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Mirrors > Home > ILE Home > Th. List > eucalgf | 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 8923 | . . . . . . . . 9 ⊢ (𝑦 ∈ ℕ → 𝑦 ≠ 0) | |
2 | 1 | adantl 277 | . . . . . . . 8 ⊢ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ) → 𝑦 ≠ 0) |
3 | 2 | neneqd 2368 | . . . . . . 7 ⊢ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ) → ¬ 𝑦 = 0) |
4 | 3 | iffalsed 3544 | . . . . . 6 ⊢ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ) → if(𝑦 = 0, 〈𝑥, 𝑦〉, 〈𝑦, (𝑥 mod 𝑦)〉) = 〈𝑦, (𝑥 mod 𝑦)〉) |
5 | nnnn0 9159 | . . . . . . . 8 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℕ0) | |
6 | 5 | adantl 277 | . . . . . . 7 ⊢ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ) → 𝑦 ∈ ℕ0) |
7 | nn0z 9249 | . . . . . . . 8 ⊢ (𝑥 ∈ ℕ0 → 𝑥 ∈ ℤ) | |
8 | zmodcl 10317 | . . . . . . . 8 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) → (𝑥 mod 𝑦) ∈ ℕ0) | |
9 | 7, 8 | sylan 283 | . . . . . . 7 ⊢ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ) → (𝑥 mod 𝑦) ∈ ℕ0) |
10 | opelxpi 4654 | . . . . . . 7 ⊢ ((𝑦 ∈ ℕ0 ∧ (𝑥 mod 𝑦) ∈ ℕ0) → 〈𝑦, (𝑥 mod 𝑦)〉 ∈ (ℕ0 × ℕ0)) | |
11 | 6, 9, 10 | syl2anc 411 | . . . . . 6 ⊢ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ) → 〈𝑦, (𝑥 mod 𝑦)〉 ∈ (ℕ0 × ℕ0)) |
12 | 4, 11 | eqeltrd 2254 | . . . . 5 ⊢ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ) → if(𝑦 = 0, 〈𝑥, 𝑦〉, 〈𝑦, (𝑥 mod 𝑦)〉) ∈ (ℕ0 × ℕ0)) |
13 | 12 | adantlr 477 | . . . 4 ⊢ (((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑦 ∈ ℕ) → if(𝑦 = 0, 〈𝑥, 𝑦〉, 〈𝑦, (𝑥 mod 𝑦)〉) ∈ (ℕ0 × ℕ0)) |
14 | iftrue 3539 | . . . . . 6 ⊢ (𝑦 = 0 → if(𝑦 = 0, 〈𝑥, 𝑦〉, 〈𝑦, (𝑥 mod 𝑦)〉) = 〈𝑥, 𝑦〉) | |
15 | 14 | adantl 277 | . . . . 5 ⊢ (((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑦 = 0) → if(𝑦 = 0, 〈𝑥, 𝑦〉, 〈𝑦, (𝑥 mod 𝑦)〉) = 〈𝑥, 𝑦〉) |
16 | opelxpi 4654 | . . . . . 6 ⊢ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) → 〈𝑥, 𝑦〉 ∈ (ℕ0 × ℕ0)) | |
17 | 16 | adantr 276 | . . . . 5 ⊢ (((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑦 = 0) → 〈𝑥, 𝑦〉 ∈ (ℕ0 × ℕ0)) |
18 | 15, 17 | eqeltrd 2254 | . . . 4 ⊢ (((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑦 = 0) → if(𝑦 = 0, 〈𝑥, 𝑦〉, 〈𝑦, (𝑥 mod 𝑦)〉) ∈ (ℕ0 × ℕ0)) |
19 | simpr 110 | . . . . 5 ⊢ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) → 𝑦 ∈ ℕ0) | |
