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Theorem eucalgval2 12747
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 447 . . . 4  |-  ( ( x  =  M  /\  y  =  N )  ->  y  =  N )
21eqeq1d 2292 . . 3  |-  ( ( x  =  M  /\  y  =  N )  ->  ( y  =  0  <-> 
N  =  0 ) )
3 opeq12 3799 . . 3  |-  ( ( x  =  M  /\  y  =  N )  -> 
<. x ,  y >.  =  <. M ,  N >. )
4 oveq12 5829 . . . 4  |-  ( ( x  =  M  /\  y  =  N )  ->  ( x  mod  y
)  =  ( M  mod  N ) )
51, 4opeq12d 3805 . . 3  |-  ( ( x  =  M  /\  y  =  N )  -> 
<. y ,  ( x  mod  y ) >.  =  <. N ,  ( M  mod  N )
>. )
62, 3, 5ifbieq12d 3588 . 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 4236 . . 3  |-  <. M ,  N >.  e.  _V
9 opex 4236 . . 3  |-  <. N , 
( M  mod  N
) >.  e.  _V
108, 9ifex 3624 . 2  |-  if ( N  =  0 , 
<. M ,  N >. , 
<. N ,  ( M  mod  N ) >.
)  e.  _V
116, 7, 10ovmpt2a 5940 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 358    = wceq 1623    e. wcel 1685   ifcif 3566   <.cop 3644  (class class class)co 5820    e. cmpt2 5822   0cc0 8733   NN0cn0 9961    mod cmo 10969
This theorem is referenced by:  eucalgval  12748
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1636  ax-8 1644  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-sbc 2993  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-br 4025  df-opab 4079  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fv 5229  df-ov 5823  df-oprab 5824  df-mpt2 5825
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