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Theorem eucalgval 14211
Description: Euclid's Algorithm eucalg 14216 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 for Euclid's Algorithm. (Contributed by Paul Chapman, 31-Mar-2011.) (Revised by Mario Carneiro, 28-May-2014.)

Hypothesis
Ref Expression
eucalgval.1
Assertion
Ref Expression
eucalgval
Distinct variable group:   , ,

Proof of Theorem eucalgval
StepHypRef Expression
1 df-ov 6299 . . 3
2 xp1st 6830 . . . 4
3 xp2nd 6831 . . . 4
4 eucalgval.1 . . . . 5
54eucalgval2 14210 . . . 4
62, 3, 5syl2anc 661 . . 3
71, 6syl5eqr 2512 . 2
8 1st2nd2 6837 . . 3
98fveq2d 5875 . 2
108fveq2d 5875 . . . . 5
11 df-ov 6299 . . . . 5
1210, 11syl6eqr 2516 . . . 4
1312opeq2d 4224 . . 3
148, 13ifeq12d 3961 . 2
157, 9, 143eqtr4d 2508 1
Colors of variables: wff setvar class
Syntax hints:  ->wi 4  =wceq 1395  e.wcel 1818  ifcif 3941  <.cop 4035  X.cxp 5002  `cfv 5593  (class class class)co 6296  e.cmpt2 6298   c1st 6798   c2nd 6799  0cc0 9513   cn0 10820   cmo 11996
This theorem is referenced by:  eucalginv  14213  eucalglt  14214
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-8 1820  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-pow 4630  ax-pr 4691  ax-un 6592
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-mpt 4512  df-id 4800  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-iota 5556  df-fun 5595  df-fv 5601  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6800  df-2nd 6801
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