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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 11659 | . . . . . . . . 9 ⊢ (𝑦 ∈ ℕ → 𝑦 ≠ 0) | |
2 | 1 | adantl 485 | . . . . . . . 8 ⊢ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ) → 𝑦 ≠ 0) |
3 | 2 | neneqd 2992 | . . . . . . 7 ⊢ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ) → ¬ 𝑦 = 0) |
4 | 3 | iffalsed 4436 | . . . . . 6 ⊢ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ) → if(𝑦 = 0, 〈𝑥, 𝑦〉, 〈𝑦, (𝑥 mod 𝑦)〉) = 〈𝑦, (𝑥 mod 𝑦)〉) |
5 | nnnn0 11892 | . . . . . . . 8 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℕ0) | |
6 | 5 | adantl 485 | . . . . . . 7 ⊢ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ) → 𝑦 ∈ ℕ0) |
7 | nn0z 11993 | . . . . . . . 8 ⊢ (𝑥 ∈ ℕ0 → 𝑥 ∈ ℤ) | |
8 | zmodcl 13254 | . . . . . . . 8 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) → (𝑥 mod 𝑦) ∈ ℕ0) | |
9 | 7, 8 | sylan 583 | . . . . . . 7 ⊢ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ) → (𝑥 mod 𝑦) ∈ ℕ0) |
10 | 6, 9 | opelxpd 5557 | . . . . . 6 ⊢ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ) → 〈𝑦, (𝑥 mod 𝑦)〉 ∈ (ℕ0 × ℕ0)) |
11 | 4, 10 | eqeltrd 2890 | . . . . 5 ⊢ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ) → if(𝑦 = 0, 〈𝑥, 𝑦〉, 〈𝑦, (𝑥 mod 𝑦)〉) ∈ (ℕ0 × ℕ0)) |
12 | 11 | adantlr 714 | . . . 4 ⊢ (((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑦 ∈ ℕ) → if(𝑦 = 0, 〈𝑥, 𝑦〉, 〈𝑦, (𝑥 mod 𝑦)〉) ∈ (ℕ0 × ℕ0)) |
13 | iftrue 4431 | . . . . . 6 ⊢ (𝑦 = 0 → if(𝑦 = 0, 〈𝑥, 𝑦〉, 〈𝑦, (𝑥 mod 𝑦)〉) = 〈𝑥, 𝑦〉) | |
14 | 13 | adantl 485 | . . . . 5 ⊢ (((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑦 = 0) → if(𝑦 = 0, 〈𝑥, 𝑦〉, 〈𝑦, (𝑥 mod 𝑦)〉) = 〈𝑥, 𝑦〉) |
15 | opelxpi 5556 | . . . . . 6 ⊢ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) → 〈𝑥, 𝑦〉 ∈ (ℕ0 × ℕ0)) | |
16 | 15 | adantr 484 | . . . . 5 ⊢ (((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑦 = 0) → 〈𝑥, 𝑦〉 ∈ (ℕ0 × ℕ0)) |
17 | 14, 16 | eqeltrd 2890 | . . . 4 ⊢ (((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑦 = 0) → if(𝑦 = 0, 〈𝑥, 𝑦〉, 〈𝑦, (𝑥 mod 𝑦)〉) ∈ (ℕ0 × ℕ0)) |
18 | simpr 488 | . . . . 5 ⊢ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) → 𝑦 ∈ ℕ0) | |
19 | elnn0 11887 | . . . . 5 ⊢ (𝑦 ∈ ℕ0 ↔ (𝑦 ∈ ℕ ∨ 𝑦 = 0)) | |
20 | 18, 19 | sylib 221 | . . . 4 ⊢ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) → (𝑦 ∈ ℕ ∨ 𝑦 = 0)) |
21 | 12, 17, 20 | mpjaodan 956 | . . 3 ⊢ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) → if(𝑦 = 0, 〈𝑥, 𝑦〉, 〈𝑦, (𝑥 mod 𝑦)〉) ∈ (ℕ0 × ℕ0)) |
22 | 21 | rgen2 3168 | . 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 7748 | . 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 233 | 1 ⊢ 𝐸:(ℕ0 × ℕ0)⟶(ℕ0 × ℕ0) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 399 ∨ wo 844 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ∀wral 3106 ifcif 4425 〈cop 4531 × cxp 5517 ⟶wf 6320 (class class class)co 7135 ∈ cmpo 7137 0cc0 10526 ℕcn 11625 ℕ0cn0 11885 ℤcz 11969 mod cmo 13232 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-sup 8890 df-inf 8891 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-fl 13157 df-mod 13233 |
This theorem is referenced by: eucalgcvga 15920 eucalg 15921 |
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