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Theorem gcdval 11033
Description: The value of the  gcd operator.  ( M  gcd  N ) is the greatest common divisor of  M and  N. If  M and  N are both  0, the result is defined conventionally as  0. (Contributed by Paul Chapman, 21-Mar-2011.) (Revised by Mario Carneiro, 10-Nov-2013.)
Assertion
Ref Expression
gcdval  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  gcd  N
)  =  if ( ( M  =  0  /\  N  =  0 ) ,  0 ,  sup ( { n  e.  ZZ  |  ( n 
||  M  /\  n  ||  N ) } ,  RR ,  <  ) ) )
Distinct variable groups:    n, M    n, N

Proof of Theorem gcdval
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr 108 . . . . 5  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =  0  /\  N  =  0 ) )  -> 
( M  =  0  /\  N  =  0 ) )
21iftrued 3396 . . . 4  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =  0  /\  N  =  0 ) )  ->  if ( ( M  =  0  /\  N  =  0 ) ,  0 ,  sup ( { n  e.  ZZ  | 
( n  ||  M  /\  n  ||  N ) } ,  RR ,  <  ) )  =  0 )
3 0nn0 8658 . . . 4  |-  0  e.  NN0
42, 3syl6eqel 2178 . . 3  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =  0  /\  N  =  0 ) )  ->  if ( ( M  =  0  /\  N  =  0 ) ,  0 ,  sup ( { n  e.  ZZ  | 
( n  ||  M  /\  n  ||  N ) } ,  RR ,  <  ) )  e.  NN0 )
5 simpr 108 . . . . . 6  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  /\  N  =  0 ) )  ->  -.  ( M  =  0  /\  N  =  0 ) )
65iffalsed 3399 . . . . 5  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  /\  N  =  0 ) )  ->  if ( ( M  =  0  /\  N  =  0 ) ,  0 ,  sup ( { n  e.  ZZ  |  ( n  ||  M  /\  n  ||  N
) } ,  RR ,  <  ) )  =  sup ( { n  e.  ZZ  |  ( n 
||  M  /\  n  ||  N ) } ,  RR ,  <  ) )
7 gcdsupcl 11032 . . . . 5  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  /\  N  =  0 ) )  ->  sup ( { n  e.  ZZ  |  ( n 
||  M  /\  n  ||  N ) } ,  RR ,  <  )  e.  NN )
86, 7eqeltrd 2164 . . . 4  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  /\  N  =  0 ) )  ->  if ( ( M  =  0  /\  N  =  0 ) ,  0 ,  sup ( { n  e.  ZZ  |  ( n  ||  M  /\  n  ||  N
) } ,  RR ,  <  ) )  e.  NN )
98nnnn0d 8696 . . 3  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  /\  N  =  0 ) )  ->  if ( ( M  =  0  /\  N  =  0 ) ,  0 ,  sup ( { n  e.  ZZ  |  ( n  ||  M  /\  n  ||  N
) } ,  RR ,  <  ) )  e. 
NN0 )
10 gcdmndc 11022 . . . 4  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  -> DECID  ( M  =  0  /\  N  =  0 ) )
11 exmiddc 782 . . . 4  |-  (DECID  ( M  =  0  /\  N  =  0 )  -> 
( ( M  =  0  /\  N  =  0 )  \/  -.  ( M  =  0  /\  N  =  0
) ) )
1210, 11syl 14 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( M  =  0  /\  N  =  0 )  \/  -.  ( M  =  0  /\  N  =  0
) ) )
134, 9, 12mpjaodan 747 . 2  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  if ( ( M  =  0  /\  N  =  0 ) ,  0 ,  sup ( { n  e.  ZZ  |  ( n  ||  M  /\  n  ||  N
) } ,  RR ,  <  ) )  e. 
NN0 )
14 eqeq1 2094 . . . . 5  |-  ( x  =  M  ->  (
x  =  0  <->  M  =  0 ) )
1514anbi1d 453 . . . 4  |-  ( x  =  M  ->  (
( x  =  0  /\  y  =  0 )  <->  ( M  =  0  /\  y  =  0 ) ) )
16 breq2 3841 . . . . . . 7  |-  ( x  =  M  ->  (
n  ||  x  <->  n  ||  M
) )
1716anbi1d 453 . . . . . 6  |-  ( x  =  M  ->  (
( n  ||  x  /\  n  ||  y )  <-> 
( n  ||  M  /\  n  ||  y ) ) )
1817rabbidv 2608 . . . . 5  |-  ( x  =  M  ->  { n  e.  ZZ  |  ( n 
||  x  /\  n  ||  y ) }  =  { n  e.  ZZ  |  ( n  ||  M  /\  n  ||  y
) } )
1918supeq1d 6661 . . . 4  |-  ( x  =  M  ->  sup ( { n  e.  ZZ  |  ( n  ||  x  /\  n  ||  y
) } ,  RR ,  <  )  =  sup ( { n  e.  ZZ  |  ( n  ||  M  /\  n  ||  y
) } ,  RR ,  <  ) )
2015, 19ifbieq2d 3411 . . 3  |-  ( x  =  M  ->  if ( ( x  =  0  /\  y  =  0 ) ,  0 ,  sup ( { n  e.  ZZ  | 
( n  ||  x  /\  n  ||  y ) } ,  RR ,  <  ) )  =  if ( ( M  =  0  /\  y  =  0 ) ,  0 ,  sup ( { n  e.  ZZ  | 
