Theorem eucalgval2 16515

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 2726	. . 3 ⊢ ((𝑥 = 𝑀 ∧ 𝑦 = 𝑁) → (𝑦 = 0 ↔ 𝑁 = 0))
3		opeq12 4867	. . 3 ⊢ ((𝑥 = 𝑀 ∧ 𝑦 = 𝑁) → ⟨𝑥, 𝑦⟩ = ⟨𝑀, 𝑁⟩)
4		oveq12 7410	. . . 4 ⊢ ((𝑥 = 𝑀 ∧ 𝑦 = 𝑁) → (𝑥 mod 𝑦) = (𝑀 mod 𝑁))
5	1, 4	opeq12d 4873	. . 3 ⊢ ((𝑥 = 𝑀 ∧ 𝑦 = 𝑁) → ⟨𝑦, (𝑥 mod 𝑦)⟩ = ⟨𝑁, (𝑀 mod 𝑁)⟩)
6	2, 3, 5	ifbieq12d 4548	. 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 5454	. . 3 ⊢ ⟨𝑀, 𝑁⟩ ∈ V
9		opex 5454	. . 3 ⊢ ⟨𝑁, (𝑀 mod 𝑁)⟩ ∈ V
10	8, 9	ifex 4570	. 2 ⊢ if(𝑁 = 0, ⟨𝑀, 𝑁⟩, ⟨𝑁, (𝑀 mod 𝑁)⟩) ∈ V
11	6, 7, 10	ovmpoa 7555	1 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (𝑀𝐸𝑁) = if(𝑁 = 0, ⟨𝑀, 𝑁⟩, ⟨𝑁, (𝑀 mod 𝑁)⟩))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098  ifcif 4520  ⟨cop 4626  (class class class)co 7401   ∈ cmpo 7403  0cc0 11106  ℕ₀cn0 12469   mod cmo 13831

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pr 5417

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-sbc 3770  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-br 5139  df-opab 5201  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-iota 6485  df-fun 6535  df-fv 6541  df-ov 7404  df-oprab 7405  df-mpo 7406

This theorem is referenced by:  eucalgval  16516