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Theorem eucalgval 16559

Description: Euclid's Algorithm eucalg 16564 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) → (𝐸‘𝑋) = if((2^nd ‘𝑋) = 0, 𝑋, ⟨(2^nd ‘𝑋), ( mod ‘𝑋)⟩))

Distinct variable group: 𝑥,𝑦,𝑋 Allowed substitution hints: 𝐸(𝑥,𝑦)

**Proof of Theorem eucalgval**
Step	Hyp	Ref	Expression
1		df-ov 7393	. . 3 ⊢ ((1^st ‘𝑋)𝐸(2^nd ‘𝑋)) = (𝐸‘⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩)
2		xp1st 8003	. . . 4 ⊢ (𝑋 ∈ (ℕ₀ × ℕ₀) → (1^st ‘𝑋) ∈ ℕ₀)
3		xp2nd 8004	. . . 4 ⊢ (𝑋 ∈ (ℕ₀ × ℕ₀) → (2^nd ‘𝑋) ∈ ℕ₀)
4		eucalgval.1	. . . . 5 ⊢ 𝐸 = (𝑥 ∈ ℕ₀, 𝑦 ∈ ℕ₀ ↦ if(𝑦 = 0, ⟨𝑥, 𝑦⟩, ⟨𝑦, (𝑥 mod 𝑦)⟩))
5	4	eucalgval2 16558	. . . 4 ⊢ (((1^st ‘𝑋) ∈ ℕ₀ ∧ (2^nd ‘𝑋) ∈ ℕ₀) → ((1^st ‘𝑋)𝐸(2^nd ‘𝑋)) = if((2^nd ‘𝑋) = 0, ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩, ⟨(2^nd ‘𝑋), ((1^st ‘𝑋) mod (2^nd ‘𝑋))⟩))
6	2, 3, 5	syl2anc 584	. . 3 ⊢ (𝑋 ∈ (ℕ₀ × ℕ₀) → ((1^st ‘𝑋)𝐸(2^nd ‘𝑋)) = if((2^nd ‘𝑋) = 0, ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩, ⟨(2^nd ‘𝑋), ((1^st ‘𝑋) mod (2^nd ‘𝑋))⟩))
7	1, 6	eqtr3id 2779	. 2 ⊢ (𝑋 ∈ (ℕ₀ × ℕ₀) → (𝐸‘⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩) = if((2^nd ‘𝑋) = 0, ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩, ⟨(2^nd ‘𝑋), ((1^st ‘𝑋) mod (2^nd ‘𝑋))⟩))
8		1st2nd2 8010	. . 3 ⊢ (𝑋 ∈ (ℕ₀ × ℕ₀) → 𝑋 = ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩)
9	8	fveq2d 6865	. 2 ⊢ (𝑋 ∈ (ℕ₀ × ℕ₀) → (𝐸‘𝑋) = (𝐸‘⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩))
10	8	fveq2d 6865	. . . . 5 ⊢ (𝑋 ∈ (ℕ₀ × ℕ₀) → ( mod ‘𝑋) = ( mod ‘⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩))
11		df-ov 7393	. . . . 5 ⊢ ((1^st ‘𝑋) mod (2^nd ‘𝑋)) = ( mod ‘⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩)
12	10, 11	eqtr4di 2783	. . . 4 ⊢ (𝑋 ∈ (ℕ₀ × ℕ₀) → ( mod ‘𝑋) = ((1^st ‘𝑋) mod (2^nd ‘𝑋)))
13	12	opeq2d 4847	. . 3 ⊢ (𝑋 ∈ (ℕ₀ × ℕ₀) → ⟨(2^nd ‘𝑋), ( mod ‘𝑋)⟩ = ⟨(2^nd ‘𝑋), ((1^st ‘𝑋) mod (2^nd ‘𝑋))⟩)
14	8, 13	ifeq12d 4513	. 2 ⊢ (𝑋 ∈ (ℕ₀ × ℕ₀) → if((2^nd ‘𝑋) = 0, 𝑋, ⟨(2^nd ‘𝑋), ( mod ‘𝑋)⟩) = if((2^nd ‘𝑋) = 0, ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩, ⟨(2^nd ‘𝑋), ((1^st ‘𝑋) mod (2^nd ‘𝑋))⟩))
15	7, 9, 14	3eqtr4d 2775	1 ⊢ (𝑋 ∈ (ℕ₀ × ℕ₀) → (𝐸‘𝑋) = if((2^nd ‘𝑋) = 0, 𝑋, ⟨(2^nd ‘𝑋), ( mod ‘𝑋)⟩))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2109 ifcif 4491 ⟨cop 4598 × cxp 5639 ‘cfv 6514 (class class class)co 7390 ∈ cmpo 7392 1^st c1st 7969 2^nd c2nd 7970 0cc0 11075 ℕ₀cn0 12449 mod cmo 13838

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-sep 5254 ax-nul 5264 ax-pr 5390 ax-un 7714

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ral 3046 df-rex 3055 df-rab 3409 df-v 3452 df-sbc 3757 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-nul 4300 df-if 4492 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4875 df-br 5111 df-opab 5173 df-mpt 5192 df-id 5536 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-iota 6467 df-fun 6516 df-fv 6522 df-ov 7393 df-oprab 7394 df-mpo 7395 df-1st 7971 df-2nd 7972

This theorem is referenced by: eucalginv 16561 eucalglt 16562

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