Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > eucalgcvga | Structured version Visualization version 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 7090 | . . . . . . 7 ⊢ (𝐴 ∈ (ℕ0 × ℕ0) → (2nd ‘𝐴) ∈ ℕ0) | |
3 | 1, 2 | syl5eqel 2692 | . . . . . 6 ⊢ (𝐴 ∈ (ℕ0 × ℕ0) → 𝑁 ∈ ℕ0) |
4 | eluznn0 11633 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐾 ∈ (ℤ≥‘𝑁)) → 𝐾 ∈ ℕ0) | |
5 | 3, 4 | sylan 487 | . . . . 5 ⊢ ((𝐴 ∈ (ℕ0 × ℕ0) ∧ 𝐾 ∈ (ℤ≥‘𝑁)) → 𝐾 ∈ ℕ0) |
6 | nn0uz 11598 | . . . . . . 7 ⊢ ℕ0 = (ℤ≥‘0) | |
7 | eucalg.2 | . . . . . . 7 ⊢ 𝑅 = seq0((𝐸 ∘ 1st ), (ℕ0 × {𝐴})) | |
8 | 0zd 11266 | . . . . . . 7 ⊢ (𝐴 ∈ (ℕ0 × ℕ0) → 0 ∈ ℤ) | |
9 | id 22 | . . . . . . 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 15134 | . . . . . . . 8 ⊢ 𝐸:(ℕ0 × ℕ0)⟶(ℕ0 × ℕ0) |
12 | 11 | a1i 11 | . . . . . . 7 ⊢ (𝐴 ∈ (ℕ0 × ℕ0) → 𝐸:(ℕ0 × ℕ0)⟶(ℕ0 × ℕ0)) |
13 | 6, 7, 8, 9, 12 | algrf 15124 | . . . . . 6 ⊢ (𝐴 ∈ (ℕ0 × ℕ0) → 𝑅:ℕ0⟶(ℕ0 × ℕ0)) |
14 | 13 | ffvelrnda 6267 | . . . . 5 ⊢ ((𝐴 ∈ (ℕ0 × ℕ0) ∧ 𝐾 ∈ ℕ0) → (𝑅‘𝐾) ∈ (ℕ0 × ℕ0)) |
15 | 5, 14 | syldan 486 | . . . 4 ⊢ ((𝐴 ∈ (ℕ0 × ℕ0) ∧ 𝐾 ∈ (ℤ≥‘𝑁)) → (𝑅‘𝐾) ∈ (ℕ0 × ℕ0)) |
16 | fvres 6117 | . . . 4 ⊢ ((𝑅‘𝐾) ∈ (ℕ0 × ℕ0) → ((2nd ↾ (ℕ0 × ℕ0))‘(𝑅‘𝐾)) = (2nd ‘(𝑅‘𝐾))) | |
17 | 15, 16 | syl 17 | . . 3 ⊢ ((𝐴 ∈ (ℕ0 × ℕ0) ∧ 𝐾 ∈ (ℤ≥‘𝑁)) → ((2nd ↾ (ℕ0 × ℕ0))‘(𝑅‘𝐾)) = (2nd ‘(𝑅‘𝐾))) |
18 | simpl 472 | . . . 4 ⊢ ((𝐴 ∈ (ℕ0 × ℕ0) ∧ 𝐾 ∈ (ℤ≥‘𝑁)) → 𝐴 ∈ (ℕ0 × ℕ0)) | |
19 | fvres 6117 | . . . . . . . 8 ⊢ (𝐴 ∈ (ℕ0 × ℕ0) → ((2nd ↾ (ℕ0 × ℕ0))‘𝐴) = (2nd ‘𝐴)) | |
20 | 19, 1 | syl6eqr 2662 | . . . . . . 7 ⊢ (𝐴 ∈ (ℕ0 × ℕ0) → ((2nd ↾ (ℕ0 × ℕ0))‘𝐴) = 𝑁) |
21 | 20 | fveq2d 6107 | . . . . . 6 ⊢ (𝐴 ∈ (ℕ0 × ℕ0) → (ℤ≥‘((2nd ↾ (ℕ0 × ℕ0))‘𝐴)) = (ℤ≥‘𝑁)) |
22 | 21 | eleq2d 2673 | . . . . 5 ⊢ (𝐴 ∈ (ℕ0 × ℕ0) → (𝐾 ∈ (ℤ≥‘((2nd ↾ (ℕ0 × ℕ0))‘𝐴)) ↔ 𝐾 ∈ (ℤ≥‘𝑁))) |
23 | 22 | biimpar 501 | . . . 4 ⊢ ((𝐴 ∈ (ℕ0 × ℕ0) ∧ 𝐾 ∈ (ℤ≥‘𝑁)) → 𝐾 ∈ (ℤ≥‘((2nd ↾ (ℕ0 × ℕ0))‘𝐴))) |
24 | f2ndres 7082 | . . . . 5 ⊢ (2nd ↾ (ℕ0 × ℕ0)):(ℕ0 × ℕ0)⟶ℕ0 | |
