Theorem ialginv 10429

Description: If 𝐼 is an invariant of 𝐹, its value is unchanged after any number of iterations of 𝐹. (Contributed by Paul Chapman, 31-Mar-2011.)

Hypotheses

Ref	Expression
alginv.1	⊢ 𝑅 = seq0((𝐹 ∘ 1st ), (ℕ0 × {𝐴}), 𝑆)
alginv.2	⊢ 𝐹:𝑆⟶𝑆
alginv.3	⊢ 𝐼 Fn 𝑆
alginv.4	⊢ (𝑥 ∈ 𝑆 → (𝐼‘(𝐹‘𝑥)) = (𝐼‘𝑥))
alginv.s	⊢ 𝑆 ∈ 𝑉

Assertion

Ref	Expression
ialginv	⊢ ((𝐴 ∈ 𝑆 ∧ 𝐾 ∈ ℕ0) → (𝐼‘(𝑅‘𝐾)) = (𝐼‘(𝑅‘0)))

Distinct variable groups:   𝑥,𝐹   𝑥,𝐼   𝑥,𝑅   𝑥,𝑆

Allowed substitution hints:   𝐴(𝑥)   𝐾(𝑥)   𝑉(𝑥)

Proof of Theorem ialginv

Dummy variables 𝑧 𝑘 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 5198	. . . . . 6 ⊢ (𝑧 = 0 → (𝑅‘𝑧) = (𝑅‘0))
2	1	fveq2d 5202	. . . . 5 ⊢ (𝑧 = 0 → (𝐼‘(𝑅‘𝑧)) = (𝐼‘(𝑅‘0)))
3	2	eqeq1d 2089	. . . 4 ⊢ (𝑧 = 0 → ((𝐼‘(𝑅‘𝑧)) = (𝐼‘(𝑅‘0)) ↔ (𝐼‘(𝑅‘0)) = (𝐼‘(𝑅‘0))))
4	3	imbi2d 228	. . 3 ⊢ (𝑧 = 0 → ((𝐴 ∈ 𝑆 → (𝐼‘(𝑅‘𝑧)) = (𝐼‘(𝑅‘0))) ↔ (𝐴 ∈ 𝑆 → (𝐼‘(𝑅‘0)) = (𝐼‘(𝑅‘0)))))
5		fveq2 5198	. . . . . 6 ⊢ (𝑧 = 𝑘 → (𝑅‘𝑧) = (𝑅‘𝑘))
6	5	fveq2d 5202	. . . . 5 ⊢ (𝑧 = 𝑘 → (𝐼‘(𝑅‘𝑧)) = (𝐼‘(𝑅‘𝑘)))
7	6	eqeq1d 2089	. . . 4 ⊢ (𝑧 = 𝑘 → ((𝐼‘(𝑅‘𝑧)) = (𝐼‘(𝑅‘0)) ↔ (𝐼‘(𝑅‘𝑘)) = (𝐼‘(𝑅‘0))))
8	7	imbi2d 228	. . 3 ⊢ (𝑧 = 𝑘 → ((𝐴 ∈ 𝑆 → (𝐼‘(𝑅‘𝑧)) = (𝐼‘(𝑅‘0))) ↔ (𝐴 ∈ 𝑆 → (𝐼‘(𝑅‘𝑘)) = (𝐼‘(𝑅‘0)))))
9		fveq2 5198	. . . . . 6 ⊢ (𝑧 = (𝑘 + 1) → (𝑅‘𝑧) = (𝑅‘(𝑘 + 1)))
10	9	fveq2d 5202	. . . . 5 ⊢ (𝑧 = (𝑘 + 1) → (𝐼‘(𝑅‘𝑧)) = (𝐼‘(𝑅‘(𝑘 + 1))))
11	10	eqeq1d 2089	. . . 4 ⊢ (𝑧 = (𝑘 + 1) → ((𝐼‘(𝑅‘𝑧)) = (𝐼‘(𝑅‘0)) ↔ (𝐼‘(𝑅‘(𝑘 + 1))) = (𝐼‘(𝑅‘0))))
12	11	imbi2d 228	. . 3 ⊢ (𝑧 = (𝑘 + 1) → ((𝐴 ∈ 𝑆 → (𝐼‘(𝑅‘𝑧)) = (𝐼‘(𝑅‘0))) ↔ (𝐴 ∈ 𝑆 → (𝐼‘(𝑅‘(𝑘 + 1))) = (𝐼‘(𝑅‘0)))))
13		fveq2 5198	. . . . . 6 ⊢ (𝑧 = 𝐾 → (𝑅‘𝑧) = (𝑅‘𝐾))
14	13	fveq2d 5202	. . . . 5 ⊢ (𝑧 = 𝐾 → (𝐼‘(𝑅‘𝑧)) = (𝐼‘(𝑅‘𝐾)))
15	14	eqeq1d 2089	. . . 4 ⊢ (𝑧 = 𝐾 → ((𝐼‘(𝑅‘𝑧)) = (𝐼‘(𝑅‘0)) ↔ (𝐼‘(𝑅‘𝐾)) = (𝐼‘(𝑅‘0))))
16	15	imbi2d 228	. . 3 ⊢ (𝑧 = 𝐾 → ((𝐴 ∈ 𝑆 → (𝐼‘(𝑅‘𝑧)) = (𝐼‘(𝑅‘0))) ↔ (𝐴 ∈ 𝑆 → (𝐼‘(𝑅‘𝐾)) = (𝐼‘(𝑅‘0)))))
17		eqidd 2082	. . 3 ⊢ (𝐴 ∈ 𝑆 → (𝐼‘(𝑅‘0)) = (𝐼‘(𝑅‘0)))
18		nn0uz 8653	. . . . . . . . . 10 ⊢ ℕ0 = (ℤ≥‘0)
19		alginv.1	. . . . . . . . . 10 ⊢ 𝑅 = seq0((𝐹 ∘ 1st ), (ℕ0 × {𝐴}), 𝑆)
20		0zd 8363	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑆 → 0 ∈ ℤ)
21		id 19	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑆 → 𝐴 ∈ 𝑆)
22		alginv.2	. . . . . . . . . . 11 ⊢ 𝐹:𝑆⟶𝑆
23	22	a1i 9	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑆 → 𝐹:𝑆⟶𝑆)
24		alginv.s	. . . . . . . . . . 11 ⊢ 𝑆 ∈ 𝑉
25	24	a1i 9	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑆 → 𝑆 ∈ 𝑉)
26	18, 19, 20, 21, 23, 25	ialgrp1 10428	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑘 ∈ ℕ0) → (𝑅‘(𝑘 + 1)) = (𝐹‘(𝑅‘𝑘)))
