Theorem eucalgcvga 16272

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 = (2nd ‘𝐴)

Assertion

Ref	Expression
eucalgcvga	⊢ (𝐴 ∈ (ℕ0 × ℕ0) → (𝐾 ∈ (ℤ≥‘𝑁) → (2nd ‘(𝑅‘𝐾)) = 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.

Step	Hyp	Ref	Expression
1		eucalgcvga.3	. . . . . . 7 ⊢ 𝑁 = (2nd ‘𝐴)
2		xp2nd 7850	. . . . . . 7 ⊢ (𝐴 ∈ (ℕ0 × ℕ0) → (2nd ‘𝐴) ∈ ℕ0)
3	1, 2	eqeltrid 2844	. . . . . 6 ⊢ (𝐴 ∈ (ℕ0 × ℕ0) → 𝑁 ∈ ℕ0)
4		eluznn0 12639	. . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐾 ∈ (ℤ≥‘𝑁)) → 𝐾 ∈ ℕ0)
5	3, 4	sylan 579	. . . . 5 ⊢ ((𝐴 ∈ (ℕ0 × ℕ0) ∧ 𝐾 ∈ (ℤ≥‘𝑁)) → 𝐾 ∈ ℕ0)
6		nn0uz 12602	. . . . . . 7 ⊢ ℕ0 = (ℤ≥‘0)
7		eucalg.2	. . . . . . 7 ⊢ 𝑅 = seq0((𝐸 ∘ 1st ), (ℕ0 × {𝐴}))
8		0zd 12314	. . . . . . 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 16269	. . . . . . . 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 16259	. . . . . 6 ⊢ (𝐴 ∈ (ℕ0 × ℕ0) → 𝑅:ℕ0⟶(ℕ0 × ℕ0))
14	13	ffvelrnda 6955	. . . . 5 ⊢ ((𝐴 ∈ (ℕ0 × ℕ0) ∧ 𝐾 ∈ ℕ0) → (𝑅‘𝐾) ∈ (ℕ0 × ℕ0))
15	5, 14	syldan 590	. . . 4 ⊢ ((𝐴 ∈ (ℕ0 × ℕ0) ∧ 𝐾 ∈ (ℤ≥‘𝑁)) → (𝑅‘𝐾) ∈ (ℕ0 × ℕ0))
16	15	fvresd 6788	. . 3 ⊢ ((𝐴 ∈ (ℕ0 × ℕ0) ∧ 𝐾 ∈ (ℤ≥‘𝑁)) → ((2nd ↾ (ℕ0 × ℕ0))‘(𝑅‘𝐾)) = (2nd ‘(𝑅‘𝐾)))
17		simpl 482	. . . 4 ⊢ ((𝐴 ∈ (ℕ0 × ℕ0) ∧ 𝐾 ∈ (ℤ≥‘𝑁)) → 𝐴 ∈ (ℕ0 × ℕ0))
18		fvres 6787	. . . . . . . 8 ⊢ (𝐴 ∈ (ℕ0 × ℕ0) → ((2nd ↾ (ℕ0 × ℕ0))‘𝐴) = (2nd ‘𝐴))
19	18, 1	eqtr4di 2797	. . . . . . 7 ⊢ (𝐴 ∈ (ℕ0 × ℕ0) → ((2nd ↾ (ℕ0 × ℕ0))‘𝐴) = 𝑁)
20	19	fveq2d 6772	. . . . . 6 ⊢ (𝐴 ∈ (ℕ0 × ℕ0) → (ℤ≥‘((2nd ↾ (ℕ0 × ℕ0))‘𝐴)) = (ℤ≥‘𝑁))
21	20	eleq2d 2825	. . . . 5 ⊢ (𝐴 ∈ (ℕ0 × ℕ0) → (𝐾 ∈ (ℤ≥‘((2nd ↾ (ℕ0 × ℕ0))‘𝐴)) ↔ 𝐾 ∈ (ℤ≥‘𝑁)))
22	21	biimpar 477	. . . 4 ⊢ ((𝐴 ∈ (ℕ0 × ℕ0) ∧ 𝐾 ∈ (ℤ≥‘𝑁)) → 𝐾 ∈ (ℤ≥‘((2nd ↾ (ℕ0 × ℕ0))‘𝐴)))
23		f2ndres 7842	. . . . 5 ⊢ (2nd ↾ (ℕ0 × ℕ0)):(ℕ0 × ℕ0)⟶ℕ0
24	10	eucalglt 16271	. . . . . 6 ⊢ (𝑧 ∈ (ℕ0 × ℕ0) → ((2nd ‘(𝐸‘𝑧)) ≠ 0 → (2nd ‘(𝐸‘𝑧)) < (2nd ‘𝑧)))
25	11	ffvelrni 6954	. . . . . . . 8 ⊢ (𝑧 ∈ (ℕ0 × ℕ0) → (𝐸‘𝑧) ∈ (ℕ0 × ℕ0))
26	25	fvresd 6788	. . . . . . 7 ⊢ (𝑧 ∈ (ℕ0 × ℕ0) → ((2nd ↾ (ℕ0 × ℕ0))‘(𝐸‘𝑧)) = (2nd ‘(𝐸‘𝑧)))
27	26	neeq1d 3004	. . . . . 6 ⊢ (𝑧 ∈ (ℕ0 × ℕ0) → (((2nd ↾ (ℕ0 × ℕ0))‘(𝐸‘𝑧)) ≠ 0 ↔ (2nd ‘(𝐸‘𝑧)) ≠ 0))
28		fvres 6787	. . . . . . 7 ⊢ (𝑧 ∈ (ℕ0 × ℕ0) → ((2nd ↾ (ℕ0 × ℕ0))‘𝑧) = (2nd ‘𝑧))
29	26, 28	breq12d 5091	. . . . . 6 ⊢ (𝑧 ∈ (ℕ0 × ℕ0) → (((2nd ↾ (ℕ0 × ℕ0))‘(𝐸‘𝑧)) < ((2nd ↾ (ℕ0 × ℕ0))‘𝑧) ↔ (2nd ‘(𝐸‘𝑧)) < (2nd ‘𝑧)))
30	24, 27, 29	3imtr4d 293	. . . . 5 ⊢ (𝑧 ∈ (ℕ0 × ℕ0) → (((2nd ↾ (ℕ0 × ℕ0))‘(𝐸‘𝑧)) ≠ 0 → ((2nd ↾ (ℕ0 × ℕ0))‘(𝐸‘𝑧)) < ((2nd ↾ (ℕ0 × ℕ0))‘𝑧)))
31		eqid 2739	. . . . 5 ⊢ ((2nd ↾ (ℕ0 × ℕ0))‘𝐴) = ((2nd ↾ (ℕ0 × ℕ0))‘𝐴)
32	11, 7, 23, 30, 31	algcvga 16265	. . . 4 ⊢ (𝐴 ∈ (ℕ0 × ℕ0) → (𝐾 ∈ (ℤ≥‘((2nd ↾ (ℕ0 × ℕ0))‘𝐴)) → ((2nd ↾ (ℕ0 × ℕ0))‘(𝑅‘𝐾)) = 0))
33	17, 22, 32	sylc 65	. . 3 ⊢ ((𝐴 ∈ (ℕ0 × ℕ0) ∧ 𝐾 ∈ (ℤ≥‘𝑁)) → ((2nd ↾ (ℕ0 × ℕ0))‘(𝑅‘𝐾)) = 0)
34	16, 33	eqtr3d 2781	. 2 ⊢ ((𝐴 ∈ (ℕ0 × ℕ0) ∧ 𝐾 ∈ (ℤ≥‘𝑁)) → (2nd ‘(𝑅‘𝐾)) = 0)
35	34	ex 412	1 ⊢ (𝐴 ∈ (ℕ0 × ℕ0) → (𝐾 ∈ (ℤ≥‘𝑁) → (2nd ‘(𝑅‘𝐾)) = 0))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2109   ≠ wne 2944  ifcif 4464  {csn 4566  ⟨cop 4572   class class class wbr 5078   × cxp 5586   ↾ cres 5590   ∘ ccom 5592  ⟶wf 6426  ‘cfv 6430  (class class class)co 7268   ∈ cmpo 7270  1^st c1st 7815  2^nd c2nd 7816  0cc0 10855   < clt 10993  ℕ₀cn0 12216  ℤ_≥cuz 12564   mod cmo 13570  seqcseq 13702

