Theorem eucalgval2 16788

Description: The value of the step function 𝐸 for Euclid's Algorithm on an ordered pair. (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
eucalgval2	⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (𝑀𝐸𝑁) = if(𝑁 = 0, ⟨𝑀, 𝑁⟩, ⟨𝑁, (𝑀 mod 𝑁)⟩))

Distinct variable groups:   𝑥,𝑦,𝑀   𝑥,𝑁,𝑦

Allowed substitution hints:   𝐸(𝑥, 𝑦)

Proof of Theorem eucalgval2

Step	Hyp	Ref	Expression
1		simpr 490	. . . 4 ⊢ ((𝑥 = 𝑀 ∧ 𝑦 = 𝑁) → 𝑦 = 𝑁)
2	1	eqeq1d 2765	. . 3 ⊢ ((𝑥 = 𝑀 ∧ 𝑦 = 𝑁) → (𝑦 = 0 ↔ 𝑁 = 0))
3		opeq12 4845	. . 3 ⊢ ((𝑥 = 𝑀 ∧ 𝑦 = 𝑁) → ⟨𝑥, 𝑦⟩ = ⟨𝑀, 𝑁⟩)
4		oveq12 7455	. . . 4 ⊢ ((𝑥 = 𝑀 ∧ 𝑦 = 𝑁) → (𝑥 mod 𝑦) = (𝑀 mod 𝑁))
5	1, 4	opeq12d 4851	. . 3 ⊢ ((𝑥 = 𝑀 ∧ 𝑦 = 𝑁) → ⟨𝑦, (𝑥 mod 𝑦)⟩ = ⟨𝑁, (𝑀 mod 𝑁)⟩)
6	2, 3, 5	ifbieq12d 4514	. 2 ⊢ ((𝑥 = 𝑀 ∧ 𝑦 = 𝑁) → if(𝑦 = 0, ⟨𝑥, 𝑦⟩, ⟨𝑦, (𝑥 mod 𝑦)⟩) = if(𝑁 = 0, ⟨𝑀, 𝑁⟩, ⟨𝑁, (𝑀 mod 𝑁)⟩))
7		eucalgval.1	. 2 ⊢ 𝐸 = (𝑥 ∈ ℕ0, 𝑦 ∈ ℕ0 ↦ if(𝑦 = 0, ⟨𝑥, 𝑦⟩, ⟨𝑦, (𝑥 mod 𝑦)⟩))
8		opex 5437	. . 3 ⊢ ⟨𝑀, 𝑁⟩ ∈ V
9		opex 5437	. . 3 ⊢ ⟨𝑁, (𝑀 mod 𝑁)⟩ ∈ V
10	8, 9	ifex 4536	. 2 ⊢ if(𝑁 = 0, ⟨𝑀, 𝑁⟩, ⟨𝑁, (𝑀 mod 𝑁)⟩) ∈ V
11	6, 7, 10	ovmpoa 7602	1 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (𝑀𝐸𝑁) = if(𝑁 = 0, ⟨𝑀, 𝑁⟩, ⟨𝑁, (𝑀 mod 𝑁)⟩))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ifcif 4485  ⟨cop 4838  (class class class)co 7446   ∈ cmpo 7448  0cc0 11230  ℕ₀cn0 12636   mod cmo 14039

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5253  ax-pr 5396

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5172  df-id 5551  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-iota 6515  df-fun 6563  df-fv 6569  df-ov 7449  df-oprab 7450  df-mpo 7451

This theorem is used by:  eucalgval  16789