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Mirrors > Home > MPE Home > Th. List > eucalgf | Structured version Visualization version GIF version |
Description: Domain and codomain of the step function 𝐸 for Euclid's Algorithm. (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 |
---|---|
eucalgf | ⊢ 𝐸:(ℕ0 × ℕ0)⟶(ℕ0 × ℕ0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnne0 12145 | . . . . . . . . 9 ⊢ (𝑦 ∈ ℕ → 𝑦 ≠ 0) | |
2 | 1 | adantl 482 | . . . . . . . 8 ⊢ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ) → 𝑦 ≠ 0) |
3 | 2 | neneqd 2946 | . . . . . . 7 ⊢ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ) → ¬ 𝑦 = 0) |
4 | 3 | iffalsed 4495 | . . . . . 6 ⊢ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ) → if(𝑦 = 0, 〈𝑥, 𝑦〉, 〈𝑦, (𝑥 mod 𝑦)〉) = 〈𝑦, (𝑥 mod 𝑦)〉) |
5 | nnnn0 12378 | . . . . . . . 8 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℕ0) | |
6 | 5 | adantl 482 | . . . . . . 7 ⊢ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ) → 𝑦 ∈ ℕ0) |
7 | nn0z 12482 | . . . . . . . 8 ⊢ (𝑥 ∈ ℕ0 → 𝑥 ∈ ℤ) | |
8 | zmodcl 13750 | . . . . . . . 8 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) → (𝑥 mod 𝑦) ∈ ℕ0) | |
9 | 7, 8 | sylan 580 | . . . . . . 7 ⊢ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ) → (𝑥 mod 𝑦) ∈ ℕ0) |
10 | 6, 9 | opelxpd 5669 | . . . . . 6 ⊢ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ) → 〈𝑦, (𝑥 mod 𝑦)〉 ∈ (ℕ0 × ℕ0)) |
11 | 4, 10 | eqeltrd 2838 | . . . . 5 ⊢ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ) → if(𝑦 = 0, 〈𝑥, 𝑦〉, 〈𝑦, (𝑥 mod 𝑦)〉) ∈ (ℕ0 × ℕ0)) |
12 | 11 | adantlr 713 | . . . 4 ⊢ (((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑦 ∈ ℕ) → if(𝑦 = 0, 〈𝑥, 𝑦〉, 〈𝑦, (𝑥 mod 𝑦)〉) ∈ (ℕ0 × ℕ0)) |
13 | iftrue 4490 | . . . . . 6 ⊢ (𝑦 = 0 → if(𝑦 = 0, 〈𝑥, 𝑦〉, 〈𝑦, (𝑥 mod 𝑦)〉) = 〈𝑥, 𝑦〉) | |
14 | 13 | adantl 482 | . . . . 5 ⊢ (((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑦 = 0) → if(𝑦 = 0, 〈𝑥, 𝑦〉, 〈𝑦, (𝑥 mod 𝑦)〉) = 〈𝑥, 𝑦〉) |
15 | opelxpi 5668 | . . . . . 6 ⊢ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) → 〈𝑥, 𝑦〉 ∈ (ℕ0 × ℕ0)) | |
16 | 15 | adantr 481 | . . . . 5 ⊢ (((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑦 = 0) → 〈𝑥, 𝑦〉 ∈ (ℕ0 × ℕ0)) |
17 | 14, 16 | eqeltrd 2838 | . . . 4 ⊢ (((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑦 = 0) → if(𝑦 = 0, 〈𝑥, 𝑦〉, 〈𝑦, (𝑥 mod 𝑦)〉) ∈ (ℕ0 × ℕ0)) |
18 | simpr 485 | . . . . 5 ⊢ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) → 𝑦 ∈ ℕ0) | |
19 | elnn0 12373 | . . . . 5 ⊢ (𝑦 ∈ ℕ0 ↔ (𝑦 ∈ ℕ ∨ 𝑦 = 0)) | |
20 | 18, 19 | sylib 217 | . . . 4 ⊢ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) → (𝑦 ∈ ℕ ∨ 𝑦 = 0)) |
21 | 12, 17, 20 | mpjaodan 957 | . . 3 ⊢ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) → if(𝑦 = 0, 〈𝑥, 𝑦〉, 〈𝑦, (𝑥 mod 𝑦)〉) ∈ (ℕ0 × ℕ0)) |
22 | 21 | rgen2 3192 | . 2 ⊢ ∀𝑥 ∈ ℕ0 ∀𝑦 ∈ ℕ0 if(𝑦 = 0, 〈𝑥, 𝑦〉, 〈𝑦, (𝑥 mod 𝑦)〉) ∈ (ℕ0 × ℕ0) |
23 | eucalgval.1 | . . 3 ⊢ 𝐸 = (𝑥 ∈ ℕ0, 𝑦 ∈ ℕ0 ↦ if(𝑦 = 0, 〈𝑥, 𝑦〉, 〈𝑦, (𝑥 mod 𝑦)〉)) | |
24 | 23 | fmpo 7992 | . 2 ⊢ (∀𝑥 ∈ ℕ0 ∀𝑦 ∈ ℕ0 if(𝑦 = 0, 〈𝑥, 𝑦〉, 〈𝑦, (𝑥 mod 𝑦)〉) ∈ (ℕ0 × ℕ0) ↔ 𝐸:(ℕ0 × ℕ0)⟶(ℕ0 × ℕ0)) |
25 | 22, 24 | mpbi 229 | 1 ⊢ 𝐸:(ℕ0 × ℕ0)⟶(ℕ0 × ℕ0) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 ∨ wo 845 = wceq 1541 ∈ wcel 2106 ≠ wne 2941 ∀wral 3062 ifcif 4484 〈cop 4590 × cxp 5629 ⟶wf 6489 (class class class)co 7351 ∈ cmpo 7353 0cc0 11009 ℕcn 12111 ℕ0cn0 12371 ℤcz 12457 mod cmo 13728 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 ax-pre-sup 11087 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7307 df-ov 7354 df-oprab 7355 df-mpo 7356 df-om 7795 df-1st 7913 df-2nd 7914 df-frecs 8204 df-wrecs 8235 df-recs 8309 df-rdg 8348 df-er 8606 df-en 8842 df-dom 8843 df-sdom 8844 df-sup 9336 df-inf 9337 df-pnf 11149 df-mnf 11150 df-xr 11151 df-ltxr 11152 df-le 11153 df-sub 11345 df-neg 11346 df-div 11771 df-nn 12112 df-n0 12372 df-z 12458 df-uz 12722 df-rp 12870 df-fl 13651 df-mod 13729 |
This theorem is referenced by: eucalgcvga 16421 eucalg 16422 |
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