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Theorem eucalgval2 12625
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 449 . . . 4  |-  ( ( x  =  M  /\  y  =  N )  ->  y  =  N )
21eqeq1d 2261 . . 3  |-  ( ( x  =  M  /\  y  =  N )  ->  ( y  =  0  <-> 
N  =  0 ) )
3 opeq12 3698 . . 3  |-  ( ( x  =  M  /\  y  =  N )  -> 
<. x ,  y >.  =  <. M ,  N >. )
4 oveq12 5719 . . . 4  |-  ( ( x  =  M  /\  y  =  N )  ->  ( x  mod  y
)  =  ( M  mod  N ) )
51, 4opeq12d 3704 . . 3  |-  ( ( x  =  M  /\  y  =  N )  -> 
<. y ,  ( x  mod  y ) >.  =  <. N ,  ( M  mod  N )
>. )
62, 3, 5ifbieq12d 3492 . 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 4130 . . 3  |-  <. M ,  N >.  e.  _V
9 opex 4130 . . 3  |-  <. N , 
( M  mod  N
) >.  e.  _V
108, 9ifex 3528 . 2  |-  if ( N  =  0 , 
<. M ,  N >. , 
<. N ,  ( M  mod  N ) >.
)  e.  _V
116, 7, 10ovmpt2a 5830 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 6    /\ wa 360    = wceq 1619    e. wcel 1621   ifcif 3470   <.cop 3547  (class class class)co 5710    e. cmpt2 5712   0cc0 8617   NN0cn0 9844    mod cmo 10851
This theorem is referenced by:  eucalgval  12626
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pr 4108
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-rab 2516  df-v 2729  df-sbc 2922  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-br 3921  df-opab 3975  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715
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