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Theorem eucalgval 15342
Description: Euclid's Algorithm eucalg 15347 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 𝐸 for Euclid's Algorithm. (Contributed by Paul Chapman, 31-Mar-2011.) (Revised by Mario Carneiro, 28-May-2014.)

Hypothesis
Ref Expression
eucalgval.1 𝐸 = (𝑥 ∈ ℕ0, 𝑦 ∈ ℕ0 ↦ if(𝑦 = 0, ⟨𝑥, 𝑦⟩, ⟨𝑦, (𝑥 mod 𝑦)⟩))
Assertion
Ref Expression
eucalgval (𝑋 ∈ (ℕ0 × ℕ0) → (𝐸𝑋) = if((2nd𝑋) = 0, 𝑋, ⟨(2nd𝑋), ( mod ‘𝑋)⟩))
Distinct variable group:   𝑥,𝑦,𝑋
Allowed substitution hints:   𝐸(𝑥,𝑦)

Proof of Theorem eucalgval
StepHypRef Expression
1 df-ov 6693 . . 3 ((1st𝑋)𝐸(2nd𝑋)) = (𝐸‘⟨(1st𝑋), (2nd𝑋)⟩)
2 xp1st 7242 . . . 4 (𝑋 ∈ (ℕ0 × ℕ0) → (1st𝑋) ∈ ℕ0)
3 xp2nd 7243 . . . 4 (𝑋 ∈ (ℕ0 × ℕ0) → (2nd𝑋) ∈ ℕ0)
4 eucalgval.1 . . . . 5 𝐸 = (𝑥 ∈ ℕ0, 𝑦 ∈ ℕ0 ↦ if(𝑦 = 0, ⟨𝑥, 𝑦⟩, ⟨𝑦, (𝑥 mod 𝑦)⟩))
54eucalgval2 15341 . . . 4 (((1st𝑋) ∈ ℕ0 ∧ (2nd𝑋) ∈ ℕ0) → ((1st𝑋)𝐸(2nd𝑋)) = if((2nd𝑋) = 0, ⟨(1st𝑋), (2nd𝑋)⟩, ⟨(2nd𝑋), ((1st𝑋) mod (2nd𝑋))⟩))
62, 3, 5syl2anc 694 . . 3 (𝑋 ∈ (ℕ0 × ℕ0) → ((1st𝑋)𝐸(2nd𝑋)) = if((2nd𝑋) = 0, ⟨(1st𝑋), (2nd𝑋)⟩, ⟨(2nd𝑋), ((1st𝑋) mod (2nd𝑋))⟩))
71, 6syl5eqr 2699 . 2 (𝑋 ∈ (ℕ0 × ℕ0) → (𝐸‘⟨(1st𝑋), (2nd𝑋)⟩) = if((2nd𝑋) = 0, ⟨(1st𝑋), (2nd𝑋)⟩, ⟨(2nd𝑋), ((1st𝑋) mod (2nd𝑋))⟩))
8 1st2nd2 7249 . . 3 (𝑋 ∈ (ℕ0 × ℕ0) → 𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩)
98fveq2d 6233 . 2 (𝑋 ∈ (ℕ0 × ℕ0) → (𝐸𝑋) = (𝐸‘⟨(1st𝑋), (2nd𝑋)⟩))
108fveq2d 6233 . . . . 5 (𝑋 ∈ (ℕ0 × ℕ0) → ( mod ‘𝑋) = ( mod ‘⟨(1st𝑋), (2nd𝑋)⟩))
11 df-ov 6693 . . . . 5 ((1st𝑋) mod (2nd𝑋)) = ( mod ‘⟨(1st𝑋), (2nd𝑋)⟩)
1210, 11syl6eqr 2703 . . . 4 (𝑋 ∈ (ℕ0 × ℕ0) → ( mod ‘𝑋) = ((1st𝑋) mod (2nd𝑋)))
1312opeq2d 4440 . . 3 (𝑋 ∈ (ℕ0 × ℕ0) → ⟨(2nd𝑋), ( mod ‘𝑋)⟩ = ⟨(2nd𝑋), ((1st𝑋) mod (2nd𝑋))⟩)
148, 13ifeq12d 4139 . 2 (𝑋 ∈ (ℕ0 × ℕ0) → if((2nd𝑋) = 0, 𝑋, ⟨(2nd𝑋), ( mod ‘𝑋)⟩) = if((2nd𝑋) = 0, ⟨(1st𝑋), (2nd𝑋)⟩, ⟨(2nd𝑋), ((1st𝑋) mod (2nd𝑋))⟩))
157, 9, 143eqtr4d 2695 1 (𝑋 ∈ (ℕ0 × ℕ0) → (𝐸𝑋) = if((2nd𝑋) = 0, 𝑋, ⟨(2nd𝑋), ( mod ‘𝑋)⟩))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1523  wcel 2030  ifcif 4119  cop 4216   × cxp 5141  cfv 5926  (class class class)co 6690  cmpt2 6692  1st c1st 7208  2nd c2nd 7209  0cc0 9974  0cn0 11330   mod cmo 12708
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-iota 5889  df-fun 5928  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-1st 7210  df-2nd 7211
This theorem is referenced by:  eucalginv  15344  eucalglt  15345
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