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Mirrors > Home > ILE Home > Th. List > eucalgcvga | GIF version |
Description: Once Euclid's Algorithm halts after 𝑁 steps, the second element of the state remains 0 . (Contributed by Paul Chapman, 22-Jun-2011.) (Revised by Mario Carneiro, 29-May-2014.) |
Ref | Expression |
---|---|
eucalgval.1 | ⊢ 𝐸 = (𝑥 ∈ ℕ0, 𝑦 ∈ ℕ0 ↦ if(𝑦 = 0, 〈𝑥, 𝑦〉, 〈𝑦, (𝑥 mod 𝑦)〉)) |
eucalg.2 | ⊢ 𝑅 = seq0((𝐸 ∘ 1st ), (ℕ0 × {𝐴})) |
eucalgcvga.3 | ⊢ 𝑁 = (2nd ‘𝐴) |
Ref | Expression |
---|---|
eucalgcvga | ⊢ (𝐴 ∈ (ℕ0 × ℕ0) → (𝐾 ∈ (ℤ≥‘𝑁) → (2nd ‘(𝑅‘𝐾)) = 0)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eucalgcvga.3 | . . . . . . 7 ⊢ 𝑁 = (2nd ‘𝐴) | |
2 | xp2nd 6064 | . . . . . . 7 ⊢ (𝐴 ∈ (ℕ0 × ℕ0) → (2nd ‘𝐴) ∈ ℕ0) | |
3 | 1, 2 | eqeltrid 2226 | . . . . . 6 ⊢ (𝐴 ∈ (ℕ0 × ℕ0) → 𝑁 ∈ ℕ0) |
4 | eluznn0 9393 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐾 ∈ (ℤ≥‘𝑁)) → 𝐾 ∈ ℕ0) | |
5 | 3, 4 | sylan 281 | . . . . 5 ⊢ ((𝐴 ∈ (ℕ0 × ℕ0) ∧ 𝐾 ∈ (ℤ≥‘𝑁)) → 𝐾 ∈ ℕ0) |
6 | nn0uz 9360 | . . . . . . 7 ⊢ ℕ0 = (ℤ≥‘0) | |
7 | eucalg.2 | . . . . . . 7 ⊢ 𝑅 = seq0((𝐸 ∘ 1st ), (ℕ0 × {𝐴})) | |
8 | 0zd 9066 | . . . . . . 7 ⊢ (𝐴 ∈ (ℕ0 × ℕ0) → 0 ∈ ℤ) | |
9 | id 19 | . . . . . . 7 ⊢ (𝐴 ∈ (ℕ0 × ℕ0) → 𝐴 ∈ (ℕ0 × ℕ0)) | |
10 | eucalgval.1 | . . . . . . . . 9 ⊢ 𝐸 = (𝑥 ∈ ℕ0, 𝑦 ∈ ℕ0 ↦ if(𝑦 = 0, 〈𝑥, 𝑦〉, 〈𝑦, (𝑥 mod 𝑦)〉)) | |
11 | 10 | eucalgf 11736 | . . . . . . . 8 ⊢ 𝐸:(ℕ0 × ℕ0)⟶(ℕ0 × ℕ0) |
12 | 11 | a1i 9 | . . . . . . 7 ⊢ (𝐴 ∈ (ℕ0 × ℕ0) → 𝐸:(ℕ0 × ℕ0)⟶(ℕ0 × ℕ0)) |
13 | 6, 7, 8, 9, 12 | algrf 11726 | . . . . . 6 ⊢ (𝐴 ∈ (ℕ0 × ℕ0) → 𝑅:ℕ0⟶(ℕ0 × ℕ0)) |
14 | 13 | ffvelrnda 5555 | . . . . 5 ⊢ ((𝐴 ∈ (ℕ0 × ℕ0) ∧ 𝐾 ∈ ℕ0) → (𝑅‘𝐾) ∈ (ℕ0 × ℕ0)) |
15 | 5, 14 | syldan 280 | . . . 4 ⊢ ((𝐴 ∈ (ℕ0 × ℕ0) ∧ 𝐾 ∈ (ℤ≥‘𝑁)) → (𝑅‘𝐾) ∈ (ℕ0 × ℕ0)) |
