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Theorem eucalgcvga 15137
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 7090 . . . . . . 7 (𝐴 ∈ (ℕ0 × ℕ0) → (2nd𝐴) ∈ ℕ0)
31, 2syl5eqel 2692 . . . . . 6 (𝐴 ∈ (ℕ0 × ℕ0) → 𝑁 ∈ ℕ0)
4 eluznn0 11633 . . . . . 6 ((𝑁 ∈ ℕ0𝐾 ∈ (ℤ𝑁)) → 𝐾 ∈ ℕ0)
53, 4sylan 487 . . . . 5 ((𝐴 ∈ (ℕ0 × ℕ0) ∧ 𝐾 ∈ (ℤ𝑁)) → 𝐾 ∈ ℕ0)
6 nn0uz 11598 . . . . . . 7 0 = (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)⟩))
1110eucalgf 15134 . . . . . . . 8 𝐸:(ℕ0 × ℕ0)⟶(ℕ0 × ℕ0)
1211a1i 11 . . . . . . 7 (𝐴 ∈ (ℕ0 × ℕ0) → 𝐸:(ℕ0 × ℕ0)⟶(ℕ0 × ℕ0))
136, 7, 8, 9, 12algrf 15124 . . . . . 6 (𝐴 ∈ (ℕ0 × ℕ0) → 𝑅:ℕ0⟶(ℕ0 × ℕ0))
1413ffvelrnda 6267 . . . . 5 ((𝐴 ∈ (ℕ0 × ℕ0) ∧ 𝐾 ∈ ℕ0) → (𝑅𝐾) ∈ (ℕ0 × ℕ0))
155, 14syldan 486 . . . 4 ((𝐴 ∈ (ℕ0 × ℕ0) ∧ 𝐾 ∈ (ℤ𝑁)) → (𝑅𝐾) ∈ (ℕ0 × ℕ0))
16 fvres 6117 . . . 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 6117 . . . . . . . 8 (𝐴 ∈ (ℕ0 × ℕ0) → ((2nd ↾ (ℕ0 × ℕ0))‘𝐴) = (2nd𝐴))
2019, 1syl6eqr 2662 . . . . . . 7 (𝐴 ∈ (ℕ0 × ℕ0) → ((2nd ↾ (ℕ0 × ℕ0))‘𝐴) = 𝑁)
2120fveq2d 6107 . . . . . 6 (𝐴 ∈ (ℕ0 × ℕ0) → (ℤ‘((2nd ↾ (ℕ0 × ℕ0))‘𝐴)) = (ℤ𝑁))
2221eleq2d 2673 . . . . 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 7082 . . . . 5 (2nd ↾ (ℕ0 × ℕ0)):(ℕ0 × ℕ0)⟶ℕ0
2510eucalglt 15136 . . . . . 6 (𝑧 ∈ (ℕ0 × ℕ0) → ((2nd ‘(𝐸𝑧)) ≠ 0 → (2nd ‘(𝐸𝑧)) < (2nd𝑧)))
2611ffvelrni 6266 . . . . . . . 8 (𝑧 ∈ (ℕ0 × ℕ0) → (𝐸𝑧) ∈ (ℕ0 × ℕ0))
27 fvres 6117 . . . . . . . 8 ((𝐸𝑧) ∈ (ℕ0 × ℕ0) → ((2nd ↾ (ℕ0 × ℕ0))‘(𝐸𝑧)) = (2nd ‘(𝐸𝑧)))
2826, 27syl 17 . . . . . . 7 (𝑧 ∈ (ℕ0 × ℕ0) → ((2nd ↾ (ℕ0 × ℕ0))‘(𝐸𝑧)) = (2nd ‘(𝐸𝑧)))
2928neeq1d 2841 . . . . . 6 (𝑧 ∈ (ℕ0 × ℕ0) → (((2nd ↾ (ℕ0 × ℕ0))‘(𝐸𝑧)) ≠ 0 ↔ (2nd ‘(𝐸𝑧)) ≠ 0))
30 fvres 6117 . . . . . . 7 (𝑧 ∈ (ℕ0 × ℕ0) → ((2nd ↾ (ℕ0 × ℕ0))‘𝑧) = (2nd𝑧))
3128, 30breq12d 4596 . . . . . 6 (𝑧 ∈ (ℕ0 × ℕ0) → (((2nd ↾ (ℕ0 × ℕ0))‘(𝐸𝑧)) < ((2nd ↾ (ℕ0 × ℕ0))‘𝑧) ↔ (2nd ‘(𝐸𝑧)) < (2nd𝑧)))
3225, 29, 313imtr4d 282 . . . . 5 (𝑧 ∈ (ℕ0 × ℕ0) → (((2nd ↾ (ℕ0 × ℕ0))‘(𝐸𝑧)) ≠ 0 → ((2nd ↾ (ℕ0 × ℕ0))‘(𝐸𝑧)) < ((2nd ↾ (ℕ0 × ℕ0))‘𝑧)))
33 eqid 2610 . . . . 5 ((2nd ↾ (ℕ0 × ℕ0))‘𝐴) = ((2nd ↾ (ℕ0 × ℕ0))‘𝐴)
3411, 7, 24, 32, 33algcvga 15130 . . . 4 (𝐴 ∈ (ℕ0 × ℕ0) → (𝐾 ∈ (ℤ‘((2nd ↾ (ℕ0 × ℕ0))‘𝐴)) → ((2nd ↾ (ℕ0 × ℕ0))‘(𝑅𝐾)) = 0))
3518, 23, 34sylc 63 . . 3 ((𝐴 ∈ (ℕ0 × ℕ0) ∧ 𝐾 ∈ (ℤ𝑁)) → ((2nd ↾ (ℕ0 × ℕ0))‘(𝑅𝐾)) = 0)
3617, 35eqtr3d 2646 . 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 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  0cn0 11169  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
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