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Theorem gcdval 12687
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 2289 . . . 4  |-  ( x  =  M  ->  (
x  =  0  <->  M  =  0 ) )
21anbi1d 685 . . 3  |-  ( x  =  M  ->  (
( x  =  0  /\  y  =  0 )  <->  ( M  =  0  /\  y  =  0 ) ) )
3 breq2 4027 . . . . . 6  |-  ( x  =  M  ->  (
n  ||  x  <->  n  ||  M
) )
43anbi1d 685 . . . . 5  |-  ( x  =  M  ->  (
( n  ||  x  /\  n  ||  y )  <-> 
( n  ||  M  /\  n  ||  y ) ) )
54rabbidv 2780 . . . 4  |-  ( x  =  M  ->  { n  e.  ZZ  |  ( n 
||  x  /\  n  ||  y ) }  =  { n  e.  ZZ  |  ( n  ||  M  /\  n  ||  y
) } )
65supeq1d 7199 . . 3  |-  ( x  =  M  ->  sup ( { n  e.  ZZ  |  ( n  ||  x  /\  n  ||  y
) } ,  RR ,  <  )  =  sup ( { n  e.  ZZ  |  ( n  ||  M  /\  n  ||  y
) } ,  RR ,  <  ) )
72, 6ifbieq2d 3585 . 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 2289 . . . 4  |-  ( y  =  N  ->  (
y  =  0  <->  N  =  0 ) )
98anbi2d 684 . . 3  |-  ( y  =  N  ->  (
( M  =  0  /\  y  =  0 )  <->  ( M  =  0  /\  N  =  0 ) ) )
10 breq2 4027 . . . . . 6  |-  ( y  =  N  ->  (
n  ||  y  <->  n  ||  N
) )
1110anbi2d 684 . . . . 5  |-  ( y  =  N  ->  (
( n  ||  M  /\  n  ||  y )  <-> 
( n  ||  M  /\  n  ||  N ) ) )
1211rabbidv 2780 . . . 4  |-  ( y  =  N  ->  { n  e.  ZZ  |  ( n 
||  M  /\  n  ||  y ) }  =  { n  e.  ZZ  |  ( n  ||  M  /\  n  ||  N
) } )
1312supeq1d 7199 . . 3  |-  ( y  =  N  ->  sup ( { n  e.  ZZ  |  ( n  ||  M  /\  n  ||  y
) } ,  RR ,  <  )  =  sup ( { n  e.  ZZ  |  ( n  ||  M  /\  n  ||  N
) } ,  RR ,  <  ) )
149, 13ifbieq2d 3585 . 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 12686 . 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 8832 . . 3  |-  0  e.  _V
17 ltso 8903 . . . 4  |-  <  Or  RR
1817supex 7214 . . 3  |-  sup ( { n  e.  ZZ  |  ( n  ||  M  /\  n  ||  N
) } ,  RR ,  <  )  e.  _V
1916, 18ifex 3623 . 2  |-  if ( ( M  =  0  /\  N  =  0 ) ,  0 ,  sup ( { n  e.  ZZ  |  ( n 
||  M  /\  n  ||  N ) } ,  RR ,  <  ) )  e.  _V
207, 14, 15, 19ovmpt2 5983 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 358    = wceq 1623    e. wcel 1684   {crab 2547   ifcif 3565   class class class wbr 4023  (class class class)co 5858   supcsup 7193   RRcr 8736   0cc0 8737    < clt 8867   ZZcz 10024    || cdivides 12531    gcd cgcd 12685
This theorem is referenced by:  gcd0val  12688  gcdn0val  12689  gcdf  12698  gcdcom  12699  gcdass  12724
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-mulcl 8799  ax-i2m1 8805  ax-pre-lttri 8811  ax-pre-lttrn 8812
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-po 4314  df-so 4315  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-ltxr 8872  df-gcd 12686
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