Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  eucialgcvga GIF version

Theorem eucialgcvga 10665
 Description: Once Euclid's Algorithm halts after 𝑁 steps, the second element of the state remains 0 . (Contributed by Jim Kingdon, 11-Jan-2022.)
Hypotheses
Ref Expression
eucalgval.1 𝐸 = (𝑥 ∈ ℕ0, 𝑦 ∈ ℕ0 ↦ if(𝑦 = 0, ⟨𝑥, 𝑦⟩, ⟨𝑦, (𝑥 mod 𝑦)⟩))
eucialg.2 𝑅 = seq0((𝐸 ∘ 1st ), (ℕ0 × {𝐴}), (ℕ0 × ℕ0))
eucialgcvga.3 𝑁 = (2nd𝐴)
Assertion
Ref Expression
eucialgcvga (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 eucialgcvga
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 eucialgcvga.3 . . . . . . 7 𝑁 = (2nd𝐴)
2 xp2nd 5845 . . . . . . 7 (𝐴 ∈ (ℕ0 × ℕ0) → (2nd𝐴) ∈ ℕ0)
31, 2syl5eqel 2169 . . . . . 6 (𝐴 ∈ (ℕ0 × ℕ0) → 𝑁 ∈ ℕ0)
4 eluznn0 8837 . . . . . 6 ((𝑁 ∈ ℕ0𝐾 ∈ (ℤ𝑁)) → 𝐾 ∈ ℕ0)
53, 4sylan 277 . . . . 5 ((𝐴 ∈ (ℕ0 × ℕ0) ∧ 𝐾 ∈ (ℤ𝑁)) → 𝐾 ∈ ℕ0)
6 nn0uz 8804 . . . . . . 7 0 = (ℤ‘0)
7 eucialg.2 . . . . . . 7 𝑅 = seq0((𝐸 ∘ 1st ), (ℕ0 × {𝐴}), (ℕ0 × ℕ0))
8 0zd 8514 . . . . . . 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)⟩))
1110eucalgf 10662 . . . . . . . 8 𝐸:(ℕ0 × ℕ0)⟶(ℕ0 × ℕ0)
1211a1i 9 . . . . . . 7 (𝐴 ∈ (ℕ0 × ℕ0) → 𝐸:(ℕ0 × ℕ0)⟶(ℕ0 × ℕ0))
13 nn0ex 8431 . . . . . . . . 9 0 ∈ V
1413, 13xpex 4501 . . . . . . . 8 (ℕ0 × ℕ0) ∈ V
1514a1i 9 . . . . . . 7 (𝐴 ∈ (ℕ0 × ℕ0) → (ℕ0 × ℕ0) ∈ V)
166, 7, 8, 9, 12, 15ialgrf 10652 . . . . . 6 (𝐴 ∈ (ℕ0 × ℕ0) → 𝑅:ℕ0⟶(ℕ0 × ℕ0))
1716ffvelrnda 5355 . . . . 5 ((𝐴 ∈ (ℕ0 × ℕ0) ∧ 𝐾 ∈ ℕ0) → (𝑅𝐾) ∈ (ℕ0 × ℕ0))
185, 17syldan 276 . . . 4 ((𝐴 ∈ (ℕ0 × ℕ0) ∧ 𝐾 ∈ (ℤ𝑁)) → (𝑅𝐾) ∈ (ℕ0 × ℕ0))
19 fvres 5251 . . . 4 ((𝑅𝐾) ∈ (ℕ0 × ℕ0) → ((2nd ↾ (ℕ0 × ℕ0))‘(𝑅𝐾)) = (2nd ‘(𝑅𝐾)))
2018, 19syl 14 . . 3 ((𝐴 ∈ (ℕ0 × ℕ0) ∧ 𝐾 ∈ (ℤ𝑁)) → ((2nd ↾ (ℕ0 × ℕ0))‘(𝑅𝐾)) = (2nd ‘(𝑅𝐾)))
21 simpl 107 . . . 4 ((𝐴 ∈ (ℕ0 × ℕ0) ∧ 𝐾 ∈ (ℤ𝑁)) → 𝐴 ∈ (ℕ0 × ℕ0))
22 fvres 5251 . . . . . . . 8 (𝐴 ∈ (ℕ0 × ℕ0) → ((2nd ↾ (ℕ0 × ℕ0))‘𝐴) = (2nd𝐴))
