ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  gcdval Unicode version

Theorem gcdval 11962
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 110 . . . . 5  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =  0  /\  N  =  0 ) )  -> 
( M  =  0  /\  N  =  0 ) )
21iftrued 3543 . . . 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 9193 . . . 4  |-  0  e.  NN0
42, 3eqeltrdi 2268 . . 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 110 . . . . . 6  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  /\  N  =  0 ) )  ->  -.  ( M  =  0  /\  N  =  0 ) )
65iffalsed 3546 . . . . 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 11961 . . . . 5  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  /\  N  =  0 ) )  ->  sup ( { n  e.  ZZ  |  ( n 
||  M  /\  n  ||  N ) } ,  RR ,  <  )  e.  NN )
86, 7eqeltrd 2254 . . . 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 9231 . . 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 11947 . . . 4  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  -> DECID  ( M  =  0  /\  N  =  0 ) )
11 exmiddc 836 . . . 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 798 . 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 2184 . . . . 5  |-  ( x  =  M  ->  (
x  =  0  <->  M  =  0 ) )
1514anbi1d 465 . . . 4  |-  ( x  =  M  ->  (
( x  =  0  /\  y  =  0 )  <->  ( M  =  0  /\  y  =  0 ) ) )
16 breq2 4009 . . . . . . 7  |-  ( x  =  M  ->  (
n  ||  x  <->  n  ||  M
) )
1716anbi1d 465 . . . . . 6  |-  ( x  =  M  ->  (
( n  ||  x  /\  n  ||  y )  <-> 
( n  ||  M  /\  n  ||  y ) ) )
1817rabbidv 2728 . . . . 5  |-  ( x  =  M  ->  { n  e.  ZZ  |  ( n 
||  x  /\  n  ||  y ) }  =  { n  e.  ZZ  |  ( n  ||  M  /\  n  ||  y
) } )
1918supeq1d 6988 . . . 4  |-  ( x  =  M  ->  sup ( { n  e.  ZZ  |  ( n  ||  x  /\  n  ||  y
) } ,  RR ,  <  )  =  sup ( { n  e.  ZZ  |  ( n  ||  M  /\  n  ||  y
) } ,  RR ,  <  ) )
2015, 19ifbieq2d 3560 . . 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 2184 . . . . 5  |-  ( y  =  N  ->  (
y  =  0  <->  N  =  0 ) )
2221anbi2d 464 . . . 4  |-  ( y  =  N  ->  (
( M  =  0  /\  y  =  0 )  <->  ( M  =  0  /\  N  =  0 ) ) )
23 breq2 4009 . . . . . . 7  |-  ( y  =  N  ->  (
n  ||  y  <->  n  ||  N
) )
2423anbi2d 464 . . . . . 6  |-  ( y  =  N  ->  (
( n  ||  M  /\  n  ||  y )  <-> 
( n  ||  M  /\  n  ||  N ) ) )
2524rabbidv 2728 . . . . 5  |-  ( y  =  N  ->  { n  e.  ZZ  |  ( n 
||  M  /\  n  ||  y ) }  =  { n  e.  ZZ  |  ( n  ||  M  /\  n  ||  N
) } )
2625supeq1d 6988 . . . 4  |-  ( y  =  N  ->  sup ( { n  e.  ZZ  |  ( n  ||  M  /\  n  ||  y
) } ,  RR ,  <  )  =  sup ( { n  e.  ZZ  |  ( n  ||  M  /\  n  ||  N
) } ,  RR ,  <  ) )
2722, 26ifbieq2d 3560 . . 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 11946 . . 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, 28ovmpog 6011 . 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 1338 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 104    \/ wo 708  DECID wdc 834    = wceq 1353    e. wcel 2148   {crab 2459   ifcif 3536   class class class wbr 4005  (class class class)co 5877   supcsup 6983   RRcr 7812   0cc0 7813    < clt 7994   NNcn 8921   NN0cn0 9178   ZZcz 9255    || cdvds 11796    gcd cgcd 11945
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-mulrcl 7912  ax-addcom 7913  ax-mulcom 7914  ax-addass 7915  ax-mulass 7916  ax-distr 7917  ax-i2m1 7918  ax-0lt1 7919  ax-1rid 7920  ax-0id 7921  ax-rnegex 7922  ax-precex 7923  ax-cnre 7924  ax-pre-ltirr 7925  ax-pre-ltwlin 7926  ax-pre-lttrn 7927  ax-pre-apti 7928  ax-pre-ltadd 7929  ax-pre-mulgt0 7930  ax-pre-mulext 7931  ax-arch 7932  ax-caucvg 7933
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-if 3537  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-po 4298  df-iso 4299  df-iord 4368  df-on 4370  df-ilim 4371  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-recs 6308  df-frec 6394  df-sup 6985  df-pnf 7996  df-mnf 7997  df-xr 7998  df-ltxr 7999  df-le 8000  df-sub 8132  df-neg 8133  df-reap 8534  df-ap 8541  df-div 8632  df-inn 8922  df-2 8980  df-3 8981  df-4 8982  df-n0 9179  df-z 9256  df-uz 9531  df-q 9622  df-rp 9656  df-fz 10011  df-fzo 10145  df-fl 10272  df-mod 10325  df-seqfrec 10448  df-exp 10522  df-cj 10853  df-re 10854  df-im 10855  df-rsqrt 11009  df-abs 11010  df-dvds 11797  df-gcd 11946
This theorem is referenced by:  gcd0val  11963  gcdn0val  11964  gcdf  11975  gcdcom  11976  dfgcd2  12017  gcdass  12018
  Copyright terms: Public domain W3C validator