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Theorem eucalgval 16552
Description: Euclid's Algorithm eucalg 16557 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 7419 . . 3 ((1st𝑋)𝐸(2nd𝑋)) = (𝐸‘⟨(1st𝑋), (2nd𝑋)⟩)
2 xp1st 8023 . . . 4 (𝑋 ∈ (ℕ0 × ℕ0) → (1st𝑋) ∈ ℕ0)
3 xp2nd 8024 . . . 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 16551 . . . 4 (((1st𝑋) ∈ ℕ0 ∧ (2nd𝑋) ∈ ℕ0) → ((1st𝑋)𝐸(2nd𝑋)) = if((2nd𝑋) = 0, ⟨(1st𝑋), (2nd𝑋)⟩, ⟨(2nd𝑋), ((1st𝑋) mod (2nd𝑋))⟩))
62, 3, 5syl2anc 582 . . 3 (𝑋 ∈ (ℕ0 × ℕ0) → ((1st𝑋)𝐸(2nd𝑋)) = if((2nd𝑋) = 0, ⟨(1st𝑋), (2nd𝑋)⟩, ⟨(2nd𝑋), ((1st𝑋) mod (2nd𝑋))⟩))
71, 6eqtr3id 2779 . 2 (𝑋 ∈ (ℕ0 × ℕ0) → (𝐸‘⟨(1st𝑋), (2nd𝑋)⟩) = if((2nd𝑋) = 0, ⟨(1st𝑋), (2nd𝑋)⟩, ⟨(2nd𝑋), ((1st𝑋) mod (2nd𝑋))⟩))
8 1st2nd2 8030 . . 3 (𝑋 ∈ (ℕ0 × ℕ0) → 𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩)
98fveq2d 6896 . 2 (𝑋 ∈ (ℕ0 × ℕ0) → (𝐸𝑋) = (𝐸‘⟨(1st𝑋), (2nd𝑋)⟩))
108fveq2d 6896 . . . . 5 (𝑋 ∈ (ℕ0 × ℕ0) → ( mod ‘𝑋) = ( mod ‘⟨(1st𝑋), (2nd𝑋)⟩))
11 df-ov 7419 . . . . 5 ((1st𝑋) mod (2nd𝑋)) = ( mod ‘⟨(1st𝑋), (2nd𝑋)⟩)
1210, 11eqtr4di 2783 . . . 4 (𝑋 ∈ (ℕ0 × ℕ0) → ( mod ‘𝑋) = ((1st𝑋) mod (2nd𝑋)))
1312opeq2d 4876 . . 3 (𝑋 ∈ (ℕ0 × ℕ0) → ⟨(2nd𝑋), ( mod ‘𝑋)⟩ = ⟨(2nd𝑋), ((1st𝑋) mod (2nd𝑋))⟩)
148, 13ifeq12d 4545 . 2 (𝑋 ∈ (ℕ0 × ℕ0) → if((2nd𝑋) = 0, 𝑋, ⟨(2nd𝑋), ( mod ‘𝑋)⟩) = if((2nd𝑋) = 0, ⟨(1st𝑋), (2nd𝑋)⟩, ⟨(2nd𝑋), ((1st𝑋) mod (2nd𝑋))⟩))
157, 9, 143eqtr4d 2775 1 (𝑋 ∈ (ℕ0 × ℕ0) → (𝐸𝑋) = if((2nd𝑋) = 0, 𝑋, ⟨(2nd𝑋), ( mod ‘𝑋)⟩))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  ifcif 4524  cop 4630   × cxp 5670  cfv 6543  (class class class)co 7416  cmpo 7418  1st c1st 7989  2nd c2nd 7990  0cc0 11138  0cn0 12502   mod cmo 13866
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423  ax-un 7738
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-sbc 3769  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-iota 6495  df-fun 6545  df-fv 6551  df-ov 7419  df-oprab 7420  df-mpo 7421  df-1st 7991  df-2nd 7992
This theorem is referenced by:  eucalginv  16554  eucalglt  16555
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