16 | fvres 5445 | . . . 4 ⊢ ((𝑅‘𝐾) ∈ (ℕ0 × ℕ0) → ((2nd ↾ (ℕ0 × ℕ0))‘(𝑅‘𝐾)) = (2nd ‘(𝑅‘𝐾))) | |
17 | 15, 16 | syl 14 | . . 3 ⊢ ((𝐴 ∈ (ℕ0 × ℕ0) ∧ 𝐾 ∈ (ℤ≥‘𝑁)) → ((2nd ↾ (ℕ0 × ℕ0))‘(𝑅‘𝐾)) = (2nd ‘(𝑅‘𝐾))) |
18 | simpl 108 | . . . 4 ⊢ ((𝐴 ∈ (ℕ0 × ℕ0) ∧ 𝐾 ∈ (ℤ≥‘𝑁)) → 𝐴 ∈ (ℕ0 × ℕ0)) | |
19 | fvres 5445 | . . . . . . . 8 ⊢ (𝐴 ∈ (ℕ0 × ℕ0) → ((2nd ↾ (ℕ0 × ℕ0))‘𝐴) = (2nd ‘𝐴)) | |
20 | 19, 1 | syl6eqr 2190 | . . . . . . 7 ⊢ (𝐴 ∈ (ℕ0 × ℕ0) → ((2nd ↾ (ℕ0 × ℕ0))‘𝐴) = 𝑁) |
21 | 20 | fveq2d 5425 | . . . . . 6 ⊢ (𝐴 ∈ (ℕ0 × ℕ0) → (ℤ≥‘((2nd ↾ (ℕ0 × ℕ0))‘𝐴)) = (ℤ≥‘𝑁)) |
22 | 21 | eleq2d 2209 | . . . . 5 ⊢ (𝐴 ∈ (ℕ0 × ℕ0) → (𝐾 ∈ (ℤ≥‘((2nd ↾ (ℕ0 × ℕ0))‘𝐴)) ↔ 𝐾 ∈ (ℤ≥‘𝑁))) |
23 | 22 | biimpar 295 | . . . 4 ⊢ ((𝐴 ∈ (ℕ0 × ℕ0) ∧ 𝐾 ∈ (ℤ≥‘𝑁)) → 𝐾 ∈ (ℤ≥‘((2nd ↾ (ℕ0 × ℕ0))‘𝐴))) |
24 | f2ndres 6058 | . . . . 5 ⊢ (2nd ↾ (ℕ0 × ℕ0)):(ℕ0 × ℕ0)⟶ℕ0 | |
25 | 10 | eucalglt 11738 | . . . . . 6 ⊢ (𝑧 ∈ (ℕ0 × ℕ0) → ((2nd ‘(𝐸‘𝑧)) ≠ 0 → (2nd ‘(𝐸‘𝑧)) < (2nd ‘𝑧))) |
26 | 11 | ffvelrni 5554 | . . . . . . . 8 ⊢ (𝑧 ∈ (ℕ0 × ℕ0) → (𝐸‘𝑧) ∈ (ℕ0 × ℕ0)) |
27 | fvres 5445 | . . . . . . . 8 ⊢ ((𝐸‘𝑧) ∈ (ℕ0 × ℕ0) → ((2nd ↾ (ℕ0 × ℕ0))‘(𝐸‘𝑧)) = (2nd ‘(𝐸‘𝑧))) | |
28 | 26, 27 | syl 14 | . . . . . . 7 ⊢ (𝑧 ∈ (ℕ0 × ℕ0) → ((2nd ↾ (ℕ0 × ℕ0))‘(𝐸‘𝑧)) = (2nd ‘(𝐸‘𝑧))) |
29 | 28 | neeq1d 2326 | . . . . . 6 ⊢ (𝑧 ∈ (ℕ0 × ℕ0) → (((2nd ↾ (ℕ0 × ℕ0))‘(𝐸‘𝑧)) ≠ 0 ↔ (2nd ‘(𝐸‘𝑧)) ≠ 0)) |
30 | fvres 5445 | . . . . . . 7 ⊢ (𝑧 ∈ (ℕ0 × ℕ0) → ((2nd ↾ (ℕ0 × ℕ0))‘𝑧) = (2nd ‘𝑧)) | |
31 | 28, 30 | breq12d 3942 | . . . . . 6 ⊢ (𝑧 ∈ (ℕ0 × ℕ0) → (((2nd ↾ (ℕ0 × ℕ0))‘(𝐸‘𝑧)) < ((2nd ↾ (ℕ0 × ℕ0))‘𝑧) ↔ (2nd ‘(𝐸‘𝑧)) < (2nd ‘𝑧))) |
32 | 25, 29, 31 | 3imtr4d 202 | . . . . 5 ⊢ (𝑧 ∈ (ℕ0 × ℕ0) → (((2nd ↾ (ℕ0 × ℕ0))‘(𝐸‘𝑧)) ≠ 0 → ((2nd ↾ (ℕ0 × ℕ0))‘(𝐸‘𝑧)) < ((2nd ↾ (ℕ0 × ℕ0))‘𝑧))) |
33 | eqid 2139 | . . . . 5 ⊢ ((2nd ↾ (ℕ0 × ℕ0))‘𝐴) = ((2nd ↾ (ℕ0 × ℕ0))‘𝐴) | |
