MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  eucalgcvga Structured version   Visualization version   GIF version

Theorem eucalgcvga 15346
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.)
Hypotheses
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 = (2nd𝐴)
Assertion
Ref Expression
eucalgcvga (𝐴 ∈ (ℕ0 × ℕ0) → (𝐾 ∈ (ℤ𝑁) → (2nd ‘(𝑅𝐾)) = 0))
Distinct variable groups:   𝑥,𝑦,𝑁   𝑥,𝐴,𝑦   𝑥,𝑅
Allowed substitution hints:   𝑅(𝑦)   𝐸(𝑥,𝑦)   𝐾(𝑥,𝑦)

Proof of Theorem eucalgcvga
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 eucalgcvga.3 . . . . . . 7 𝑁 = (2nd𝐴)
2 xp2nd 7243 . . . . . . 7 (𝐴 ∈ (ℕ0 × ℕ0) → (2nd𝐴) ∈ ℕ0)
31, 2syl5eqel 2734 . . . . . 6 (𝐴 ∈ (ℕ0 × ℕ0) → 𝑁 ∈ ℕ0)
4 eluznn0 11795 . . . . . 6 ((𝑁 ∈ ℕ0𝐾 ∈ (ℤ𝑁)) → 𝐾 ∈ ℕ0)
53, 4sylan 487 . . . . 5 ((𝐴 ∈ (ℕ0 × ℕ0) ∧ 𝐾 ∈ (ℤ𝑁)) → 𝐾 ∈ ℕ0)
6 nn0uz 11760 . . . . . . 7 0 = (ℤ‘0)
7 eucalg.2 . . . . . . 7 𝑅 = seq0((𝐸 ∘ 1st ), (ℕ0 × {𝐴}))
8 0zd 11427 . . . . . . 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)⟩))
1110eucalgf 15343 . . . . . . . 8 𝐸:(ℕ0 × ℕ0)⟶(ℕ0 × ℕ0)
1211a1i 11 . . . . . . 7 (𝐴 ∈ (ℕ0 × ℕ0) → 𝐸:(ℕ0 × ℕ0)⟶(ℕ0 × ℕ0))
136, 7, 8, 9, 12algrf 15333 . . . . . 6 (𝐴 ∈ (ℕ0 × ℕ0) → 𝑅:ℕ0⟶(ℕ0 × ℕ0))
1413ffvelrnda 6399 . . . . 5 ((𝐴 ∈ (ℕ0 × ℕ0) ∧ 𝐾 ∈ ℕ0) → (𝑅𝐾) ∈ (ℕ0 × ℕ0))
155, 14syldan 486 . . . 4 ((𝐴 ∈ (ℕ0 × ℕ0) ∧ 𝐾 ∈ (ℤ𝑁)) → (𝑅𝐾) ∈ (ℕ0 × ℕ0))
16 fvres 6245 . . . 4 ((𝑅𝐾) ∈ (ℕ0 × ℕ0) → ((2nd ↾ (ℕ0 × ℕ0))‘(𝑅𝐾)) = (2nd ‘(𝑅𝐾)))
1715, 16syl 17 . . 3 ((𝐴 ∈ (ℕ0 × ℕ0) ∧ 𝐾 ∈ (ℤ𝑁)) → ((2nd ↾ (ℕ0 × ℕ0))‘(𝑅𝐾)) = (2nd ‘(𝑅𝐾)))
18 simpl 472 . . . 4 ((𝐴 ∈ (ℕ0 × ℕ0) ∧ 𝐾 ∈ (ℤ𝑁)) → 𝐴 ∈ (ℕ0 × ℕ0))
19 fvres 6245 . . . . . . . 8 (𝐴 ∈ (ℕ0 × ℕ0) → ((2nd ↾ (ℕ0 × ℕ0))‘𝐴) = (2nd𝐴))
2019, 1syl6eqr 2703 . . . . . . 7 (𝐴 ∈ (ℕ0 × ℕ0) → ((2nd ↾ (ℕ0 × ℕ0))‘𝐴) = 𝑁)
2120fveq2d 6233 . . . . . 6 (𝐴 ∈ (ℕ0 × ℕ0) → (ℤ‘((2nd ↾ (ℕ0 × ℕ0))‘𝐴)) = (ℤ𝑁))
2221eleq2d 2716 . . . . 5 (𝐴 ∈ (ℕ0 × ℕ0) → (𝐾 ∈ (ℤ‘((2nd ↾ (ℕ0 × ℕ0))‘𝐴)) ↔ 𝐾 ∈ (ℤ𝑁)))
