Theorem eucalgval2 16550

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 484	. . . 4 ⊢ ((𝑥 = 𝑀 ∧ 𝑦 = 𝑁) → 𝑦 = 𝑁)
2	1	eqeq1d 2738	. . 3 ⊢ ((𝑥 = 𝑀 ∧ 𝑦 = 𝑁) → (𝑦 = 0 ↔ 𝑁 = 0))
3		opeq12 4818	. . 3 ⊢ ((𝑥 = 𝑀 ∧ 𝑦 = 𝑁) → ⟨𝑥, 𝑦⟩ = ⟨𝑀, 𝑁⟩)
4		oveq12 7376	. . . 4 ⊢ ((𝑥 = 𝑀 ∧ 𝑦 = 𝑁) → (𝑥 mod 𝑦) = (𝑀 mod 𝑁))
5	1, 4	opeq12d 4824	. . 3 ⊢ ((𝑥 = 𝑀 ∧ 𝑦 = 𝑁) → ⟨𝑦, (𝑥 mod 𝑦)⟩ = ⟨𝑁, (𝑀 mod 𝑁)⟩)
6	2, 3, 5	ifbieq12d 4495	. 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 5416	. . 3 ⊢ ⟨𝑀, 𝑁⟩ ∈ V
9		opex 5416	. . 3 ⊢ ⟨𝑁, (𝑀 mod 𝑁)⟩ ∈ V
10	8, 9	ifex 4517	. 2 ⊢ if(𝑁 = 0, ⟨𝑀, 𝑁⟩, ⟨𝑁, (𝑀 mod 𝑁)⟩) ∈ V
11	6, 7, 10	ovmpoa 7522	1 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (𝑀𝐸𝑁) = if(𝑁 = 0, ⟨𝑀, 𝑁⟩, ⟨𝑁, (𝑀 mod 𝑁)⟩))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ifcif 4466  ⟨cop 4573  (class class class)co 7367   ∈ cmpo 7369  0cc0 11038  ℕ₀cn0 12437   mod cmo 13828

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-sbc 3729  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-iota 6454  df-fun 6500  df-fv 6506  df-ov 7370  df-oprab 7371  df-mpo 7372

This theorem is referenced by:  eucalgval  16551