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Theorem eucalgval 13063
 Description: Euclid's Algorithm eucalg 13068 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:   ,,
Allowed substitution hints:   (,)

Proof of Theorem eucalgval
StepHypRef Expression
1 df-ov 6076 . . 3
2 xp1st 6368 . . . 4
3 xp2nd 6369 . . . 4
4 eucalgval.1 . . . . 5
54eucalgval2 13062 . . . 4
62, 3, 5syl2anc 643 . . 3
71, 6syl5eqr 2481 . 2
8 1st2nd2 6378 . . 3
98fveq2d 5724 . 2
108fveq2d 5724 . . . . 5
11 df-ov 6076 . . . . 5
1210, 11syl6eqr 2485 . . . 4
1312opeq2d 3983 . . 3
148, 13ifeq12d 3747 . 2
157, 9, 143eqtr4d 2477 1
 Colors of variables: wff set class Syntax hints:   wi 4   wceq 1652   wcel 1725  cif 3731  cop 3809   cxp 4868  cfv 5446  (class class class)co 6073   cmpt2 6075  c1st 6339  c2nd 6340  cc0 8980  cn0 10211   cmo 11240 This theorem is referenced by:  eucalginv  13065  eucalglt  13066 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-13 1727  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-pow 4369  ax-pr 4395  ax-un 4693 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-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-iota 5410  df-fun 5448  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342
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