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Theorem eucalgval 11746
Description: Euclid's Algorithm eucalg 11751 computes the greatest common divisor of two nonnegative integers by repeatedly replacing the larger of them with its remainder modulo the smaller until the remainder is 0.

The value of the step function  E for Euclid's Algorithm. (Contributed by Paul Chapman, 31-Mar-2011.) (Revised by Mario Carneiro, 28-May-2014.)

Hypothesis
Ref Expression
eucalgval.1  |-  E  =  ( x  e.  NN0 ,  y  e.  NN0  |->  if ( y  =  0 , 
<. x ,  y >. ,  <. y ,  ( x  mod  y )
>. ) )
Assertion
Ref Expression
eucalgval  |-  ( X  e.  ( NN0  X.  NN0 )  ->  ( E `
 X )  =  if ( ( 2nd `  X )  =  0 ,  X ,  <. ( 2nd `  X ) ,  (  mod  `  X
) >. ) )
Distinct variable group:    x, y, X
Allowed substitution hints:    E( x, y)

Proof of Theorem eucalgval
StepHypRef Expression
1 df-ov 5777 . . 3  |-  ( ( 1st `  X ) E ( 2nd `  X
) )  =  ( E `  <. ( 1st `  X ) ,  ( 2nd `  X
) >. )
2 xp1st 6063 . . . 4  |-  ( X  e.  ( NN0  X.  NN0 )  ->  ( 1st `  X )  e.  NN0 )
3 xp2nd 6064 . . . 4  |-  ( X  e.  ( NN0  X.  NN0 )  ->  ( 2nd `  X )  e.  NN0 )
4 eucalgval.1 . . . . 5  |-  E  =  ( x  e.  NN0 ,  y  e.  NN0  |->  if ( y  =  0 , 
<. x ,  y >. ,  <. y ,  ( x  mod  y )
>. ) )
54eucalgval2 11745 . . . 4  |-  ( ( ( 1st `  X
)  e.  NN0  /\  ( 2nd `  X )  e.  NN0 )  -> 
( ( 1st `  X
) E ( 2nd `  X ) )  =  if ( ( 2nd `  X )  =  0 ,  <. ( 1st `  X
) ,  ( 2nd `  X ) >. ,  <. ( 2nd `  X ) ,  ( ( 1st `  X )  mod  ( 2nd `  X ) )
>. ) )
62, 3, 5syl2anc 408 . . 3  |-  ( X  e.  ( NN0  X.  NN0 )  ->  ( ( 1st `  X ) E ( 2nd `  X
) )  =  if ( ( 2nd `  X
)  =  0 , 
<. ( 1st `  X
) ,  ( 2nd `  X ) >. ,  <. ( 2nd `  X ) ,  ( ( 1st `  X )  mod  ( 2nd `  X ) )
>. ) )
71, 6syl5eqr 2186 . 2  |-  ( X  e.  ( NN0  X.  NN0 )  ->  ( E `
 <. ( 1st `  X
) ,  ( 2nd `  X ) >. )  =  if ( ( 2nd `  X )  =  0 ,  <. ( 1st `  X
) ,  ( 2nd `  X ) >. ,  <. ( 2nd `  X ) ,  ( ( 1st `  X )  mod  ( 2nd `  X ) )
>. ) )
8 1st2nd2 6073 . . 3  |-  ( X  e.  ( NN0  X.  NN0 )  ->  X  = 
<. ( 1st `  X
) ,  ( 2nd `  X ) >. )
98fveq2d 5425 . 2  |-  ( X  e.  ( NN0  X.  NN0 )  ->  ( E `
 X )  =  ( E `  <. ( 1st `  X ) ,  ( 2nd `  X
) >. ) )
108fveq2d 5425 . . . . 5  |-  ( X  e.  ( NN0  X.  NN0 )  ->  (  mod  `  X )  =  (  mod  `  <. ( 1st `  X ) ,  ( 2nd `  X )
>. ) )
11 df-ov 5777 . . . . 5  |-  ( ( 1st `  X )  mod  ( 2nd `  X
) )  =  (  mod  `  <. ( 1st `  X ) ,  ( 2nd `  X )
>. )
1210, 11syl6eqr 2190 . . . 4  |-  ( X  e.  ( NN0  X.  NN0 )  ->  (  mod  `  X )  =  ( ( 1st `  X
)  mod  ( 2nd `  X ) ) )
1312opeq2d 3712 . . 3  |-  ( X  e.  ( NN0  X.  NN0 )  ->  <. ( 2nd `  X ) ,  (  mod  `  X
) >.  =  <. ( 2nd `  X ) ,  ( ( 1st `  X
)  mod  ( 2nd `  X ) ) >.
)
148, 13ifeq12d 3491 . 2  |-  ( X  e.  ( NN0  X.  NN0 )  ->  if ( ( 2nd `  X
)  =  0 ,  X ,  <. ( 2nd `  X ) ,  (  mod  `  X
) >. )  =  if ( ( 2nd `  X
)  =  0 , 
<. ( 1st `  X
) ,  ( 2nd `  X ) >. ,  <. ( 2nd `  X ) ,  ( ( 1st `  X )  mod  ( 2nd `  X ) )
>. ) )
157, 9, 143eqtr4d 2182 1  |-  ( X  e.  ( NN0  X.  NN0 )  ->  ( E `
 X )  =  if ( ( 2nd `  X )  =  0 ,  X ,  <. ( 2nd `  X ) ,  (  mod  `  X
) >. ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1331    e. wcel 1480   ifcif 3474   <.cop 3530    X. cxp 4537   ` cfv 5123  (class class class)co 5774    e. cmpo 5776   1stc1st 6036   2ndc2nd 6037   0cc0 7632   NN0cn0 8989    mod cmo 10107
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7723  ax-resscn 7724  ax-1cn 7725  ax-1re 7726  ax-icn 7727  ax-addcl 7728  ax-addrcl 7729  ax-mulcl 7730  ax-mulrcl 7731  ax-addcom 7732  ax-mulcom 7733  ax-addass 7734  ax-mulass 7735  ax-distr 7736  ax-i2m1 7737  ax-0lt1 7738  ax-1rid 7739  ax-0id 7740  ax-rnegex 7741  ax-precex 7742  ax-cnre 7743  ax-pre-ltirr 7744  ax-pre-ltwlin 7745  ax-pre-lttrn 7746  ax-pre-apti 7747  ax-pre-ltadd 7748  ax-pre-mulgt0 7749  ax-pre-mulext 7750  ax-arch 7751
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-po 4218  df-iso 4219  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-pnf 7814  df-mnf 7815  df-xr 7816  df-ltxr 7817  df-le 7818  df-sub 7947  df-neg 7948  df-reap 8349  df-ap 8356  df-div 8445  df-inn 8733  df-n0 8990  df-z 9067  df-q 9424  df-rp 9454  df-fl 10055  df-mod 10108
This theorem is referenced by:  eucalginv  11748  eucalglt  11749
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