2322, 1syl6eqr 2133 . . . . . . 7 (𝐴 ∈ (ℕ0 × ℕ0) → ((2nd ↾ (ℕ0 × ℕ0))‘𝐴) = 𝑁)
2423fveq2d 5234 . . . . . 6 (𝐴 ∈ (ℕ0 × ℕ0) → (ℤ‘((2nd ↾ (ℕ0 × ℕ0))‘𝐴)) = (ℤ𝑁))
2524eleq2d 2152 . . . . 5 (𝐴 ∈ (ℕ0 × ℕ0) → (𝐾 ∈ (ℤ‘((2nd ↾ (ℕ0 × ℕ0))‘𝐴)) ↔ 𝐾 ∈ (ℤ𝑁)))
2625biimpar 291 . . . 4 ((𝐴 ∈ (ℕ0 × ℕ0) ∧ 𝐾 ∈ (ℤ𝑁)) → 𝐾 ∈ (ℤ‘((2nd ↾ (ℕ0 × ℕ0))‘𝐴)))
27 f2ndres 5839 . . . . 5 (2nd ↾ (ℕ0 × ℕ0)):(ℕ0 × ℕ0)⟶ℕ0
2810eucalglt 10664 . . . . . 6 (𝑧 ∈ (ℕ0 × ℕ0) → ((2nd ‘(𝐸𝑧)) ≠ 0 → (2nd ‘(𝐸𝑧)) < (2nd𝑧)))
2911ffvelrni 5354 . . . . . . . 8 (𝑧 ∈ (ℕ0 × ℕ0) → (𝐸𝑧) ∈ (ℕ0 × ℕ0))
30 fvres 5251 . . . . . . . 8 ((𝐸𝑧) ∈ (ℕ0 × ℕ0) → ((2nd ↾ (ℕ0 × ℕ0))‘(𝐸𝑧)) = (2nd ‘(𝐸𝑧)))
3129, 30syl 14 . . . . . . 7 (𝑧 ∈ (ℕ0 × ℕ0) → ((2nd ↾ (ℕ0 × ℕ0))‘(𝐸𝑧)) = (2nd ‘(𝐸𝑧)))
3231neeq1d 2267 . . . . . 6 (𝑧 ∈ (ℕ0 × ℕ0) → (((2nd ↾ (ℕ0 × ℕ0))‘(𝐸𝑧)) ≠ 0 ↔ (2nd ‘(𝐸𝑧)) ≠ 0))
33 fvres 5251 . . . . . . 7 (𝑧 ∈ (ℕ0 × ℕ0) → ((2nd ↾ (ℕ0 × ℕ0))‘𝑧) = (2nd𝑧))
3431, 33breq12d 3818 . . . . . 6 (𝑧 ∈ (ℕ0 × ℕ0) → (((2nd ↾ (ℕ0 × ℕ0))‘(𝐸𝑧)) < ((2nd ↾ (ℕ0 × ℕ0))‘𝑧) ↔ (2nd ‘(𝐸𝑧)) < (2nd𝑧)))
3528, 32, 343imtr4d 201 . . . . 5 (𝑧 ∈ (ℕ0 × ℕ0) → (((2nd ↾ (ℕ0 × ℕ0))‘(𝐸𝑧)) ≠ 0 → ((2nd ↾ (ℕ0 × ℕ0))‘(𝐸𝑧)) < ((2nd ↾ (ℕ0 × ℕ0))‘𝑧)))
36 eqid 2083 . . . . 5 ((2nd ↾ (ℕ0 × ℕ0))‘𝐴) = ((2nd ↾ (ℕ0 × ℕ0))‘𝐴)
3711, 7, 27, 35, 36, 14ialgcvga 10658 . . . 4 (𝐴 ∈ (ℕ0 × ℕ0) → (𝐾 ∈ (ℤ‘((2nd ↾ (ℕ0 × ℕ0))‘𝐴)) → ((2nd ↾ (ℕ0 × ℕ0))‘(𝑅𝐾)) = 0))
3821, 26, 37sylc 61 . . 3 ((𝐴 ∈ (ℕ0 × ℕ0) ∧ 𝐾 ∈ (ℤ𝑁)) → ((2nd ↾ (ℕ0 × ℕ0))‘(𝑅𝐾)) = 0)
3920, 38eqtr3d 2117 . 2 ((𝐴 ∈ (ℕ0 × ℕ0) ∧ 𝐾 ∈ (ℤ𝑁)) → (2nd ‘(𝑅𝐾)) = 0)
4039ex 113 1 (𝐴 ∈ (ℕ0 × ℕ0) → (𝐾 ∈ (ℤ𝑁) → (2nd ‘(𝑅𝐾)) = 0))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 102   = wceq 1285   ∈ wcel 1434   ≠ wne 2249  Vcvv 2610  ifcif 3368  {csn 3416  ⟨cop 3419   class class class wbr 3805   × cxp 4389   ↾ cres 4393   ∘ ccom 4395  ⟶wf 4948  ‘cfv 4952  (class class class)co 