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Mirrors > Home > MPE Home > Th. List > eucalgval2 | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
eucalgval.1 | ⊢ 𝐸 = (𝑥 ∈ ℕ0, 𝑦 ∈ ℕ0 ↦ if(𝑦 = 0, 〈𝑥, 𝑦〉, 〈𝑦, (𝑥 mod 𝑦)〉)) |
Ref | Expression |
---|---|
eucalgval2 | ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (𝑀𝐸𝑁) = if(𝑁 = 0, 〈𝑀, 𝑁〉, 〈𝑁, (𝑀 mod 𝑁)〉)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 484 | . . . 4 ⊢ ((𝑥 = 𝑀 ∧ 𝑦 = 𝑁) → 𝑦 = 𝑁) | |
2 | 1 | eqeq1d 2737 | . . 3 ⊢ ((𝑥 = 𝑀 ∧ 𝑦 = 𝑁) → (𝑦 = 0 ↔ 𝑁 = 0)) |
3 | opeq12 4880 | . . 3 ⊢ ((𝑥 = 𝑀 ∧ 𝑦 = 𝑁) → 〈𝑥, 𝑦〉 = 〈𝑀, 𝑁〉) | |
4 | oveq12 7440 | . . . 4 ⊢ ((𝑥 = 𝑀 ∧ 𝑦 = 𝑁) → (𝑥 mod 𝑦) = (𝑀 mod 𝑁)) | |
5 | 1, 4 | opeq12d 4886 | . . 3 ⊢ ((𝑥 = 𝑀 ∧ 𝑦 = 𝑁) → 〈𝑦, (𝑥 mod 𝑦)〉 = 〈𝑁, (𝑀 mod 𝑁)〉) |
6 | 2, 3, 5 | ifbieq12d 4559 | . 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 5475 | . . 3 ⊢ 〈𝑀, 𝑁〉 ∈ V | |
9 | opex 5475 | . . 3 ⊢ 〈𝑁, (𝑀 mod 𝑁)〉 ∈ V | |
10 | 8, 9 | ifex 4581 | . 2 ⊢ if(𝑁 = 0, 〈𝑀, 𝑁〉, 〈𝑁, (𝑀 mod 𝑁)〉) ∈ V |
11 | 6, 7, 10 | ovmpoa 7588 | 1 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (𝑀𝐸𝑁) = if(𝑁 = 0, 〈𝑀, 𝑁〉, 〈𝑁, (𝑀 mod 𝑁)〉)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ifcif 4531 〈cop 4637 (class class class)co 7431 ∈ cmpo 7433 0cc0 11153 ℕ0cn0 12524 mod cmo 13906 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-iota 6516 df-fun 6565 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 |
This theorem is referenced by: eucalgval 16616 |
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