20 | elnn0 9154 | . . . . 5 ⊢ (𝑦 ∈ ℕ0 ↔ (𝑦 ∈ ℕ ∨ 𝑦 = 0)) | |
21 | 19, 20 | sylib 122 | . . . 4 ⊢ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) → (𝑦 ∈ ℕ ∨ 𝑦 = 0)) |
22 | 13, 18, 21 | mpjaodan 798 | . . 3 ⊢ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) → if(𝑦 = 0, 〈𝑥, 𝑦〉, 〈𝑦, (𝑥 mod 𝑦)〉) ∈ (ℕ0 × ℕ0)) |
23 | 22 | rgen2a 2531 | . 2 ⊢ ∀𝑥 ∈ ℕ0 ∀𝑦 ∈ ℕ0 if(𝑦 = 0, 〈𝑥, 𝑦〉, 〈𝑦, (𝑥 mod 𝑦)〉) ∈ (ℕ0 × ℕ0) |
24 | eucalgval.1 | . . 3 ⊢ 𝐸 = (𝑥 ∈ ℕ0, 𝑦 ∈ ℕ0 ↦ if(𝑦 = 0, 〈𝑥, 𝑦〉, 〈𝑦, (𝑥 mod 𝑦)〉)) | |
25 | 24 | fmpo 6195 | . 2 ⊢ (∀𝑥 ∈ ℕ0 ∀𝑦 ∈ ℕ0 if(𝑦 = 0, 〈𝑥, 𝑦〉, 〈𝑦, (𝑥 mod 𝑦)〉) ∈ (ℕ0 × ℕ0) ↔ 𝐸:(ℕ0 × ℕ0)⟶(ℕ0 × ℕ0)) |
26 | 23, 25 | mpbi 145 | 1 ⊢ 𝐸:(ℕ0 × ℕ0)⟶(ℕ0 × ℕ0) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 104 ∨ wo 708 = wceq 1353 ∈ wcel 2148 ≠ wne 2347 ∀wral 2455 ifcif 3534 〈cop 3594 × cxp 4620 ⟶wf 5207 (class class class)co 5868 ∈ cmpo 5870 0cc0 7789 ℕcn 8895 ℕ0cn0 9152 ℤcz 9229 mod cmo 10295 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4205 ax-un 4429 ax-setind 4532 ax-cnex 7880 ax-resscn 7881 ax-1cn 7882 ax-1re 7883 ax-icn 7884 ax-addcl 7885 ax-addrcl 7886 ax-mulcl 7887 ax-mulrcl 7888 ax-addcom 7889 ax-mulcom 7890 ax-addass 7891 ax-mulass 7892 ax-distr 7893 ax-i2m1 7894 ax-0lt1 7895 ax-1rid 7896 ax-0id 7897 ax-rnegex 7898 ax-precex 7899 ax-cnre 7900 ax-pre-ltirr 7901 ax-pre-ltwlin 7902 ax-pre-lttrn 7903 ax-pre-apti 7904 ax-pre-ltadd 7905 ax-pre-mulgt0 7906 ax-pre-mulext 7907 ax-arch 7908 |
This theorem depends on definitions: df-bi 117 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-if 3535 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 df-iun 3886 df-br 4001 df-opab 4062 df-mpt 4063 df-id 4289 df-po 4292 df-iso 4293 df-xp 4628 df-rel 4629 df-cnv 4630 df-co 4631 df-dm 4632 df-rn 4633 df-res 4634 df-ima 4635 df-iota 5173 df-fun 5213 df-fn 5214 df-f 5215 df-fv 5219 df-riota 5824 df-ov 5871 df-oprab 5872 df-mpo 5873 df-1st 6134 df-2nd 6135 df-pnf 7971 df-mnf 7972 df-xr 7973 df-ltxr 7974 df-le 7975 df-sub 8107 df-neg 8108 df-reap 8509 df-ap 8516 df-div 8606 df-inn 8896 df-n0 9153 df-z 9230 df-q 9596 df-rp 9628 df-fl 10243 df-mod 10296 |
This theorem is referenced by: eucalgcvga 12028 eucalg 12029 |
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