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Theorem eucalgval 11735

Description: Euclid's Algorithm eucalg 11740 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 5777	. . 3 ⊢ ((1^st ‘𝑋)𝐸(2^nd ‘𝑋)) = (𝐸‘⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩)
2		xp1st 6063	. . . 4 ⊢ (𝑋 ∈ (ℕ₀ × ℕ₀) → (1^st ‘𝑋) ∈ ℕ₀)
3		xp2nd 6064	. . . 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 11734	. . . 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 408	. . 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	syl5eqr 2186	. 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 6073	. . 3 ⊢ (𝑋 ∈ (ℕ₀ × ℕ₀) → 𝑋 = ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩)
9	8	fveq2d 5425	. 2 ⊢ (𝑋 ∈ (ℕ₀ × ℕ₀) → (𝐸‘𝑋) = (𝐸‘⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩))
10	8	fveq2d 5425	. . . . 5 ⊢ (𝑋 ∈ (ℕ₀ × ℕ₀) → ( mod ‘𝑋) = ( mod ‘⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩))
11		df-ov 5777	. . . . 5 ⊢ ((1^st ‘𝑋) mod (2^nd ‘𝑋)) = ( mod ‘⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩)
12	10, 11	syl6eqr 2190	. . . 4 ⊢ (𝑋 ∈ (ℕ₀ × ℕ₀) → ( mod ‘𝑋) = ((1^st ‘𝑋) mod (2^nd ‘𝑋)))
13	12	opeq2d 3712	. . 3 ⊢ (𝑋 ∈ (ℕ₀ × ℕ₀) → ⟨(2^nd ‘𝑋), ( mod ‘𝑋)⟩ = ⟨(2^nd ‘𝑋), ((1^st ‘𝑋) mod (2^nd ‘𝑋))⟩)
14	8, 13	ifeq12d 3491	. 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 2182	1 ⊢ (𝑋 ∈ (ℕ₀ × ℕ₀) → (𝐸‘𝑋) = if((2^nd ‘𝑋) = 0, 𝑋, ⟨(2^nd ‘𝑋), ( mod ‘𝑋)⟩))

Colors of variables: wff set class

Syntax hints: → wi 4 = wceq 1331 ∈ wcel 1480 ifcif 3474 ⟨cop 3530 × cxp 4537 ‘cfv 5123 (class class class)co 5774 ∈ cmpo 5776 1^st c1st 6036 2^nd c2nd 6037 0cc0 7620 ℕ₀cn0 8977 mod cmo 10095

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-mulrcl 7719 ax-addcom 7720 ax-mulcom 7721 ax-addass 7722 ax-mulass 7723 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-1rid 7727 ax-0id 7728 ax-rnegex 7729 ax-precex 7730 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-apti 7735 ax-pre-ltadd 7736 ax-pre-mulgt0 7737 ax-pre-mulext 7738 ax-arch 7739

This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rmo 2424 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-if 3475 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-po 4218 df-iso 4219 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-sub 7935 df-neg 7936 df-reap 8337 df-ap 8344 df-div 8433 df-inn 8721 df-n0 8978 df-z 9055 df-q 9412 df-rp 9442 df-fl 10043 df-mod 10096

This theorem is referenced by: eucalginv 11737 eucalglt 11738

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