Theorem alginv 12624

Description: If 𝐼 is an invariant of 𝐹, then 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}))
alginv.2	⊢ 𝐹:𝑆⟶𝑆
alginv.3	⊢ (𝑥 ∈ 𝑆 → (𝐼‘(𝐹‘𝑥)) = (𝐼‘𝑥))

Assertion

Ref	Expression
alginv	⊢ ((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)

Proof of Theorem alginv

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

Step	Hyp	Ref	Expression
1		2fveq3 5644	. . . . 5 ⊢ (𝑧 = 0 → (𝐼‘(𝑅‘𝑧)) = (𝐼‘(𝑅‘0)))
2	1	eqeq1d 2240	. . . 4 ⊢ (𝑧 = 0 → ((𝐼‘(𝑅‘𝑧)) = (𝐼‘(𝑅‘0)) ↔ (𝐼‘(𝑅‘0)) = (𝐼‘(𝑅‘0))))
3	2	imbi2d 230	. . 3 ⊢ (𝑧 = 0 → ((𝐴 ∈ 𝑆 → (𝐼‘(𝑅‘𝑧)) = (𝐼‘(𝑅‘0))) ↔ (𝐴 ∈ 𝑆 → (𝐼‘(𝑅‘0)) = (𝐼‘(𝑅‘0)))))
4		2fveq3 5644	. . . . 5 ⊢ (𝑧 = 𝑘 → (𝐼‘(𝑅‘𝑧)) = (𝐼‘(𝑅‘𝑘)))
5	4	eqeq1d 2240	. . . 4 ⊢ (𝑧 = 𝑘 → ((𝐼‘(𝑅‘𝑧)) = (𝐼‘(𝑅‘0)) ↔ (𝐼‘(𝑅‘𝑘)) = (𝐼‘(𝑅‘0))))
6	5	imbi2d 230	. . 3 ⊢ (𝑧 = 𝑘 → ((𝐴 ∈ 𝑆 → (𝐼‘(𝑅‘𝑧)) = (𝐼‘(𝑅‘0))) ↔ (𝐴 ∈ 𝑆 → (𝐼‘(𝑅‘𝑘)) = (𝐼‘(𝑅‘0)))))
7		2fveq3 5644	. . . . 5 ⊢ (𝑧 = (𝑘 + 1) → (𝐼‘(𝑅‘𝑧)) = (𝐼‘(𝑅‘(𝑘 + 1))))
8	7	eqeq1d 2240	. . . 4 ⊢ (𝑧 = (𝑘 + 1) → ((𝐼‘(𝑅‘𝑧)) = (𝐼‘(𝑅‘0)) ↔ (𝐼‘(𝑅‘(𝑘 + 1))) = (𝐼‘(𝑅‘0))))
9	8	imbi2d 230	. . 3 ⊢ (𝑧 = (𝑘 + 1) → ((𝐴 ∈ 𝑆 → (𝐼‘(𝑅‘𝑧)) = (𝐼‘(𝑅‘0))) ↔ (𝐴 ∈ 𝑆 → (𝐼‘(𝑅‘(𝑘 + 1))) = (𝐼‘(𝑅‘0)))))
10		2fveq3 5644	. . . . 5 ⊢ (𝑧 = 𝐾 → (𝐼‘(𝑅‘𝑧)) = (𝐼‘(𝑅‘𝐾)))
11	10	eqeq1d 2240	. . . 4 ⊢ (𝑧 = 𝐾 → ((𝐼‘(𝑅‘𝑧)) = (𝐼‘(𝑅‘0)) ↔ (𝐼‘(𝑅‘𝐾)) = (𝐼‘(𝑅‘0))))
12	11	imbi2d 230	. . 3 ⊢ (𝑧 = 𝐾 → ((𝐴 ∈ 𝑆 → (𝐼‘(𝑅‘𝑧)) = (𝐼‘(𝑅‘0))) ↔ (𝐴 ∈ 𝑆 → (𝐼‘(𝑅‘𝐾)) = (𝐼‘(𝑅‘0)))))
13		eqidd 2232	. . 3 ⊢ (𝐴 ∈ 𝑆 → (𝐼‘(𝑅‘0)) = (𝐼‘(𝑅‘0)))
14		nn0uz 9791	. . . . . . . . . 10 ⊢ ℕ0 = (ℤ≥‘0)
15		alginv.1	. . . . . . . . . 10 ⊢ 𝑅 = seq0((𝐹 ∘ 1st ), (ℕ0 × {𝐴}))
16		0zd 9491	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑆 → 0 ∈ ℤ)
17		id 19	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑆 → 𝐴 ∈ 𝑆)
18		alginv.2	. . . . . . . . . . 11 ⊢ 𝐹:𝑆⟶𝑆
19	18	a1i 9	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑆 → 𝐹:𝑆⟶𝑆)
20	14, 15, 16, 17, 19	algrp1 12623	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑘 ∈ ℕ0) → (𝑅‘(𝑘 + 1)) = (𝐹‘(𝑅‘𝑘)))
21	20	fveq2d 5643	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑘 ∈ ℕ0) → (𝐼‘(𝑅‘(𝑘 + 1))) = (𝐼‘(𝐹‘(𝑅‘𝑘))))
22	14, 15, 16, 17, 19	algrf 12622	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑆 → 𝑅:ℕ0⟶𝑆)
23	22	ffvelcdmda 5782	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑘 ∈ ℕ0) → (𝑅‘𝑘) ∈ 𝑆)
