ) )
2217, 21sylan9eqr 2412 . . . 4  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( ( abs `  M )  =  -u M  /\  ( abs `  N
)  =  -u N
) )  ->  (
( abs `  M
)  gcd  ( abs `  N ) )  =  ( M  gcd  N
) )
2322ex 423 . . 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 913 . 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 14 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    \/ wo 357    /\ wa 358    = wceq 1642    e. wcel 1710   ` cfv 5337  (class class class)co 5945   RRcr 8826   -ucneg 9128   ZZcz 10116   abscabs 11815    gcd cgcd 12782
This theorem is referenced by:  absmulgcd  12823  zgcdsq  12921  lgsne0  20684
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594  ax-cnex 8883  ax-resscn 8884  ax-1cn 8885  ax-icn 8886  ax-addcl 8887  ax-addrcl 8888  ax-mulcl 8889  ax-mulrcl 8890  ax-mulcom 8891  ax-addass 8892  ax-mulass 8893  ax-distr 8894  ax-i2m1 8895  ax-1ne0 8896  ax-1rid 8897  ax-rnegex 8898  ax-rrecex 8899  ax-cnre 8900  ax-pre-lttri 8901  ax-pre-lttrn 8902  ax-pre-ltadd 8903  ax-pre-mulgt0 8904  ax-pre-sup 8905
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3909  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-tr 4195  df-eprel 4387  df-id 4391  df-po 4396  df-so 4397  df-fr 4434  df-we 4436  df-ord 4477  df-on 4478  df-lim 4479  df-suc 4480  df-om 4739  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-2nd 6210  df-riota 6391  df-recs 6475  df-rdg 6510  df-er 6747  df-en 6952  df-dom 6953  df-sdom 6954  df-sup 7284  df-pnf 8959  df-mnf 8960  df-xr 8961  df-ltxr 8962  df-le 8963  df-sub 9129  df-neg 9130  df-div 9514  df-nn 9837  df-2 9894  df-3 9895  df-n0 10058  df-z 10117  df-uz 10323  df-rp 10447  df-seq 11139  df-exp 11198  df-cj 11680  df-re 11681  df-im 11682  df-sqr 11816  df-abs 11817  df-dvds 12629  df-gcd 12783
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