Theorem eucalgcvga 11999

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 6142	. . . . . . 7 ⊢ (𝐴 ∈ (ℕ0 × ℕ0) → (2nd ‘𝐴) ∈ ℕ0)
3	1, 2	eqeltrid 2257	. . . . . 6 ⊢ (𝐴 ∈ (ℕ0 × ℕ0) → 𝑁 ∈ ℕ0)
4		eluznn0 9545	. . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐾 ∈ (ℤ≥‘𝑁)) → 𝐾 ∈ ℕ0)
5	3, 4	sylan 281	. . . . 5 ⊢ ((𝐴 ∈ (ℕ0 × ℕ0) ∧ 𝐾 ∈ (ℤ≥‘𝑁)) → 𝐾 ∈ ℕ0)
6		nn0uz 9508	. . . . . . 7 ⊢ ℕ0 = (ℤ≥‘0)
7		eucalg.2	. . . . . . 7 ⊢ 𝑅 = seq0((𝐸 ∘ 1st ), (ℕ0 × {𝐴}))
8		0zd 9211	. . . . . . 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)⟩))
11	10	eucalgf 11996	. . . . . . . 8 ⊢ 𝐸:(ℕ0 × ℕ0)⟶(ℕ0 × ℕ0)
12	11	a1i 9	. . . . . . 7 ⊢ (𝐴 ∈ (ℕ0 × ℕ0) → 𝐸:(ℕ0 × ℕ0)⟶(ℕ0 × ℕ0))
13	6, 7, 8, 9, 12	algrf 11986	. . . . . 6 ⊢ (𝐴 ∈ (ℕ0 × ℕ0) → 𝑅:ℕ0⟶(ℕ0 × ℕ0))
14	13	ffvelrnda 5628	. . . . 5 ⊢ ((𝐴 ∈ (ℕ0 × ℕ0) ∧ 𝐾 ∈ ℕ0) → (𝑅‘𝐾) ∈ (ℕ0 × ℕ0))
15	5, 14	syldan 280	. . . 4 ⊢ ((𝐴 ∈ (ℕ0 × ℕ0) ∧ 𝐾 ∈ (ℤ≥‘𝑁)) → (𝑅‘𝐾) ∈ (ℕ0 × ℕ0))
16		fvres 5518	. . . 4 ⊢ ((𝑅‘𝐾) ∈ (ℕ0 × ℕ0) → ((2nd ↾ (ℕ0 × ℕ0))‘(𝑅‘𝐾)) = (2nd ‘(𝑅‘𝐾)))
17	15, 16	syl 14	. . 3 ⊢ ((𝐴 ∈ (ℕ0 × ℕ0) ∧ 𝐾 ∈ (ℤ≥‘𝑁)) → ((2nd ↾ (ℕ0 × ℕ0))‘(𝑅‘𝐾)) = (2nd ‘(𝑅‘𝐾)))
18		simpl 108	. . . 4 ⊢ ((𝐴 ∈ (ℕ0 × ℕ0) ∧ 𝐾 ∈ (ℤ≥‘𝑁)) → 𝐴 ∈ (ℕ0 × ℕ0))
19		fvres 5518	. . . . . . . 8 ⊢ (𝐴 ∈ (ℕ0 × ℕ0) → ((2nd ↾ (ℕ0 × ℕ0))‘𝐴) = (2nd ‘𝐴))
20	19, 1	eqtr4di 2221	. . . . . . 7 ⊢ (𝐴 ∈ (ℕ0 × ℕ0) → ((2nd ↾ (ℕ0 × ℕ0))‘𝐴) = 𝑁)
21	20	fveq2d 5498	. . . . . 6 ⊢ (𝐴 ∈ (ℕ0 × ℕ0) → (ℤ≥‘((2nd ↾ (ℕ0 × ℕ0))‘𝐴)) = (ℤ≥‘𝑁))
22	21	eleq2d 2240	. . . . 5 ⊢ (𝐴 ∈ (ℕ0 × ℕ0) → (𝐾 ∈ (ℤ≥‘((2nd ↾ (ℕ0 × ℕ0))‘𝐴)) ↔ 𝐾 ∈ (ℤ≥‘𝑁)))
23	22	biimpar 295	. . . 4 ⊢ ((𝐴 ∈ (ℕ0 × ℕ0) ∧ 𝐾 ∈ (ℤ≥‘𝑁)) → 𝐾 ∈ (ℤ≥‘((2nd ↾ (ℕ0 × ℕ0))‘𝐴)))
24		f2ndres 6136	. . . . 5 ⊢ (2nd ↾ (ℕ0 × ℕ0)):(ℕ0 × ℕ0)⟶ℕ0
25	10	eucalglt 11998	. . . . . 6 ⊢ (𝑧 ∈ (ℕ0 × ℕ0) → ((2nd ‘(𝐸‘𝑧)) ≠ 0 → (2nd ‘(𝐸‘𝑧)) < (2nd ‘𝑧)))
26	11	ffvelrni 5627	. . . . . . . 8 ⊢ (𝑧 ∈ (ℕ0 × ℕ0) → (𝐸‘𝑧) ∈ (ℕ0 × ℕ0))
27		fvres 5518	. . . . . . . 8 ⊢ ((𝐸‘𝑧) ∈ (ℕ0 × ℕ0) → ((2nd ↾ (ℕ0 × ℕ0))‘(𝐸‘𝑧)) = (2nd ‘(𝐸‘𝑧)))
28	26, 27	syl 14	. . . . . . 7 ⊢ (𝑧 ∈ (ℕ0 × ℕ0) → ((2nd ↾ (ℕ0 × ℕ0))‘(𝐸‘𝑧)) = (2nd ‘(𝐸‘𝑧)))
29	28	neeq1d 2358	. . . . . 6 ⊢ (𝑧 ∈ (ℕ0 × ℕ0) → (((2nd ↾ (ℕ0 × ℕ0))‘(𝐸‘𝑧)) ≠ 0 ↔ (2nd ‘(𝐸‘𝑧)) ≠ 0))
30		fvres 5518	. . . . . . 7 ⊢ (𝑧 ∈ (ℕ0 × ℕ0) → ((2nd ↾ (ℕ0 × ℕ0))‘𝑧) = (2nd ‘𝑧))
31	28, 30	breq12d 4000	. . . . . 6 ⊢ (𝑧 ∈ (ℕ0 × ℕ0) → (((2nd ↾ (ℕ0 × ℕ0))‘(𝐸‘𝑧)) < ((2nd ↾ (ℕ0 × ℕ0))‘𝑧) ↔ (2nd ‘(𝐸‘𝑧)) < (2nd ‘𝑧)))
32	25, 29, 31	3imtr4d 202	. . . . 5 ⊢ (𝑧 ∈ (ℕ0 × ℕ0) → (((2nd ↾ (ℕ0 × ℕ0))‘(𝐸‘𝑧)) ≠ 0 → ((2nd ↾ (ℕ0 × ℕ0))‘(𝐸‘𝑧)) < ((2nd ↾ (ℕ0 × ℕ0))‘𝑧)))
33		eqid 2170	. . . . 5 ⊢ ((2nd ↾ (ℕ0 × ℕ0))‘𝐴) = ((2nd ↾ (ℕ0 × ℕ0))‘𝐴)
34	11, 7, 24, 32, 33	algcvga 11992	. . . 4 ⊢ (𝐴 ∈ (ℕ0 × ℕ0) → (𝐾 ∈ (ℤ≥‘((2nd ↾ (ℕ0 × ℕ0))‘𝐴)) → ((2nd ↾ (ℕ0 × ℕ0))‘(𝑅‘𝐾)) = 0))
35	18, 23, 34	sylc 62	. . 3 ⊢ ((𝐴 ∈ (ℕ0 × ℕ0) ∧ 𝐾 ∈ (ℤ≥‘𝑁)) → ((2nd ↾ (ℕ0 × ℕ0))‘(𝑅‘𝐾)) = 0)
36	17, 35	eqtr3d 2205	. 2 ⊢ ((𝐴 ∈ (ℕ0 × ℕ0) ∧ 𝐾 ∈ (ℤ≥‘𝑁)) → (2nd ‘(𝑅‘𝐾)) = 0)
37	36	ex 114	1 ⊢ (𝐴 ∈ (ℕ0 × ℕ0) → (𝐾 ∈ (ℤ≥‘𝑁) → (2nd ‘(𝑅‘𝐾)) = 0))