2322biimpar 501 . . . 4 ((𝐴 ∈ (ℕ0 × ℕ0) ∧ 𝐾 ∈ (ℤ𝑁)) → 𝐾 ∈ (ℤ‘((2nd ↾ (ℕ0 × ℕ0))‘𝐴)))
24 f2ndres 7235 . . . . 5 (2nd ↾ (ℕ0 × ℕ0)):(ℕ0 × ℕ0)⟶ℕ0
2510eucalglt 15345 . . . . . 6 (𝑧 ∈ (ℕ0 × ℕ0) → ((2nd ‘(𝐸𝑧)) ≠ 0 → (2nd ‘(𝐸𝑧)) < (2nd𝑧)))
2611ffvelrni 6398 . . . . . . . 8 (𝑧 ∈ (ℕ0 × ℕ0) → (𝐸𝑧) ∈ (ℕ0 × ℕ0))
27 fvres 6245 . . . . . . . 8 ((𝐸𝑧) ∈ (ℕ0 × ℕ0) → ((2nd ↾ (ℕ0 × ℕ0))‘(𝐸𝑧)) = (2nd ‘(𝐸𝑧)))
2826, 27syl 17 . . . . . . 7 (𝑧 ∈ (ℕ0 × ℕ0) → ((2nd ↾ (ℕ0 × ℕ0))‘(𝐸𝑧)) = (2nd ‘(𝐸𝑧)))
2928neeq1d 2882 . . . . . 6 (𝑧 ∈ (ℕ0 × ℕ0) → (((2nd ↾ (ℕ0 × ℕ0))‘(𝐸𝑧)) ≠ 0 ↔ (2nd ‘(𝐸𝑧)) ≠ 0))
30 fvres 6245 . . . . . . 7 (𝑧 ∈ (ℕ0 × ℕ0) → ((2nd ↾ (ℕ0 × ℕ0))‘𝑧) = (2nd𝑧))
3128, 30breq12d 4698 . . . . . 6 (𝑧 ∈ (ℕ0 × ℕ0) → (((2nd ↾ (ℕ0 × ℕ0))‘(𝐸𝑧)) < ((2nd ↾ (ℕ0 × ℕ0))‘𝑧) ↔ (2nd ‘(𝐸𝑧)) < (2nd𝑧)))
3225, 29, 313imtr4d 283 . . . . 5 (𝑧 ∈ (ℕ0 × ℕ0) → (((2nd ↾ (ℕ0 × ℕ0))‘(𝐸𝑧)) ≠ 0 → ((2nd ↾ (ℕ0 × ℕ0))‘(𝐸𝑧)) < ((2nd ↾ (ℕ0 × ℕ0))‘𝑧)))
33 eqid 2651 . . . . 5 ((2nd ↾ (ℕ0 × ℕ0))‘𝐴) = ((2nd ↾ (ℕ0 × ℕ0))‘𝐴)
3411, 7, 24, 32, 33algcvga 15339 . . . 4 (𝐴 ∈ (ℕ0 × ℕ0) → (𝐾 ∈ (ℤ‘((2nd ↾ (ℕ0 × ℕ0))‘𝐴)) → ((2nd ↾ (ℕ0 × ℕ0))‘(𝑅𝐾)) = 0))
3518, 23, 34sylc 65 . . 3 ((𝐴 ∈ (ℕ0 × ℕ0) ∧ 𝐾 ∈ (ℤ𝑁)) → ((2nd ↾ (ℕ0 × ℕ0))‘(𝑅𝐾)) = 0)
3617, 35eqtr3d 2687 . 2 ((𝐴 ∈ (ℕ0 × ℕ0) ∧ 𝐾 ∈ (ℤ𝑁)) → (2nd ‘(𝑅𝐾)) = 0)
3736ex 449 1 (𝐴 ∈ (ℕ0 × ℕ0) → (𝐾 ∈ (ℤ𝑁) → (2nd ‘(𝑅𝐾)) = 0))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  wne 2823  ifcif 4119  {csn 4210  cop 4216   class class class wbr 4685   × cxp 5141  cres 5145  ccom 5147  wf 5922  cfv 5926  (class class class)co 6690  cmpt2 6692  1st c1st 7208  2nd c2nd 7209  0cc0 9974   < clt 10112  0cn0 11330  cuz 11725   mod cmo 12708  seqcseq 12841
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-sup 8389  df-inf 8390  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-n0 11331  df-z 11416  df-uz 11726  df-rp 11871  df-fz 12365  df-fl 12633  df-mod 12709  df-seq 12842
This theorem is referenced by:  eucalg  15347
  Copyright terms: Public domain W3C validator