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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 8982 | . . . . . . . . 9 ⊢ (𝑦 ∈ ℕ → 𝑦 ≠ 0) | |
2 | 1 | adantl 277 | . . . . . . . 8 ⊢ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ) → 𝑦 ≠ 0) |
3 | 2 | neneqd 2381 | . . . . . . 7 ⊢ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ) → ¬ 𝑦 = 0) |
4 | 3 | iffalsed 3559 | . . . . . 6 ⊢ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ) → if(𝑦 = 0, 〈𝑥, 𝑦〉, 〈𝑦, (𝑥 mod 𝑦)〉) = 〈𝑦, (𝑥 mod 𝑦)〉) |
5 | nnnn0 9218 | . . . . . . . 8 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℕ0) | |
6 | 5 | adantl 277 | . . . . . . 7 ⊢ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ) → 𝑦 ∈ ℕ0) |
7 | nn0z 9308 | . . . . . . . 8 ⊢ (𝑥 ∈ ℕ0 → 𝑥 ∈ ℤ) | |
8 | zmodcl 10381 | . . . . . . . 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 4679 | . . . . . . 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 2266 | . . . . 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 3554 | . . . . . 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 4679 | . . . . . 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 2266 | . . . 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 9213 | . . . . 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 799 | . . 3 ⊢ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) → if(𝑦 = 0, 〈𝑥, 𝑦〉, 〈𝑦, (𝑥 mod 𝑦)〉) ∈ (ℕ0 × ℕ0)) |
23 | 22 | rgen2a 2544 | . 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 6230 | . 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 709 = wceq 1364 ∈ wcel 2160 ≠ wne 2360 ∀wral 2468 ifcif 3549 〈cop 3613 × cxp 4645 ⟶wf 5234 (class class class)co 5900 ∈ cmpo 5902 0cc0 7846 ℕcn 8954 ℕ0cn0 9211 ℤcz 9288 mod cmo 10359 |
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 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4139 ax-pow 4195 ax-pr 4230 ax-un 4454 ax-setind 4557 ax-cnex 7937 ax-resscn 7938 ax-1cn 7939 ax-1re 7940 ax-icn 7941 ax-addcl 7942 ax-addrcl 7943 ax-mulcl 7944 ax-mulrcl 7945 ax-addcom 7946 ax-mulcom 7947 ax-addass 7948 ax-mulass 7949 ax-distr 7950 ax-i2m1 7951 ax-0lt1 7952 ax-1rid 7953 ax-0id 7954 ax-rnegex 7955 ax-precex 7956 ax-cnre 7957 ax-pre-ltirr 7958 ax-pre-ltwlin 7959 ax-pre-lttrn 7960 ax-pre-apti 7961 ax-pre-ltadd 7962 ax-pre-mulgt0 7963 ax-pre-mulext 7964 ax-arch 7965 |
This theorem depends on definitions: df-bi 117 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rmo 2476 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-if 3550 df-pw 3595 df-sn 3616 df-pr 3617 df-op 3619 df-uni 3828 df-int 3863 df-iun 3906 df-br 4022 df-opab 4083 df-mpt 4084 df-id 4314 df-po 4317 df-iso 4318 df-xp 4653 df-rel 4654 df-cnv 4655 df-co 4656 df-dm 4657 df-rn 4658 df-res 4659 df-ima 4660 df-iota 5199 df-fun 5240 df-fn 5241 df-f 5242 df-fv 5246 df-riota 5855 df-ov 5903 df-oprab 5904 df-mpo 5905 df-1st 6169 df-2nd 6170 df-pnf 8029 df-mnf 8030 df-xr 8031 df-ltxr 8032 df-le 8033 df-sub 8165 df-neg 8166 df-reap 8567 df-ap 8574 df-div 8665 df-inn 8955 df-n0 9212 df-z 9289 df-q 9656 df-rp 9690 df-fl 10307 df-mod 10360 |
This theorem is referenced by: eucalgcvga 12101 eucalg 12102 |
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