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Theorem gcdval 12963
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 eqeq1 2410 . . . 4  |-  ( x  =  M  ->  (
x  =  0  <->  M  =  0 ) )
21anbi1d 686 . . 3  |-  ( x  =  M  ->  (
( x  =  0  /\  y  =  0 )  <->  ( M  =  0  /\  y  =  0 ) ) )
3 breq2 4176 . . . . . 6  |-  ( x  =  M  ->  (
n  ||  x  <->  n  ||  M
) )
43anbi1d 686 . . . . 5  |-  ( x  =  M  ->  (
( n  ||  x  /\  n  ||  y )  <-> 
( n  ||  M  /\  n  ||  y ) ) )
54rabbidv 2908 . . . 4  |-  ( x  =  M  ->  { n  e.  ZZ  |  ( n 
||  x  /\  n  ||  y ) }  =  { n  e.  ZZ  |  ( n  ||  M  /\  n  ||  y
) } )
65supeq1d 7409 . . 3  |-  ( x  =  M  ->  sup ( { n  e.  ZZ  |  ( n  ||  x  /\  n  ||  y
) } ,  RR ,  <  )  =  sup ( { n  e.  ZZ  |  ( n  ||  M  /\  n  ||  y
) } ,  RR ,  <  ) )
72, 6ifbieq2d 3719 . 2  |-  ( 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 ,  <  ) ) )
8 eqeq1 2410 . . . 4  |-  ( y  =  N  ->  (
y  =  0  <->  N  =  0 ) )
98anbi2d 685 . . 3  |-  ( y  =  N  ->  (
( M  =  0  /\  y  =  0 )  <->  ( M  =  0  /\  N  =  0 ) ) )
10 breq2 4176 . . . . . 6  |-  ( y  =  N  ->  (
n  ||  y  <->  n  ||  N
) )
1110anbi2d 685 . . . . 5  |-  ( y  =  N  ->  (
( n  ||  M  /\  n  ||  y )  <-> 
( n  ||  M  /\  n  ||  N ) ) )
1211rabbidv 2908 . . . 4  |-  ( y  =  N  ->  { n  e.  ZZ  |  ( n 
||  M  /\  n  ||  y ) }  =  { n  e.  ZZ  |  ( n  ||  M  /\  n  ||  N
) } )
1312supeq1d 7409 . . 3  |-  ( y  =  N  ->  sup ( { n  e.  ZZ  |  ( n  ||  M  /\  n  ||  y
) } ,  RR ,  <  )  =  sup ( { n  e.  ZZ  |  ( n  ||  M  /\  n  ||  N
) } ,  RR ,  <  ) )
149, 13ifbieq2d 3719 . 2  |-  ( 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 ,  <  ) ) )
15 df-gcd 12962 . 2  |-  gcd  =  ( x  e.  ZZ ,  y  e.  ZZ  |->  if ( ( x  =  0  /\  y  =  0 ) ,  0 ,  sup ( { n  e.  ZZ  | 
( n  ||  x  /\  n  ||  y ) } ,  RR ,  <  ) ) )
16 c0ex 9041 . . 3  |-  0  e.  _V
17 ltso 9112 . . . 4  |-  <  Or  RR
1817supex 7424 . . 3  |-  sup ( { n  e.  ZZ  |  ( n  ||  M  /\  n  ||  N
) } ,  RR ,  <  )  e.  _V
1916, 18ifex 3757 . 2  |-  if ( ( M  =  0  /\  N  =  0 ) ,  0 ,  sup ( { n  e.  ZZ  |  ( n 
||  M  /\  n  ||  N ) } ,  RR ,  <  ) )  e.  _V
207, 14, 15, 19ovmpt2 6168 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:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   {crab 2670   ifcif 3699   class class class wbr 4172  (class class class)co 6040   supcsup 7403   RRcr 8945   0cc0 8946    < clt 9076   ZZcz 10238    || cdivides 12807    gcd cgcd 12961
This theorem is referenced by:  gcd0val  12964  gcdn0val  12965  gcdf  12974  gcdcom  12975  gcdass  13000
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-mulcl 9008  ax-i2m1 9014  ax-pre-lttri 9020  ax-pre-lttrn 9021
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-po 4463  df-so 4464  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-sup 7404  df-pnf 9078  df-mnf 9079  df-ltxr 9081  df-gcd 12962
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