( n  ||  M  /\  n  ||  y ) } ,  RR ,  <  ) ) )
21 eqeq1 2094 . . . . 5  |-  ( y  =  N  ->  (
y  =  0  <->  N  =  0 ) )
2221anbi2d 452 . . . 4  |-  ( y  =  N  ->  (
( M  =  0  /\  y  =  0 )  <->  ( M  =  0  /\  N  =  0 ) ) )
23 breq2 3841 . . . . . . 7  |-  ( y  =  N  ->  (
n  ||  y  <->  n  ||  N
) )
2423anbi2d 452 . . . . . 6  |-  ( y  =  N  ->  (
( n  ||  M  /\  n  ||  y )  <-> 
( n  ||  M  /\  n  ||  N ) ) )
2524rabbidv 2608 . . . . 5  |-  ( y  =  N  ->  { n  e.  ZZ  |  ( n 
||  M  /\  n  ||  y ) }  =  { n  e.  ZZ  |  ( n  ||  M  /\  n  ||  N
) } )
2625supeq1d 6661 . . . 4  |-  ( y  =  N  ->  sup ( { n  e.  ZZ  |  ( n  ||  M  /\  n  ||  y
) } ,  RR ,  <  )  =  sup ( { n  e.  ZZ  |  ( n  ||  M  /\  n  ||  N
) } ,  RR ,  <  ) )
2722, 26ifbieq2d 3411 . . 3  |-  ( y  =  N  ->  if ( ( M  =  0  /\  y  =  0 ) ,  0 ,  sup ( { n  e.  ZZ  | 
( n  ||  M  /\  n  ||  y ) } ,  RR ,  <  ) )  =  if ( ( M  =  0  /\  N  =  0 ) ,  0 ,  sup ( { n  e.  ZZ  | 
( n  ||  M  /\  n  ||  N ) } ,  RR ,  <  ) ) )
28 df-gcd 11021 . . 3  |-  gcd  =  ( x  e.  ZZ ,  y  e.  ZZ  |->  if ( ( x  =  0  /\  y  =  0 ) ,  0 ,  sup ( { n  e.  ZZ  | 
( n  ||  x  /\  n  ||  y ) } ,  RR ,  <  ) ) )
2920, 27, 28ovmpt2g 5761 . 2  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  if ( ( M  =  0  /\  N  =  0 ) ,  0 ,  sup ( { n  e.  ZZ  | 
( n  ||  M  /\  n  ||  N ) } ,  RR ,  <  ) )  e.  NN0 )  ->  ( M  gcd  N )  =  if ( ( M  =  0  /\  N  =  0 ) ,  0 ,  sup ( { n  e.  ZZ  |  ( n 
||  M  /\  n  ||  N ) } ,  RR ,  <  ) ) )
3013, 29mpd3an3 1274 1  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  gcd  N
)  =  if ( ( M  =  0  /\  N  =  0 ) ,  0 ,  sup ( { n  e.  ZZ  |  ( n 
||  M  /\  n  ||  N ) } ,  RR ,  <  ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    \/ wo 664  DECID wdc 780    = wceq 1289    e. wcel 1438   {crab 2363   ifcif 3389   class class class wbr 3837  (class class class)co 5634   supcsup 6656   RRcr 7328   0cc0 7329    < clt 7501   NNcn 8394   NN0cn0 8643   ZZcz 8720    || cdvds 10878    gcd cgcd 11020
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3946  ax-sep 3949  ax-nul 3957  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-iinf 4393  ax-cnex 7415  ax-resscn 7416  ax-1cn 7417  ax-1re 7418  ax-icn 7419  ax-addcl 7420  ax-addrcl 7421  ax-mulcl 7422  ax-mulrcl 7423  ax-addcom 7424  ax-mulcom 7425  ax-addass 7426  ax-mulass 7427  ax-distr 7428  ax-i2m1 7429  ax-0lt1 7430  ax-1rid 7431  ax-0id 7432  ax-rnegex 7433  ax-precex 7434  ax-cnre 7435  ax-pre-ltirr 7436  ax-pre-ltwlin 7437  ax-pre-lttrn 7438  ax-pre-apti 7439  ax-pre-ltadd 7440  ax-pre-mulgt0 7441  ax-pre-mulext 7442  ax-arch 7443  ax-caucvg 7444
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rmo 2367  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-if 3390  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-iun 3727  df-br 3838  df-opab 3892  df-mpt 3893  df-tr 3929  df-id 4111  df-po 4114  df-iso 4115  df-iord 4184  df-on 4186  df-ilim 4187  df-suc 4189  df-iom 4396  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010  df-riota 5590  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-1st 5893  df-2nd 5894  df-recs 6052  df-frec 6138  df-sup 6658  df-pnf 7503  df-mnf 7504  df-xr 7505  df-ltxr 7506  df-le 7507  df-sub 7634  df-neg 7635  df-reap 8028  df-ap 8035  df-div 8114  df-inn 8395  df-2 8452  df-3 8453  df-4 8454  df-n0 8644  df-z 8721  df-uz 8989  df-q 9074  df-rp 9104  df-fz 9394  df-fzo 9519  df-fl 9642  df-mod 9695  df-iseq 9818  df-seq3 9819  df-exp 9920  df-cj 10241  df-re 10242  df-im 10243  df-rsqrt 10396  df-abs 10397  df-dvds 10879  df-gcd 11021
This theorem is referenced by:  gcd0val  11034  gcdn0val  11035  gcdf  11046  gcdcom  11047  dfgcd2  11085  gcdass  11086
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