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Theorem eucalgval 16485

Description: Euclid's Algorithm eucalg 16490 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 7344	. . 3 ⊢ ((1^st ‘𝑋)𝐸(2^nd ‘𝑋)) = (𝐸‘⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩)
2		xp1st 7948	. . . 4 ⊢ (𝑋 ∈ (ℕ₀ × ℕ₀) → (1^st ‘𝑋) ∈ ℕ₀)
3		xp2nd 7949	. . . 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 16484	. . . 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 7955	. . 3 ⊢ (𝑋 ∈ (ℕ₀ × ℕ₀) → 𝑋 = ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩)
9	8	fveq2d 6821	. 2 ⊢ (𝑋 ∈ (ℕ₀ × ℕ₀) → (𝐸‘𝑋) = (𝐸‘⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩))
10	8	fveq2d 6821	. . . . 5 ⊢ (𝑋 ∈ (ℕ₀ × ℕ₀) → ( mod ‘𝑋) = ( mod ‘⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩))
11		df-ov 7344	. . . . 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 4830	. . 3 ⊢ (𝑋 ∈ (ℕ₀ × ℕ₀) → ⟨(2^nd ‘𝑋), ( mod ‘𝑋)⟩ = ⟨(2^nd ‘𝑋), ((1^st ‘𝑋) mod (2^nd ‘𝑋))⟩)
14	8, 13	ifeq12d 4495	. 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 1541 ∈ wcel 2110 ifcif 4473 ⟨cop 4580 × cxp 5612 ‘cfv 6477 (class class class)co 7341 ∈ cmpo 7343 1^st c1st 7914 2^nd c2nd 7915 0cc0 10998 ℕ₀cn0 12373 mod cmo 13765

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2112 ax-9 2120 ax-10 2143 ax-11 2159 ax-12 2179 ax-ext 2702 ax-sep 5232 ax-nul 5242 ax-pr 5368 ax-un 7663

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2067 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 3394 df-v 3436 df-sbc 3740 df-dif 3903 df-un 3905 df-in 3907 df-ss 3917 df-nul 4282 df-if 4474 df-sn 4575 df-pr 4577 df-op 4581 df-uni 4858 df-br 5090 df-opab 5152 df-mpt 5171 df-id 5509 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-iota 6433 df-fun 6479 df-fv 6485 df-ov 7344 df-oprab 7345 df-mpo 7346 df-1st 7916 df-2nd 7917

This theorem is referenced by: eucalginv 16487 eucalglt 16488

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