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1348   ∈ wcel 2141   ≠ wne 2340  ifcif 3525  {csn 3581  ⟨cop 3584   class class class wbr 3987   × cxp 4607   ↾ cres 4611   ∘ ccom 4613  ⟶wf 5192  ‘cfv 5196  (class class class)co 5850   ∈ cmpo 5852  1^st c1st 6114  2^nd c2nd 6115  0cc0 7761   < clt 7941  ℕ₀cn0 9122  ℤ_≥cuz 9474   mod cmo 10265  seqcseq 10388

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4102  ax-sep 4105  ax-nul 4113  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-iinf 4570  ax-cnex 7852  ax-resscn 7853  ax-1cn 7854  ax-1re 7855  ax-icn 7856  ax-addcl 7857  ax-addrcl 7858  ax-mulcl 7859  ax-mulrcl 7860  ax-addcom 7861  ax-mulcom 7862  ax-addass 7863  ax-mulass 7864  ax-distr 7865  ax-i2m1 7866  ax-0lt1 7867  ax-1rid 7868  ax-0id 7869  ax-rnegex 7870  ax-precex 7871  ax-cnre 7872  ax-pre-ltirr 7873  ax-pre-ltwlin 7874  ax-pre-lttrn 7875  ax-pre-apti 7876  ax-pre-ltadd 7877  ax-pre-mulgt0 7878  ax-pre-mulext 7879  ax-arch 7880

This theorem depends on definitions:  df-bi 116  df-stab 826  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3526  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-int 3830  df-iun 3873  df-br 3988  df-opab 4049  df-mpt 4050  df-tr 4086  df-id 4276  df-po 4279  df-iso 4280  df-iord 4349  df-on 4351  df-ilim 4352  df-suc 4354  df-iom 4573  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203  df-fv 5204  df-riota 5806  df-ov 5853  df-oprab 5854  df-mpo 5855  df-1st 6116  df-2nd 6117  df-recs 6281  df-frec 6367  df-pnf 7943  df-mnf 7944  df-xr 7945  df-ltxr 7946  df-le 7947  df-sub 8079  df-neg 8080  df-reap 8481  df-ap 8488  df-div 8577  df-inn 8866  df-n0 9123  df-z 9200  df-uz 9475  df-q 9566  df-rp 9598  df-fl 10213  df-mod 10266  df-seqfrec 10389

This theorem is referenced by:  eucalg  12000