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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 479 | . . . 4 ⊢ ((𝑥 = 𝑀 ∧ 𝑦 = 𝑁) → 𝑦 = 𝑁) | |
2 | 1 | eqeq1d 2762 | . . 3 ⊢ ((𝑥 = 𝑀 ∧ 𝑦 = 𝑁) → (𝑦 = 0 ↔ 𝑁 = 0)) |
3 | opeq12 4555 | . . 3 ⊢ ((𝑥 = 𝑀 ∧ 𝑦 = 𝑁) → 〈𝑥, 𝑦〉 = 〈𝑀, 𝑁〉) | |
4 | oveq12 6823 | . . . 4 ⊢ ((𝑥 = 𝑀 ∧ 𝑦 = 𝑁) → (𝑥 mod 𝑦) = (𝑀 mod 𝑁)) | |
5 | 1, 4 | opeq12d 4561 | . . 3 ⊢ ((𝑥 = 𝑀 ∧ 𝑦 = 𝑁) → 〈𝑦, (𝑥 mod 𝑦)〉 = 〈𝑁, (𝑀 mod 𝑁)〉) |
6 | 2, 3, 5 | ifbieq12d 4257 | . 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 5081 | . . 3 ⊢ 〈𝑀, 𝑁〉 ∈ V | |
9 | opex 5081 | . . 3 ⊢ 〈𝑁, (𝑀 mod 𝑁)〉 ∈ V | |
10 | 8, 9 | ifex 4300 | . 2 ⊢ if(𝑁 = 0, 〈𝑀, 𝑁〉, 〈𝑁, (𝑀 mod 𝑁)〉) ∈ V |
11 | 6, 7, 10 | ovmpt2a 6957 | 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 383 = wceq 1632 ∈ wcel 2139 ifcif 4230 〈cop 4327 (class class class)co 6814 ↦ cmpt2 6816 0cc0 10148 ℕ0cn0 11504 mod cmo 12882 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-sep 4933 ax-nul 4941 ax-pr 5055 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ral 3055 df-rex 3056 df-rab 3059 df-v 3342 df-sbc 3577 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-nul 4059 df-if 4231 df-sn 4322 df-pr 4324 df-op 4328 df-uni 4589 df-br 4805 df-opab 4865 df-id 5174 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-iota 6012 df-fun 6051 df-fv 6057 df-ov 6817 df-oprab 6818 df-mpt2 6819 |
This theorem is referenced by: eucalgval 15517 |
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