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710  ax-sep 5226  ax-nul 5233  ax-pow 5291  ax-pr 5355  ax-un 7579  ax-cnex 10911  ax-resscn 10912  ax-1cn 10913  ax-icn 10914  ax-addcl 10915  ax-addrcl 10916  ax-mulcl 10917  ax-mulrcl 10918  ax-mulcom 10919  ax-addass 10920  ax-mulass 10921  ax-distr 10922  ax-i2m1 10923  ax-1ne0 10924  ax-1rid 10925  ax-rnegex 10926  ax-rrecex 10927  ax-cnre 10928  ax-pre-lttri 10929  ax-pre-lttrn 10930  ax-pre-ltadd 10931  ax-pre-mulgt0 10932  ax-pre-sup 10933

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-nf 1790  df-sb 2071  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3070  df-rex 3071  df-reu 3072  df-rmo 3073  df-rab 3074  df-v 3432  df-sbc 3720  df-csb 3837  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-pss 3910  df-nul 4262  df-if 4465  df-pw 4540  df-sn 4567  df-pr 4569  df-tp 4571  df-op 4573  df-uni 4845  df-iun 4931  df-br 5079  df-opab 5141  df-mpt 5162  df-tr 5196  df-id 5488  df-eprel 5494  df-po 5502  df-so 5503  df-fr 5543  df-we 5545  df-xp 5594  df-rel 5595  df-cnv 5596  df-co 5597  df-dm 5598  df-rn 5599  df-res 5600  df-ima 5601  df-pred 6199  df-ord 6266  df-on 6267  df-lim 6268  df-suc 6269  df-iota 6388  df-fun 6432  df-fn 6433  df-f 6434  df-f1 6435  df-fo 6436  df-f1o 6437  df-fv 6438  df-riota 7225  df-ov 7271  df-oprab 7272  df-mpo 7273  df-om 7701  df-1st 7817  df-2nd 7818  df-frecs 8081  df-wrecs 8112  df-recs 8186  df-rdg 8225  df-er 8472  df-en 8708  df-dom 8709  df-sdom 8710  df-sup 9162  df-inf 9163  df-pnf 10995  df-mnf 10996  df-xr 10997  df-ltxr 10998  df-le 10999  df-sub 11190  df-neg 11191  df-div 11616  df-nn 11957  df-n0 12217  df-z 12303  df-uz 12565  df-rp 12713  df-fz 13222  df-fl 13493  df-mod 13571  df-seq 13703

This theorem is referenced by:  eucalg  16273