25 | 10 | eucalglt 15136 | . . . . . 6 ⊢ (𝑧 ∈ (ℕ0 × ℕ0) → ((2nd ‘(𝐸‘𝑧)) ≠ 0 → (2nd ‘(𝐸‘𝑧)) < (2nd ‘𝑧))) |
26 | 11 | ffvelrni 6266 | . . . . . . . 8 ⊢ (𝑧 ∈ (ℕ0 × ℕ0) → (𝐸‘𝑧) ∈ (ℕ0 × ℕ0)) |
27 | fvres 6117 | . . . . . . . 8 ⊢ ((𝐸‘𝑧) ∈ (ℕ0 × ℕ0) → ((2nd ↾ (ℕ0 × ℕ0))‘(𝐸‘𝑧)) = (2nd ‘(𝐸‘𝑧))) | |
28 | 26, 27 | syl 17 | . . . . . . 7 ⊢ (𝑧 ∈ (ℕ0 × ℕ0) → ((2nd ↾ (ℕ0 × ℕ0))‘(𝐸‘𝑧)) = (2nd ‘(𝐸‘𝑧))) |
29 | 28 | neeq1d 2841 | . . . . . 6 ⊢ (𝑧 ∈ (ℕ0 × ℕ0) → (((2nd ↾ (ℕ0 × ℕ0))‘(𝐸‘𝑧)) ≠ 0 ↔ (2nd ‘(𝐸‘𝑧)) ≠ 0)) |
30 | fvres 6117 | . . . . . . 7 ⊢ (𝑧 ∈ (ℕ0 × ℕ0) → ((2nd ↾ (ℕ0 × ℕ0))‘𝑧) = (2nd ‘𝑧)) | |
31 | 28, 30 | breq12d 4596 | . . . . . 6 ⊢ (𝑧 ∈ (ℕ0 × ℕ0) → (((2nd ↾ (ℕ0 × ℕ0))‘(𝐸‘𝑧)) < ((2nd ↾ (ℕ0 × ℕ0))‘𝑧) ↔ (2nd ‘(𝐸‘𝑧)) < (2nd ‘𝑧))) |
32 | 25, 29, 31 | 3imtr4d 282 | . . . . 5 ⊢ (𝑧 ∈ (ℕ0 × ℕ0) → (((2nd ↾ (ℕ0 × ℕ0))‘(𝐸‘𝑧)) ≠ 0 → ((2nd ↾ (ℕ0 × ℕ0))‘(𝐸‘𝑧)) < ((2nd ↾ (ℕ0 × ℕ0))‘𝑧))) |
33 | eqid 2610 | . . . . 5 ⊢ ((2nd ↾ (ℕ0 × ℕ0))‘𝐴) = ((2nd ↾ (ℕ0 × ℕ0))‘𝐴) | |
34 | 11, 7, 24, 32, 33 | algcvga 15130 | . . . 4 ⊢ (𝐴 ∈ (ℕ0 × ℕ0) → (𝐾 ∈ (ℤ≥‘((2nd ↾ (ℕ0 × ℕ0))‘𝐴)) → ((2nd ↾ (ℕ0 × ℕ0))‘(𝑅‘𝐾)) = 0)) |
35 | 18, 23, 34 | sylc 63 | . . 3 ⊢ ((𝐴 ∈ (ℕ0 × ℕ0) ∧ 𝐾 ∈ (ℤ≥‘𝑁)) → ((2nd ↾ (ℕ0 × ℕ0))‘(𝑅‘𝐾)) = 0) |
36 | 17, 35 | eqtr3d 2646 | . 2 ⊢ ((𝐴 ∈ (ℕ0 × ℕ0) ∧ 𝐾 ∈ (ℤ≥‘𝑁)) → (2nd ‘(𝑅‘𝐾)) = 0) |
37 | 36 | ex 449 | 1 ⊢ (𝐴 ∈ (ℕ0 × ℕ0) → (𝐾 ∈ (ℤ≥‘𝑁) → (2nd ‘(𝑅‘𝐾)) = 0)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ifcif 4036 {csn 4125 〈cop 4131 class class class wbr 4583 × cxp 5036 ↾ cres 5040 ∘ ccom 5042 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ↦ cmpt2 6551 1st c1st 7057 2nd c2nd 7058 0cc0 9815 < clt 9953 ℕ0cn0 11169 ℤ≥cuz 11563 mod cmo 12530 seqcseq 12663 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-sup 8231 df-inf 8232 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-n0 11170 df-z 11255 df-uz 11564 df-rp 11709 df-fz 12198 df-fl 12455 df-mod 12531 df-seq 12664 |
This theorem is referenced by: eucalg 15138 |
Copyright terms: Public domain | W3C validator |