Theorem eucialgcvga 10440

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 = (2nd ‘𝐴)

Assertion

Ref	Expression
eucialgcvga	⊢ (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 eucialgcvga

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eucialgcvga.3	. . . . . . 7 ⊢ 𝑁 = (2nd ‘𝐴)
2		xp2nd 5813	. . . . . . 7 ⊢ (𝐴 ∈ (ℕ0 × ℕ0) → (2nd ‘𝐴) ∈ ℕ0)
3	1, 2	syl5eqel 2165	. . . . . 6 ⊢ (𝐴 ∈ (ℕ0 × ℕ0) → 𝑁 ∈ ℕ0)
4		eluznn0 8686	. . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐾 ∈ (ℤ≥‘𝑁)) → 𝐾 ∈ ℕ0)
5	3, 4	sylan 277	. . . . 5 ⊢ ((𝐴 ∈ (ℕ0 × ℕ0) ∧ 𝐾 ∈ (ℤ≥‘𝑁)) → 𝐾 ∈ ℕ0)
6		nn0uz 8653	. . . . . . 7 ⊢ ℕ0 = (ℤ≥‘0)
7		eucialg.2	. . . . . . 7 ⊢ 𝑅 = seq0((𝐸 ∘ 1st ), (ℕ0 × {𝐴}), (ℕ0 × ℕ0))
8		0zd 8363	. . . . . . 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 10437	. . . . . . . 8 ⊢ 𝐸:(ℕ0 × ℕ0)⟶(ℕ0 × ℕ0)
12	11	a1i 9	. . . . . . 7 ⊢ (𝐴 ∈ (ℕ0 × ℕ0) → 𝐸:(ℕ0 × ℕ0)⟶(ℕ0 × ℕ0))
13		nn0ex 8294	. . . . . . . . 9 ⊢ ℕ0 ∈ V
14	13, 13	xpex 4471	. . . . . . . 8 ⊢ (ℕ0 × ℕ0) ∈ V
15	14	a1i 9	. . . . . . 7 ⊢ (𝐴 ∈ (ℕ0 × ℕ0) → (ℕ0 × ℕ0) ∈ V)
16	6, 7, 8, 9, 12, 15	ialgrf 10427	. . . . . 6 ⊢ (𝐴 ∈ (ℕ0 × ℕ0) → 𝑅:ℕ0⟶(ℕ0 × ℕ0))
17	16	ffvelrnda 5323	. . . . 5 ⊢ ((𝐴 ∈ (ℕ0 × ℕ0) ∧ 𝐾 ∈ ℕ0) → (𝑅‘𝐾) ∈ (ℕ0 × ℕ0))
18	5, 17	syldan 276	. . . 4 ⊢ ((𝐴 ∈ (ℕ0 × ℕ0) ∧ 𝐾 ∈ (ℤ≥‘𝑁)) → (𝑅‘𝐾) ∈ (ℕ0 × ℕ0))
19		fvres 5219	. . . 4 ⊢ ((𝑅‘𝐾) ∈ (ℕ0 × ℕ0) → ((2nd ↾ (ℕ0 × ℕ0))‘(𝑅‘𝐾)) = (2nd ‘(𝑅‘𝐾)))
20	18, 19	syl 14	. . 3 ⊢ ((𝐴 ∈ (ℕ0 × ℕ0) ∧ 𝐾 ∈ (ℤ≥‘𝑁)) → ((2nd ↾ (ℕ0 × ℕ0))‘(𝑅‘𝐾)) = (2nd ‘(𝑅‘𝐾)))
21		simpl 107	. . . 4 ⊢ ((𝐴 ∈ (ℕ0 × ℕ0) ∧ 𝐾 ∈ (ℤ≥‘𝑁)) → 𝐴 ∈ (ℕ0 × ℕ0))
22		fvres 5219	. . . . . . . 8 ⊢ (𝐴 ∈ (ℕ0 × ℕ0) → ((2nd ↾ (ℕ0 × ℕ0))‘𝐴) = (2nd ‘𝐴))
23	22, 1	syl6eqr 2131	. . . . . . 7 ⊢ (𝐴 ∈ (ℕ0 × ℕ0) → ((2nd ↾ (ℕ0 × ℕ0))‘𝐴) = 𝑁)
24	23	fveq2d 5202	. . . . . 6 ⊢ (𝐴 ∈ (ℕ0 × ℕ0) → (ℤ≥‘((2nd ↾ (ℕ0 × ℕ0))‘𝐴)) = (ℤ≥‘𝑁))
25	24	eleq2d 2148	. . . . 5 ⊢ (𝐴 ∈ (ℕ0 × ℕ0) → (𝐾 ∈ (ℤ≥‘((2nd ↾ (ℕ0 × ℕ0))‘𝐴)) ↔ 𝐾 ∈ (ℤ≥‘𝑁)))
26	25	biimpar 291	. . . 4 ⊢ ((𝐴 ∈ (ℕ0 × ℕ0) ∧ 𝐾 ∈ (ℤ≥‘𝑁)) → 𝐾 ∈ (ℤ≥‘((2nd ↾ (ℕ0 × ℕ0))‘𝐴)))
27		f2ndres 5807	. . . . 5 ⊢ (2nd ↾ (ℕ0 × ℕ0)):(ℕ0 × ℕ0)⟶ℕ0
28	10	eucalglt 10439	. . . . . 6 ⊢ (𝑧 ∈ (ℕ0 × ℕ0) → ((2nd ‘(𝐸‘𝑧)) ≠ 0 → (2nd ‘(𝐸‘𝑧)) < (2nd ‘𝑧)))
29	11	ffvelrni 5322	. . . . . . . 8 ⊢ (𝑧 ∈ (ℕ0 × ℕ0) → (𝐸‘𝑧) ∈ (ℕ0 × ℕ0))
30		fvres 5219	. . . . . . . 8 ⊢ ((𝐸‘𝑧) ∈ (ℕ0 × ℕ0) → ((2nd ↾ (ℕ0 × ℕ0))‘(𝐸‘𝑧)) = (2nd ‘(𝐸‘𝑧)))
31	29, 30	syl 14	. . . . . . 7 ⊢ (𝑧 ∈ (ℕ0 × ℕ0) → ((2nd ↾ (ℕ0 × ℕ0))‘(𝐸‘𝑧)) = (2nd ‘(𝐸‘𝑧)))
32	31	neeq1d 2263	. . . . . 6 ⊢ (𝑧 ∈ (ℕ0 × ℕ0) → (((2nd ↾ (ℕ0 × ℕ0))‘(𝐸‘𝑧)) ≠ 0 ↔ (2nd ‘(𝐸‘𝑧)) ≠ 0))
33		fvres 5219	. . . . . . 7 ⊢ (𝑧 ∈ (ℕ0 × ℕ0) → ((2nd ↾ (ℕ0 × ℕ0))‘𝑧) = (2nd ‘𝑧))
34	31, 33	breq12d 3798	. . . . . 6 ⊢ (𝑧 ∈ (ℕ0 × ℕ0) → (((2nd ↾ (ℕ0 × ℕ0))‘(𝐸‘𝑧)) < ((2nd ↾ (ℕ0 × ℕ0))‘𝑧) ↔ (2nd ‘(𝐸‘𝑧)) < (2nd ‘𝑧)))
35	28, 32, 34	3imtr4d 201	. . . . 5 ⊢ (𝑧 ∈ (ℕ0 × ℕ0) → (((2nd ↾ (ℕ0 × ℕ0))‘(𝐸‘𝑧)) ≠ 0 → ((2nd ↾ (ℕ0 × ℕ0))‘(𝐸‘𝑧)) < ((2nd ↾ (ℕ0 × ℕ0))‘𝑧)))
36		eqid 2081	. . . . 5 ⊢ ((2nd ↾ (ℕ0 × ℕ0))‘𝐴) = ((2nd ↾ (ℕ0 × ℕ0))‘𝐴)
37	11, 7, 27, 35, 36, 14	ialgcvga 10433	. . . 4 ⊢ (𝐴 ∈ (ℕ0 × ℕ0) → (𝐾 ∈ (ℤ≥‘((2nd ↾ (ℕ0 × ℕ0))‘𝐴)) → ((2nd ↾ (ℕ0 × ℕ0))‘(𝑅‘𝐾)) = 0))
38	21, 26, 37	sylc 61	. . 3 ⊢ ((𝐴 ∈ (ℕ0 × ℕ0) ∧ 𝐾 ∈ (ℤ≥‘𝑁)) → ((2nd ↾ (ℕ0 × ℕ0))‘(𝑅‘𝐾)) = 0)
39	20, 38	eqtr3d 2115	. 2 ⊢ ((𝐴 ∈ (ℕ0 × ℕ0) ∧ 𝐾 ∈ (ℤ≥‘𝑁)) → (2nd ‘(𝑅‘𝐾)) = 0)
40	39	ex 113	1 ⊢ (𝐴 ∈ (ℕ0 × ℕ0) → (𝐾 ∈ (ℤ≥‘𝑁) → (2nd ‘(𝑅‘𝐾)) = 0))

