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Theorem eucalgval2 15502
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 ∈ ℕ0𝑁 ∈ 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
StepHypRef Expression
1 simpr 471 . . . 4 ((𝑥 = 𝑀𝑦 = 𝑁) → 𝑦 = 𝑁)
21eqeq1d 2773 . . 3 ((𝑥 = 𝑀𝑦 = 𝑁) → (𝑦 = 0 ↔ 𝑁 = 0))
3 opeq12 4542 . . 3 ((𝑥 = 𝑀𝑦 = 𝑁) → ⟨𝑥, 𝑦⟩ = ⟨𝑀, 𝑁⟩)
4 oveq12 6805 . . . 4 ((𝑥 = 𝑀𝑦 = 𝑁) → (𝑥 mod 𝑦) = (𝑀 mod 𝑁))
51, 4opeq12d 4548 . . 3 ((𝑥 = 𝑀𝑦 = 𝑁) → ⟨𝑦, (𝑥 mod 𝑦)⟩ = ⟨𝑁, (𝑀 mod 𝑁)⟩)
62, 3, 5ifbieq12d 4253 . 2 ((𝑥 = 𝑀𝑦 = 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 5061 . . 3 𝑀, 𝑁⟩ ∈ V
9 opex 5061 . . 3 𝑁, (𝑀 mod 𝑁)⟩ ∈ V
108, 9ifex 4296 . 2 if(𝑁 = 0, ⟨𝑀, 𝑁⟩, ⟨𝑁, (𝑀 mod 𝑁)⟩) ∈ V
116, 7, 10ovmpt2a 6942 1 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → (𝑀𝐸𝑁) = if(𝑁 = 0, ⟨𝑀, 𝑁⟩, ⟨𝑁, (𝑀 mod 𝑁)⟩))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  ifcif 4226  cop 4323  (class class class)co 6796  cmpt2 6798  0cc0 10142  0cn0 11499   mod cmo 12876
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-iota 5993  df-fun 6032  df-fv 6038  df-ov 6799  df-oprab 6800  df-mpt2 6801
This theorem is referenced by:  eucalgval  15503
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