34 | 11, 7, 24, 32, 33 | algcvga 11732 | . . . 4 ⊢ (𝐴 ∈ (ℕ0 × ℕ0) → (𝐾 ∈ (ℤ≥‘((2nd ↾ (ℕ0 × ℕ0))‘𝐴)) → ((2nd ↾ (ℕ0 × ℕ0))‘(𝑅‘𝐾)) = 0)) |
35 | 18, 23, 34 | sylc 62 | . . 3 ⊢ ((𝐴 ∈ (ℕ0 × ℕ0) ∧ 𝐾 ∈ (ℤ≥‘𝑁)) → ((2nd ↾ (ℕ0 × ℕ0))‘(𝑅‘𝐾)) = 0) |
36 | 17, 35 | eqtr3d 2174 | . 2 ⊢ ((𝐴 ∈ (ℕ0 × ℕ0) ∧ 𝐾 ∈ (ℤ≥‘𝑁)) → (2nd ‘(𝑅‘𝐾)) = 0) |
37 | 36 | ex 114 | 1 ⊢ (𝐴 ∈ (ℕ0 × ℕ0) → (𝐾 ∈ (ℤ≥‘𝑁) → (2nd ‘(𝑅‘𝐾)) = 0)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1331 ∈ wcel 1480 ≠ wne 2308 ifcif 3474 {csn 3527 〈cop 3530 class class class wbr 3929 × cxp 4537 ↾ cres 4541 ∘ ccom 4543 ⟶wf 5119 ‘cfv 5123 (class class class)co 5774 ∈ cmpo 5776 1st c1st 6036 2nd c2nd 6037 0cc0 7620 < clt 7800 ℕ0cn0 8977 ℤ≥cuz 9326 mod cmo 10095 seqcseq 10218 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-mulrcl 7719 ax-addcom 7720 ax-mulcom 7721 ax-addass 7722 ax-mulass 7723 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-1rid 7727 ax-0id 7728 ax-rnegex 7729 ax-precex 7730 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-apti 7735 ax-pre-ltadd 7736 ax-pre-mulgt0 7737 ax-pre-mulext 7738 ax-arch 7739 |
This theorem depends on definitions: df-bi 116 df-stab 816 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rmo 2424 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-if 3475 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-po 4218 df-iso 4219 df-iord 4288 df-on 4290 df-ilim 4291 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-recs 6202 df-frec 6288 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-sub 7935 df-neg 7936 df-reap 8337 df-ap 8344 df-div 8433 df-inn 8721 df-n0 8978 df-z 9055 df-uz 9327 df-q 9412 df-rp 9442 df-fl 10043 df-mod 10096 df-seqfrec 10219 |
This theorem is referenced by: eucalg 11740 |
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