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Theorem eucalgcvga 15918
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 7711 . . . . . . 7 (𝐴 ∈ (ℕ0 × ℕ0) → (2nd𝐴) ∈ ℕ0)
31, 2eqeltrid 2914 . . . . . 6 (𝐴 ∈ (ℕ0 × ℕ0) → 𝑁 ∈ ℕ0)
4 eluznn0 12305 . . . . . 6 ((𝑁 ∈ ℕ0𝐾 ∈ (ℤ𝑁)) → 𝐾 ∈ ℕ0)
53, 4sylan 580 . . . . 5 ((𝐴 ∈ (ℕ0 × ℕ0) ∧ 𝐾 ∈ (ℤ𝑁)) → 𝐾 ∈ ℕ0)
6 nn0uz 12268 . . . . . . 7 0 = (ℤ‘0)
7 eucalg.2 . . . . . . 7 𝑅 = seq0((𝐸 ∘ 1st ), (ℕ0 × {𝐴}))
8 0zd 11981 . . . . . . 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 15915 . . . . . . . 8 𝐸:(ℕ0 × ℕ0)⟶(ℕ0 × ℕ0)
1211a1i 11 . . . . . . 7 (𝐴 ∈ (ℕ0 × ℕ0) → 𝐸:(ℕ0 × ℕ0)⟶(ℕ0 × ℕ0))
136, 7, 8, 9, 12algrf 15905 . . . . . 6 (𝐴 ∈ (ℕ0 × ℕ0) → 𝑅:ℕ0⟶(ℕ0 × ℕ0))
1413ffvelrnda 6843 . . . . 5 ((𝐴 ∈ (ℕ0 × ℕ0) ∧ 𝐾 ∈ ℕ0) → (𝑅𝐾) ∈ (ℕ0 × ℕ0))
155, 14syldan 591 . . . 4 ((𝐴 ∈ (ℕ0 × ℕ0) ∧ 𝐾 ∈ (ℤ𝑁)) → (𝑅𝐾) ∈ (ℕ0 × ℕ0))
1615fvresd 6683 . . 3 ((𝐴 ∈ (ℕ0 × ℕ0) ∧ 𝐾 ∈ (ℤ𝑁)) → ((2nd ↾ (ℕ0 × ℕ0))‘(𝑅𝐾)) = (2nd ‘(𝑅𝐾)))
17 simpl 483 . . . 4 ((𝐴 ∈ (ℕ0 × ℕ0) ∧ 𝐾 ∈ (ℤ𝑁)) → 𝐴 ∈ (ℕ0 × ℕ0))
18 fvres 6682 . . . . . . . 8 (𝐴 ∈ (ℕ0 × ℕ0) → ((2nd ↾ (ℕ0 × ℕ0))‘𝐴) = (2nd𝐴))
1918, 1syl6eqr 2871 . . . . . . 7 (𝐴 ∈ (ℕ0 × ℕ0) → ((2nd ↾ (ℕ0 × ℕ0))‘𝐴) = 𝑁)
2019fveq2d 6667 . . . . . 6 (𝐴 ∈ (ℕ0 × ℕ0) → (ℤ‘((2nd ↾ (ℕ0 × ℕ0))‘𝐴)) = (ℤ𝑁))
2120eleq2d 2895 . . . . 5 (𝐴 ∈ (ℕ0 × ℕ0) → (𝐾 ∈ (ℤ‘((2nd ↾ (ℕ0 × ℕ0))‘𝐴)) ↔ 𝐾 ∈ (ℤ𝑁)))
2221biimpar 478 . . . 4 ((𝐴 ∈ (ℕ0 × ℕ0) ∧ 𝐾 ∈ (ℤ𝑁)) → 𝐾 ∈ (ℤ‘((2nd ↾ (ℕ0 × ℕ0))‘𝐴)))
23 f2ndres 7703 . . . . 5 (2nd ↾ (ℕ0 × ℕ0)):(ℕ0 × ℕ0)⟶ℕ0
2410eucalglt 15917 . . . . . 6 (𝑧 ∈ (ℕ0 × ℕ0) → ((2nd ‘(𝐸𝑧)) ≠ 0 → (2nd ‘(𝐸𝑧)) < (2nd𝑧)))
2511ffvelrni 6842 . . . . . . . 8 (𝑧 ∈ (ℕ0 × ℕ0) → (𝐸𝑧) ∈ (ℕ0 × ℕ0))
2625fvresd 6683 . . . . . . 7 (𝑧 ∈ (ℕ0 × ℕ0) → ((2nd ↾ (ℕ0 × ℕ0))‘(𝐸𝑧)) = (2nd ‘(𝐸𝑧)))