Syntax hints:   → wi 4   ∧ wa 102   = wceq 1284   ∈ wcel 1433   ≠ wne 2245  Vcvv 2601  ifcif 3351  {csn 3398  ⟨cop 3401   class class class wbr 3785   × cxp 4361   ↾ cres 4365   ∘ ccom 4367  ⟶wf 4918  ‘cfv 4922  (class class class)co 5532   ↦ cmpt2 5534  1^st c1st 5785  2^nd c2nd 5786  0cc0 6981   < clt 7153  ℕ₀cn0 8288  ℤ_≥cuz 8619   mod cmo 9324  seqcseq 9431

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 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-iinf 4329  ax-cnex 7067  ax-resscn 7068  ax-1cn 7069  ax-1re 7070  ax-icn 7071  ax-addcl 7072  ax-addrcl 7073  ax-mulcl 7074  ax-mulrcl 7075  ax-addcom 7076  ax-mulcom 7077  ax-addass 7078  ax-mulass 7079  ax-distr 7080  ax-i2m1 7081  ax-0lt1 7082  ax-1rid 7083  ax-0id 7084  ax-rnegex 7085  ax-precex 7086  ax-cnre 7087  ax-pre-ltirr 7088  ax-pre-ltwlin 7089  ax-pre-lttrn 7090  ax-pre-apti 7091  ax-pre-ltadd 7092  ax-pre-mulgt0 7093  ax-pre-mulext 7094  ax-arch 7095

This theorem depends on definitions:  df-bi 115  df-dc 776  df-3or 920  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-nel 2340  df-ral 2353  df-rex 2354  df-reu 2355  df-rmo 2356  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-if 3352  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-tr 3876  df-id 4048  df-po 4051  df-iso 4052  df-iord 4121  df-on 4123  df-suc 4126  df-iom 4332  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-riota 5488  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-1st 5787  df-2nd 5788  df-recs 5943  df-frec 6001  df-pnf 7155  df-mnf 7156  df-xr 7157  df-ltxr 7158  df-le 7159  df-sub 7281  df-neg 7282  df-reap 7675  df-ap 7682  df-div 7761  df-inn 8040  df-n0 8289  df-z 8352  df-uz 8620  df-q 8705  df-rp 8735  df-fl 9274  df-mod 9325  df-iseq 9432

This theorem is referenced by:  eucialg  10441