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Theorem gcdval 12703
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 2302 . . . 4  |-  ( x  =  M  ->  (
x  =  0  <->  M  =  0 ) )
21anbi1d 685 . . 3  |-  ( x  =  M  ->  (
( x  =  0  /\  y  =  0 )  <->  ( M  =  0  /\  y  =  0 ) ) )
3 breq2 4043 . . . . . 6  |-  ( x  =  M  ->  (
n  ||  x  <->  n  ||  M
) )
43anbi1d 685 . . . . 5  |-  ( x  =  M  ->  (
( n  ||  x  /\  n  ||  y )  <-> 
( n  ||  M  /\  n  ||  y ) ) )
54rabbidv 2793 . . . 4  |-  ( x  =  M  ->  { n  e.  ZZ  |  ( n 
||  x  /\  n  ||  y ) }  =  { n  e.  ZZ  |  ( n  ||  M  /\  n  ||  y
) } )
65supeq1d 7215 . . 3  |-  ( x  =  M  ->  sup ( { n  e.  ZZ  |  ( n  ||  x  /\  n  ||  y
) } ,  RR ,  <  )  =  sup ( { n  e.  ZZ  |  ( n  ||  M  /\  n  ||  y
) } ,  RR ,  <  ) )
72, 6ifbieq2d 3598 . 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 2302 . . . 4  |-  ( y  =  N  ->  (
y  =  0  <->  N  =  0 ) )
98anbi2d 684 . . 3  |-  ( y  =  N  ->  (
( M  =  0  /\  y  =  0 )  <->  ( M  =  0  /\  N  =  0 ) ) )
10 breq2 4043 . . . . . 6  |-  ( y  =  N  ->  (
n  ||  y  <->  n  ||  N
) )
1110anbi2d 684 . . . . 5  |-  ( y  =  N  ->  (
( n  ||  M  /\  n  ||  y )  <-> 
( n  ||  M  /\  n  ||  N ) ) )
1211rabbidv 2793 . . . 4  |-  ( y  =  N  ->  { n  e.  ZZ  |  ( n 
||  M  /\  n  ||  y ) }  =  { n  e.  ZZ  |  ( n  ||  M  /\  n  ||  N
) } )
1312supeq1d 7215 . . 3  |-  ( y  =  N  ->  sup ( { n  e.  ZZ  |  ( n  ||  M  /\  n  ||  y
) } ,  RR ,  <  )  =  sup ( { n  e.  ZZ  |  ( n  ||  M  /\  n  ||  N
) } ,  RR ,  <  ) )
149, 13ifbieq2d 3598 . 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 12702 . 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 8848 . . 3  |-  0  e.  _V
17 ltso 8919 . . . 4  |-  <  Or  RR
1817supex 7230 . . 3  |-  sup ( { n  e.  ZZ  |  ( n  ||  M  /\  n  ||  N
) } ,  RR ,  <  )  e.  _V
1916, 18ifex 3636 . 2  |-  if ( ( M  =  0  /\  N  =  0 ) ,  0 ,  sup ( { n  e.  ZZ  |  ( n 
||  M  /\  n  ||  N ) } ,  RR ,  <  ) )  e.  _V
207, 14, 15, 19ovmpt2 5999 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 1632    e. wcel 1696   {crab 2560   ifcif 3578   class class class wbr 4039  (class class class)co 5874   supcsup 7209   RRcr 8752   0cc0 8753    < clt 8883   ZZcz 10040    || cdivides 12547    gcd cgcd 12701
This theorem is referenced by:  gcd0val  12704  gcdn0val  12705  gcdf  12714  gcdcom  12715  gcdass  12740
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-mulcl 8815  ax-i2m1 8821  ax-pre-lttri 8827  ax-pre-lttrn 8828
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-po 4330  df-so 4331  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-sup 7210  df-pnf 8885  df-mnf 8886  df-ltxr 8888  df-gcd 12702
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