2726neeq1d 3072 . . . . . 6 (𝑧 ∈ (ℕ0 × ℕ0) → (((2nd ↾ (ℕ0 × ℕ0))‘(𝐸𝑧)) ≠ 0 ↔ (2nd ‘(𝐸𝑧)) ≠ 0))
28 fvres 6682 . . . . . . 7 (𝑧 ∈ (ℕ0 × ℕ0) → ((2nd ↾ (ℕ0 × ℕ0))‘𝑧) = (2nd𝑧))
2926, 28breq12d 5070 . . . . . 6 (𝑧 ∈ (ℕ0 × ℕ0) → (((2nd ↾ (ℕ0 × ℕ0))‘(𝐸𝑧)) < ((2nd ↾ (ℕ0 × ℕ0))‘𝑧) ↔ (2nd ‘(𝐸𝑧)) < (2nd𝑧)))
3024, 27, 293imtr4d 295 . . . . 5 (𝑧 ∈ (ℕ0 × ℕ0) → (((2nd ↾ (ℕ0 × ℕ0))‘(𝐸𝑧)) ≠ 0 → ((2nd ↾ (ℕ0 × ℕ0))‘(𝐸𝑧)) < ((2nd ↾ (ℕ0 × ℕ0))‘𝑧)))
31 eqid 2818 . . . . 5 ((2nd ↾ (ℕ0 × ℕ0))‘𝐴) = ((2nd ↾ (ℕ0 × ℕ0))‘𝐴)
3211, 7, 23, 30, 31algcvga 15911 . . . 4 (𝐴 ∈ (ℕ0 × ℕ0) → (𝐾 ∈ (ℤ‘((2nd ↾ (ℕ0 × ℕ0))‘𝐴)) → ((2nd ↾ (ℕ0 × ℕ0))‘(𝑅𝐾)) = 0))
3317, 22, 32sylc 65 . . 3 ((𝐴 ∈ (ℕ0 × ℕ0) ∧ 𝐾 ∈ (ℤ𝑁)) → ((2nd ↾ (ℕ0 × ℕ0))‘(𝑅𝐾)) = 0)
3416, 33eqtr3d 2855 . 2 ((𝐴 ∈ (ℕ0 × ℕ0) ∧ 𝐾 ∈ (ℤ𝑁)) → (2nd ‘(𝑅𝐾)) = 0)
3534ex 413 1 (𝐴 ∈ (ℕ0 × ℕ0) → (𝐾 ∈ (ℤ𝑁) → (2nd ‘(𝑅𝐾)) = 0))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  wne 3013  ifcif 4463  {csn 4557  cop 4563   class class class wbr 5057   × cxp 5546  cres 5550  ccom 5552  wf 6344  cfv 6348  (class class class)co 7145  cmpo 7147  1st c1st 7676  2nd c2nd 7677  0cc0 10525   < clt 10663  0cn0 11885  cuz 12231   mod cmo 13225  seqcseq 13357
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602  ax-pre-sup 10603
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-er 8278  df-en 8498  df-dom 8499  df-sdom 8500  df-sup 8894  df-inf 8895  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-div 11286  df-nn 11627  df-n0 11886  df-z 11970  df-uz 12232  df-rp 12378  df-fz 12881  df-fl 13150  df-mod 13226  df-seq 13358
This theorem is referenced by:  eucalg  15919
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