eucalgval - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > eucalgval		Structured version Visualization version GIF version

Theorem eucalgval 16758

Description: Euclid's Algorithm eucalg 16763 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 7423	. . 3 ⊢ ((1^st ‘𝑋)𝐸(2^nd ‘𝑋)) = (𝐸‘⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩)
2		xp1st 8033	. . . 4 ⊢ (𝑋 ∈ (ℕ₀ × ℕ₀) → (1^st ‘𝑋) ∈ ℕ₀)
3		xp2nd 8034	. . . 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 16757	. . . 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 596	. . 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 2809	. 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 8040	. . 3 ⊢ (𝑋 ∈ (ℕ₀ × ℕ₀) → 𝑋 = ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩)
9	8	fveq2d 6889	. 2 ⊢ (𝑋 ∈ (ℕ₀ × ℕ₀) → (𝐸‘𝑋) = (𝐸‘⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩))
10	8	fveq2d 6889	. . . . 5 ⊢ (𝑋 ∈ (ℕ₀ × ℕ₀) → ( mod ‘𝑋) = ( mod ‘⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩))
11		df-ov 7423	. . . . 5 ⊢ ((1^st ‘𝑋) mod (2^nd ‘𝑋)) = ( mod ‘⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩)
12	10, 11	eqtr4di 2813	. . . 4 ⊢ (𝑋 ∈ (ℕ₀ × ℕ₀) → ( mod ‘𝑋) = ((1^st ‘𝑋) mod (2^nd ‘𝑋)))
13	12	opeq2d 4839	. . 3 ⊢ (𝑋 ∈ (ℕ₀ × ℕ₀) → ⟨(2^nd ‘𝑋), ( mod ‘𝑋)⟩ = ⟨(2^nd ‘𝑋), ((1^st ‘𝑋) mod (2^nd ‘𝑋))⟩)
14	8, 13	ifeq12d 4503	. 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 2805	1 ⊢ (𝑋 ∈ (ℕ₀ × ℕ₀) → (𝐸‘𝑋) = if((2^nd ‘𝑋) = 0, 𝑋, ⟨(2^nd ‘𝑋), ( mod ‘𝑋)⟩))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 = wceq 1569 ∈ wcel 2144 ifcif 4481 ⟨cop 4589 × cxp 5648 ‘cfv 6538 (class class class)co 7420 ∈ cmpo 7422 1^st c1st 7999 2^nd c2nd 8000 0cc0 11201 ℕ₀cn0 12607 mod cmo 14010

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-sep 5248 ax-nul 5259 ax-pr 5390 ax-un 7751

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-ral 3077 df-rex 3087 df-rab 3413 df-v 3452 df-sbc 3739 df-ss 3896 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-sn 4584 df-pr 4586 df-op 4590 df-uni 4867 df-br 5103 df-opab 5167 df-mpt 5186 df-id 5545 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-iota 6494 df-fun 6540 df-fv 6546 df-ov 7423 df-oprab 7424 df-mpo 7425 df-1st 8001 df-2nd 8002

This theorem is used by: eucalginv 16760 eucalglt 16761

W3C validator