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Theorem eucalgval2 13060
Description: The value of the step function  E for Euclid's Algorithm on an ordered pair. (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
eucalgval2  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( M E N )  =  if ( N  =  0 , 
<. M ,  N >. , 
<. N ,  ( M  mod  N ) >.
) )
Distinct variable groups:    x, y, M    x, N, y
Allowed substitution hints:    E( x, y)

Proof of Theorem eucalgval2
StepHypRef Expression
1 simpr 448 . . . 4  |-  ( ( x  =  M  /\  y  =  N )  ->  y  =  N )
21eqeq1d 2443 . . 3  |-  ( ( x  =  M  /\  y  =  N )  ->  ( y  =  0  <-> 
N  =  0 ) )
3 opeq12 3978 . . 3  |-  ( ( x  =  M  /\  y  =  N )  -> 
<. x ,  y >.  =  <. M ,  N >. )
4 oveq12 6081 . . . 4  |-  ( ( x  =  M  /\  y  =  N )  ->  ( x  mod  y
)  =  ( M  mod  N ) )
51, 4opeq12d 3984 . . 3  |-  ( ( x  =  M  /\  y  =  N )  -> 
<. y ,  ( x  mod  y ) >.  =  <. N ,  ( M  mod  N )
>. )
62, 3, 5ifbieq12d 3753 . 2  |-  ( ( x  =  M  /\  y  =  N )  ->  if ( y  =  0 ,  <. x ,  y >. ,  <. y ,  ( x  mod  y ) >. )  =  if ( N  =  0 ,  <. M ,  N >. ,  <. N , 
( M  mod  N
) >. ) )
7 eucalgval.1 . 2  |-  E  =  ( x  e.  NN0 ,  y  e.  NN0  |->  if ( y  =  0 , 
<. x ,  y >. ,  <. y ,  ( x  mod  y )
>. ) )
8 opex 4419 . . 3  |-  <. M ,  N >.  e.  _V
9 opex 4419 . . 3  |-  <. N , 
( M  mod  N
) >.  e.  _V
108, 9ifex 3789 . 2  |-  if ( N  =  0 , 
<. M ,  N >. , 
<. N ,  ( M  mod  N ) >.
)  e.  _V
116, 7, 10ovmpt2a 6195 1  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( M E N )  =  if ( N  =  0 , 
<. M ,  N >. , 
<. N ,  ( M  mod  N ) >.
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   ifcif 3731   <.cop 3809  (class class class)co 6072    e. cmpt2 6074   0cc0 8979   NN0cn0 10210    mod cmo 11238
This theorem is referenced by:  eucalgval  13061
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-id 4490  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-iota 5409  df-fun 5447  df-fv 5453  df-ov 6075  df-oprab 6076  df-mpt2 6077
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