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Theorem eucalgval 12815

Description: Euclid's Algorithm eucalg 12820 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 6082	. . 3 ⊢ ((1^st ‘𝑋)𝐸(2^nd ‘𝑋)) = (𝐸‘⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩)
2		xp1st 6393	. . . 4 ⊢ (𝑋 ∈ (ℕ₀ × ℕ₀) → (1^st ‘𝑋) ∈ ℕ₀)
3		xp2nd 6394	. . . 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 12814	. . . 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 415	. . 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 2285	. 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 6403	. . 3 ⊢ (𝑋 ∈ (ℕ₀ × ℕ₀) → 𝑋 = ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩)
9	8	fveq2d 5697	. 2 ⊢ (𝑋 ∈ (ℕ₀ × ℕ₀) → (𝐸‘𝑋) = (𝐸‘⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩))
10	8	fveq2d 5697	. . . . 5 ⊢ (𝑋 ∈ (ℕ₀ × ℕ₀) → ( mod ‘𝑋) = ( mod ‘⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩))
11		df-ov 6082	. . . . 5 ⊢ ((1^st ‘𝑋) mod (2^nd ‘𝑋)) = ( mod ‘⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩)
12	10, 11	eqtr4di 2289	. . . 4 ⊢ (𝑋 ∈ (ℕ₀ × ℕ₀) → ( mod ‘𝑋) = ((1^st ‘𝑋) mod (2^nd ‘𝑋)))
13	12	opeq2d 3909	. . 3 ⊢ (𝑋 ∈ (ℕ₀ × ℕ₀) → ⟨(2^nd ‘𝑋), ( mod ‘𝑋)⟩ = ⟨(2^nd ‘𝑋), ((1^st ‘𝑋) mod (2^nd ‘𝑋))⟩)
14	8, 13	ifeq12d 3660	. 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 2281	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 1402 ∈ wcel 2209 ifcif 3638 ⟨cop 3711 × cxp 4770 ‘cfv 5375 (class class class)co 6079 ∈ cmpo 6081 1^st c1st 6366 2^nd c2nd 6367 0cc0 8173 ℕ₀cn0 9546 mod cmo 10742

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-sep 4247 ax-pow 4309 ax-pr 4344 ax-un 4576 ax-setind 4682 ax-cnex 8264 ax-resscn 8265 ax-1cn 8266 ax-1re 8267 ax-icn 8268 ax-addcl 8269 ax-addrcl 8270 ax-mulcl 8271 ax-mulrcl 8272 ax-addcom 8273 ax-mulcom 8274 ax-addass 8275 ax-mulass 8276 ax-distr 8277 ax-i2m1 8278 ax-0lt1 8279 ax-1rid 8280 ax-0id 8281 ax-rnegex 8282 ax-precex 8283 ax-cnre 8284 ax-pre-ltirr 8285 ax-pre-ltwlin 8286 ax-pre-lttrn 8287 ax-pre-apti 8288 ax-pre-ltadd 8289 ax-pre-mulgt0 8290 ax-pre-mulext 8291 ax-arch 8292

This theorem depends on definitions: df-bi 117 df-dc 847 df-3or 1010 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-ral 2533 df-rex 2534 df-reu 2535 df-rmo 2536 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-nul 3521 df-if 3639 df-pw 3690 df-sn 3714 df-pr 3715 df-op 3717 df-uni 3934 df-int 3969 df-iun 4012 df-br 4129 df-opab 4191 df-mpt 4192 df-id 4436 df-po 4439 df-iso 4440 df-xp 4778 df-rel 4779 df-cnv 4780 df-co 4781 df-dm 4782 df-rn 4783 df-res 4784 df-ima 4785 df-iota 5335 df-fun 5377 df-fn 5378 df-f 5379 df-fv 5383 df-riota 6032 df-ov 6082 df-oprab 6083 df-mpo 6084 df-1st 6368 df-2nd 6369 df-pnf 8356 df-mnf 8357 df-xr 8358 df-ltxr 8359 df-le 8360 df-sub 8493 df-neg 8494 df-reap 8897 df-ap 8904 df-div 8997 df-inn 9288 df-n0 9547 df-z 9628 df-q 10003 df-rp 10038 df-fl 10688 df-mod 10743

This theorem is referenced by: eucalginv 12817 eucalglt 12818

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