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Theorem eucalgval 14059
Description: Euclid's Algorithm eucalg 14064 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 6278 . . 3  |-  ( ( 1st `  X ) E ( 2nd `  X
) )  =  ( E `  <. ( 1st `  X ) ,  ( 2nd `  X
) >. )
2 xp1st 6804 . . . 4  |-  ( X  e.  ( NN0  X.  NN0 )  ->  ( 1st `  X )  e.  NN0 )
3 xp2nd 6805 . . . 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 14058 . . . 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 661 . . 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 2515 . 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 6811 . . 3  |-  ( X  e.  ( NN0  X.  NN0 )  ->  X  = 
<. ( 1st `  X
) ,  ( 2nd `  X ) >. )
98fveq2d 5861 . 2  |-  ( X  e.  ( NN0  X.  NN0 )  ->  ( E `
 X )  =  ( E `  <. ( 1st `  X ) ,  ( 2nd `  X
) >. ) )
108fveq2d 5861 . . . . 5  |-  ( X  e.  ( NN0  X.  NN0 )  ->  (  mod  `  X )  =  (  mod  `  <. ( 1st `  X ) ,  ( 2nd `  X )
>. ) )
11 df-ov 6278 . . . . 5  |-  ( ( 1st `  X )  mod  ( 2nd `  X
) )  =  (  mod  `  <. ( 1st `  X ) ,  ( 2nd `  X )
>. )
1210, 11syl6eqr 2519 . . . 4  |-  ( X  e.  ( NN0  X.  NN0 )  ->  (  mod  `  X )  =  ( ( 1st `  X
)  mod  ( 2nd `  X ) ) )
1312opeq2d 4213 . . 3  |-  ( X  e.  ( NN0  X.  NN0 )  ->  <. ( 2nd `  X ) ,  (  mod  `  X
) >.  =  <. ( 2nd `  X ) ,  ( ( 1st `  X
)  mod  ( 2nd `  X ) ) >.
)
148, 13ifeq12d 3952 . 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 2511 1  |-  ( X  e.  ( NN0  X.  NN0 )  ->  ( E `
 X )  =  if ( ( 2nd `  X )  =  0 ,  X ,  <. ( 2nd `  X ) ,  (  mod  `  X
) >. ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1374    e. wcel 1762   ifcif 3932   <.cop 4026    X. cxp 4990   ` cfv 5579  (class class class)co 6275    |-> cmpt2 6277   1stc1st 6772   2ndc2nd 6773   0cc0 9481   NN0cn0 10784    mod cmo 11952
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-iota 5542  df-fun 5581  df-fv 5587  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-1st 6774  df-2nd 6775
This theorem is referenced by:  eucalginv  14061  eucalglt  14062
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