5564   ↦ cmpt2 5566  1st c1st 5817  2nd c2nd 5818  0cc0 7113   < clt 7285  ℕ0cn0 8425  ℤ≥cuz 8770   mod cmo 9474  seqcseq 9591 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3913  ax-sep 3916  ax-nul 3924  ax-pow 3968  ax-pr 3992  ax-un 4216  ax-setind 4308  ax-iinf 4357  ax-cnex 7199  ax-resscn 7200  ax-1cn 7201  ax-1re 7202  ax-icn 7203  ax-addcl 7204  ax-addrcl 7205  ax-mulcl 7206  ax-mulrcl 7207  ax-addcom 7208  ax-mulcom 7209  ax-addass 7210  ax-mulass 7211  ax-distr 7212  ax-i2m1 7213  ax-0lt1 7214  ax-1rid 7215  ax-0id 7216  ax-rnegex 7217  ax-precex 7218  ax-cnre 7219  ax-pre-ltirr 7220  ax-pre-ltwlin 7221  ax-pre-lttrn 7222  ax-pre-apti 7223  ax-pre-ltadd 7224  ax-pre-mulgt0 7225  ax-pre-mulext 7226  ax-arch 7227 This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-nel 2345  df-ral 2358  df-rex 2359  df-reu 2360  df-rmo 2361  df-rab 2362  df-v 2612  df-sbc 2825  df-csb 2918  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-nul 3268  df-if 3369  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-int 3657  df-iun 3700  df-br 3806  df-opab 3860  df-mpt 3861  df-tr 3896  df-id 4076  df-po 4079  df-iso 4080  df-iord 4149  df-on 4151  df-ilim 4152  df-suc 4154  df-iom 4360  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-res 4403  df-ima 4404  df-iota 4917  df-fun 4954  df-fn 4955  df-f 4956  df-f1 4957  df-fo 4958  df-f1o 4959  df-fv 4960  df-riota 5520  df-ov 5567  df-oprab 5568  df-mpt2 5569  df-1st 5819  df-2nd 5820  df-recs 5975  df-frec 6061  df-pnf 7287  df-mnf 7288  df-xr 7289  df-ltxr 7290  df-le 7291  df-sub 7418  df-neg 7419  df-reap 7812  df-ap 7819  df-div 7898  df-inn 8177  df-n0 8426  df-z 8503  df-uz 8771  df-q 8856  df-rp 8886  df-fl 9422  df-mod 9475  df-iseq 9592 This theorem is referenced by:  eucialg  10666
 Copyright terms: Public domain W3C validator