24		2fveq3 5644	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑅‘𝑘) → (𝐼‘(𝐹‘𝑥)) = (𝐼‘(𝐹‘(𝑅‘𝑘))))
25		fveq2 5639	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑅‘𝑘) → (𝐼‘𝑥) = (𝐼‘(𝑅‘𝑘)))
26	24, 25	eqeq12d 2246	. . . . . . . . . 10 ⊢ (𝑥 = (𝑅‘𝑘) → ((𝐼‘(𝐹‘𝑥)) = (𝐼‘𝑥) ↔ (𝐼‘(𝐹‘(𝑅‘𝑘))) = (𝐼‘(𝑅‘𝑘))))
27		alginv.3	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝑆 → (𝐼‘(𝐹‘𝑥)) = (𝐼‘𝑥))
28	26, 27	vtoclga 2870	. . . . . . . . 9 ⊢ ((𝑅‘𝑘) ∈ 𝑆 → (𝐼‘(𝐹‘(𝑅‘𝑘))) = (𝐼‘(𝑅‘𝑘)))
29	23, 28	syl 14	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑘 ∈ ℕ0) → (𝐼‘(𝐹‘(𝑅‘𝑘))) = (𝐼‘(𝑅‘𝑘)))
30	21, 29	eqtrd 2264	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑘 ∈ ℕ0) → (𝐼‘(𝑅‘(𝑘 + 1))) = (𝐼‘(𝑅‘𝑘)))
31	30	eqeq1d 2240	. . . . . 6 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑘 ∈ ℕ0) → ((𝐼‘(𝑅‘(𝑘 + 1))) = (𝐼‘(𝑅‘0)) ↔ (𝐼‘(𝑅‘𝑘)) = (𝐼‘(𝑅‘0))))
32	31	biimprd 158	. . . . 5 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑘 ∈ ℕ0) → ((𝐼‘(𝑅‘𝑘)) = (𝐼‘(𝑅‘0)) → (𝐼‘(𝑅‘(𝑘 + 1))) = (𝐼‘(𝑅‘0))))
33	32	expcom 116	. . . 4 ⊢ (𝑘 ∈ ℕ0 → (𝐴 ∈ 𝑆 → ((𝐼‘(𝑅‘𝑘)) = (𝐼‘(𝑅‘0)) → (𝐼‘(𝑅‘(𝑘 + 1))) = (𝐼‘(𝑅‘0)))))
34	33	a2d 26	. . 3 ⊢ (𝑘 ∈ ℕ0 → ((𝐴 ∈ 𝑆 → (𝐼‘(𝑅‘𝑘)) = (𝐼‘(𝑅‘0))) → (𝐴 ∈ 𝑆 → (𝐼‘(𝑅‘(𝑘 + 1))) = (𝐼‘(𝑅‘0)))))
35	3, 6, 9, 12, 13, 34	nn0ind 9594	. 2 ⊢ (𝐾 ∈ ℕ0 → (𝐴 ∈ 𝑆 → (𝐼‘(𝑅‘𝐾)) = (𝐼‘(𝑅‘0))))
36	35	impcom 125	1 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐾 ∈ ℕ0) → (𝐼‘(𝑅‘𝐾)) = (𝐼‘(𝑅‘0)))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1397   ∈ wcel 2202  {csn 3669   × cxp 4723   ∘ ccom 4729  ⟶wf 5322  ‘cfv 5326  (class class class)co 6018  1^st c1st 6301  0cc0 8032  1c1 8033   + caddc 8035  ℕ₀cn0 9402  seqcseq 10710

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-addcom 8132  ax-addass 8134  ax-distr 8136  ax-i2m1 8137  ax-0lt1 8138  ax-0id 8140  ax-rnegex 8141  ax-cnre 8143  ax-pre-ltirr 8144  ax-pre-ltwlin 8145  ax-pre-lttrn 8146  ax-pre-ltadd 8148

This theorem depends on definitions:  df-bi 117  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-recs 6471  df-frec 6557  df-pnf 8216  df-mnf 8217  df-xr 8218  df-ltxr 8219  df-le 8220  df-sub 8352  df-neg 8353  df-inn 9144  df-n0 9403  df-z 9480  df-uz 9756  df-seqfrec 10711

This theorem is referenced by:  eucalg  12636