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Theorem eucalgval2 14210
 Description: The value of the step function 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
Assertion
Ref Expression
eucalgval2
Distinct variable groups:   ,,M   ,N,

Proof of Theorem eucalgval2
StepHypRef Expression
1 simpr 461 . . . 4
21eqeq1d 2459 . . 3
3 opeq12 4219 . . 3
4 oveq12 6305 . . . 4
51, 4opeq12d 4225 . . 3
62, 3, 5ifbieq12d 3968 . 2
7 eucalgval.1 . 2
8 opex 4716 . . 3
9 opex 4716 . . 3
108, 9ifex 4010 . 2
116, 7, 10ovmpt2a 6433 1
 Colors of variables: wff setvar class Syntax hints:  ->wi 4  /\wa 369  =wceq 1395  e.wcel 1818  ifcif 3941  <.cop 4035  (class class class)co 6296  e.cmpt2 6298  0cc0 9513   cn0 10820   cmo 11996 This theorem is referenced by:  eucalgval  14211 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-uni 4250  df-br 4453  df-opab 4511  df-id 4800  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-iota 5556  df-fun 5595  df-fv 5601  df-ov 6299  df-oprab 6300  df-mpt2 6301
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