27	26	fveq2d 5202	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑘 ∈ ℕ0) → (𝐼‘(𝑅‘(𝑘 + 1))) = (𝐼‘(𝐹‘(𝑅‘𝑘))))
28	18, 19, 20, 21, 23, 25	ialgrf 10427	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑆 → 𝑅:ℕ0⟶𝑆)
29	28	ffvelrnda 5323	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑘 ∈ ℕ0) → (𝑅‘𝑘) ∈ 𝑆)
30		fveq2 5198	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝑅‘𝑘) → (𝐹‘𝑥) = (𝐹‘(𝑅‘𝑘)))
31	30	fveq2d 5202	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑅‘𝑘) → (𝐼‘(𝐹‘𝑥)) = (𝐼‘(𝐹‘(𝑅‘𝑘))))
32		fveq2 5198	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑅‘𝑘) → (𝐼‘𝑥) = (𝐼‘(𝑅‘𝑘)))
33	31, 32	eqeq12d 2095	. . . . . . . . . 10 ⊢ (𝑥 = (𝑅‘𝑘) → ((𝐼‘(𝐹‘𝑥)) = (𝐼‘𝑥) ↔ (𝐼‘(𝐹‘(𝑅‘𝑘))) = (𝐼‘(𝑅‘𝑘))))
34		alginv.4	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝑆 → (𝐼‘(𝐹‘𝑥)) = (𝐼‘𝑥))
35	33, 34	vtoclga 2664	. . . . . . . . 9 ⊢ ((𝑅‘𝑘) ∈ 𝑆 → (𝐼‘(𝐹‘(𝑅‘𝑘))) = (𝐼‘(𝑅‘𝑘)))
36	29, 35	syl 14	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑘 ∈ ℕ0) → (𝐼‘(𝐹‘(𝑅‘𝑘))) = (𝐼‘(𝑅‘𝑘)))
37	27, 36	eqtrd 2113	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑘 ∈ ℕ0) → (𝐼‘(𝑅‘(𝑘 + 1))) = (𝐼‘(𝑅‘𝑘)))
38	37	eqeq1d 2089	. . . . . 6 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑘 ∈ ℕ0) → ((𝐼‘(𝑅‘(𝑘 + 1))) = (𝐼‘(𝑅‘0)) ↔ (𝐼‘(𝑅‘𝑘)) = (𝐼‘(𝑅‘0))))
39	38	biimprd 156	. . . . 5 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑘 ∈ ℕ0) → ((𝐼‘(𝑅‘𝑘)) = (𝐼‘(𝑅‘0)) → (𝐼‘(𝑅‘(𝑘 + 1))) = (𝐼‘(𝑅‘0))))
40	39	expcom 114	. . . 4 ⊢ (𝑘 ∈ ℕ0 → (𝐴 ∈ 𝑆 → ((𝐼‘(𝑅‘𝑘)) = (𝐼‘(𝑅‘0)) → (𝐼‘(𝑅‘(𝑘 + 1))) = (𝐼‘(𝑅‘0)))))
41	40	a2d 26	. . 3 ⊢ (𝑘 ∈ ℕ0 → ((𝐴 ∈ 𝑆 → (𝐼‘(𝑅‘𝑘)) = (𝐼‘(𝑅‘0))) → (𝐴 ∈ 𝑆 → (𝐼‘(𝑅‘(𝑘 + 1))) = (𝐼‘(𝑅‘0)))))
42	4, 8, 12, 16, 17, 41	nn0ind 8461	. 2 ⊢ (𝐾 ∈ ℕ0 → (𝐴 ∈ 𝑆 → (𝐼‘(𝑅‘𝐾)) = (𝐼‘(𝑅‘0))))
43	42	impcom 123	1 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐾 ∈ ℕ0) → (𝐼‘(𝑅‘𝐾)) = (𝐼‘(𝑅‘0)))

Syntax hints:   → wi 4   ∧ wa 102   = wceq 1284   ∈ wcel 1433  {csn 3398   × cxp 4361   ∘ ccom 4367   Fn wfn 4917  ⟶wf 4918  ‘cfv 4922  (class class class)co 5532  1^st c1st 5785  0cc0 6981  1c1 6982   + caddc 6984  ℕ₀cn0 8288  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-addcom 7076  ax-addass 7078  ax-distr 7080  ax-i2m1 7081  ax-0lt1 7082  ax-0id 7084  ax-rnegex 7085  ax-cnre 7087  ax-pre-ltirr 7088  ax-pre-ltwlin 7089  ax-pre-lttrn 7090  ax-pre-ltadd 7092

This theorem depends on definitions:  df-bi 115  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-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-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-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-inn 8040  df-n0 8289  df-z 8352  df-uz 8620  df-iseq 9432

This theorem is referenced by:  eucialg  10441