Theorem itcovalt2 46023

Description: The value of the function that returns the n-th iterate of the "times 2 plus a constant" function with regard to composition. (Contributed by AV, 7-May-2024.)

Hypothesis

Ref	Expression
itcovalt2.f	⊢ 𝐹 = (𝑛 ∈ ℕ0 ↦ ((2 · 𝑛) + 𝐶))

Assertion

Ref	Expression
itcovalt2	⊢ ((𝐼 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) → ((IterComp‘𝐹)‘𝐼) = (𝑛 ∈ ℕ0 ↦ (((𝑛 + 𝐶) · (2↑𝐼)) − 𝐶)))

Distinct variable groups:   𝐶,𝑛   𝑛,𝐼

Allowed substitution hint:   𝐹(𝑛)

Proof of Theorem itcovalt2

Dummy variables 𝑦 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6774	. . . . 5 ⊢ (𝑥 = 0 → ((IterComp‘𝐹)‘𝑥) = ((IterComp‘𝐹)‘0))
2		oveq2 7283	. . . . . . . 8 ⊢ (𝑥 = 0 → (2↑𝑥) = (2↑0))
3	2	oveq2d 7291	. . . . . . 7 ⊢ (𝑥 = 0 → ((𝑛 + 𝐶) · (2↑𝑥)) = ((𝑛 + 𝐶) · (2↑0)))
4	3	oveq1d 7290	. . . . . 6 ⊢ (𝑥 = 0 → (((𝑛 + 𝐶) · (2↑𝑥)) − 𝐶) = (((𝑛 + 𝐶) · (2↑0)) − 𝐶))
5	4	mpteq2dv 5176	. . . . 5 ⊢ (𝑥 = 0 → (𝑛 ∈ ℕ0 ↦ (((𝑛 + 𝐶) · (2↑𝑥)) − 𝐶)) = (𝑛 ∈ ℕ0 ↦ (((𝑛 + 𝐶) · (2↑0)) − 𝐶)))
6	1, 5	eqeq12d 2754	. . . 4 ⊢ (𝑥 = 0 → (((IterComp‘𝐹)‘𝑥) = (𝑛 ∈ ℕ0 ↦ (((𝑛 + 𝐶) · (2↑𝑥)) − 𝐶)) ↔ ((IterComp‘𝐹)‘0) = (𝑛 ∈ ℕ0 ↦ (((𝑛 + 𝐶) · (2↑0)) − 𝐶))))
7	6	imbi2d 341	. . 3 ⊢ (𝑥 = 0 → ((𝐶 ∈ ℕ0 → ((IterComp‘𝐹)‘𝑥) = (𝑛 ∈ ℕ0 ↦ (((𝑛 + 𝐶) · (2↑𝑥)) − 𝐶))) ↔ (𝐶 ∈ ℕ0 → ((IterComp‘𝐹)‘0) = (𝑛 ∈ ℕ0 ↦ (((𝑛 + 𝐶) · (2↑0)) − 𝐶)))))
8		fveq2 6774	. . . . 5 ⊢ (𝑥 = 𝑦 → ((IterComp‘𝐹)‘𝑥) = ((IterComp‘𝐹)‘𝑦))
9		oveq2 7283	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (2↑𝑥) = (2↑𝑦))
10	9	oveq2d 7291	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ((𝑛 + 𝐶) · (2↑𝑥)) = ((𝑛 + 𝐶) · (2↑𝑦)))
11	10	oveq1d 7290	. . . . . 6 ⊢ (𝑥 = 𝑦 → (((𝑛 + 𝐶) · (2↑𝑥)) − 𝐶) = (((𝑛 + 𝐶) · (2↑𝑦)) − 𝐶))
12	11	mpteq2dv 5176	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑛 ∈ ℕ0 ↦ (((𝑛 + 𝐶) · (2↑𝑥)) − 𝐶)) = (𝑛 ∈ ℕ0 ↦ (((𝑛 + 𝐶) · (2↑𝑦)) − 𝐶)))
13	8, 12	eqeq12d 2754	. . . 4 ⊢ (𝑥 = 𝑦 → (((IterComp‘𝐹)‘𝑥) = (𝑛 ∈ ℕ0 ↦ (((𝑛 + 𝐶) · (2↑𝑥)) − 𝐶)) ↔ ((IterComp‘𝐹)‘𝑦) = (𝑛 ∈ ℕ0 ↦ (((𝑛 + 𝐶) · (2↑𝑦)) − 𝐶))))
14	13	imbi2d 341	. . 3 ⊢ (𝑥 = 𝑦 → ((𝐶 ∈ ℕ0 → ((IterComp‘𝐹)‘𝑥) = (𝑛 ∈ ℕ0 ↦ (((𝑛 + 𝐶) · (2↑𝑥)) − 𝐶))) ↔ (𝐶 ∈ ℕ0 → ((IterComp‘𝐹)‘𝑦) = (𝑛 ∈ ℕ0 ↦ (((𝑛 + 𝐶) · (2↑𝑦)) − 𝐶)))))
15		fveq2 6774	. . . . 5 ⊢ (𝑥 = (𝑦 + 1) → ((IterComp‘𝐹)‘𝑥) = ((IterComp‘𝐹)‘(𝑦 + 1)))
16		oveq2 7283	. . . . . . . 8 ⊢ (𝑥 = (𝑦 + 1) → (2↑𝑥) = (2↑(𝑦 + 1)))
17	16	oveq2d 7291	. . . . . . 7 ⊢ (𝑥 = (𝑦 + 1) → ((𝑛 + 𝐶) · (2↑𝑥)) = ((𝑛 + 𝐶) · (2↑(𝑦 + 1))))
18	17	oveq1d 7290	. . . . . 6 ⊢ (𝑥 = (𝑦 + 1) → (((𝑛 + 𝐶) · (2↑𝑥)) − 𝐶) = (((𝑛 + 𝐶) · (2↑(𝑦 + 1))) − 𝐶))
19	18	mpteq2dv 5176	. . . . 5 ⊢ (𝑥 = (𝑦 + 1) → (𝑛 ∈ ℕ0 ↦ (((𝑛 + 𝐶) · (2↑𝑥)) − 𝐶)) = (𝑛 ∈ ℕ0 ↦ (((𝑛 + 𝐶) · (2↑(𝑦 + 1))) − 𝐶)))
20	15, 19	eqeq12d 2754	. . . 4 ⊢ (𝑥 = (𝑦 + 1) → (((IterComp‘𝐹)‘𝑥) = (𝑛 ∈ ℕ0 ↦ (((𝑛 + 𝐶) · (2↑𝑥)) − 𝐶)) ↔ ((IterComp‘𝐹)‘(𝑦 + 1)) = (𝑛 ∈ ℕ0 ↦ (((𝑛 + 𝐶) · (2↑(𝑦 + 1))) − 𝐶))))
21	20	imbi2d 341	. . 3 ⊢ (𝑥 = (𝑦 + 1) → ((𝐶 ∈ ℕ0 → ((IterComp‘𝐹)‘𝑥) = (𝑛 ∈ ℕ0 ↦ (((𝑛 + 𝐶) · (2↑𝑥)) − 𝐶))) ↔ (𝐶 ∈ ℕ0 → ((IterComp‘𝐹)‘(𝑦 + 1)) = (𝑛 ∈ ℕ0 ↦ (((𝑛 + 𝐶) · (2↑(𝑦 + 1))) − 𝐶)))))
22		fveq2 6774	. . . . 5 ⊢ (𝑥 = 𝐼 → ((IterComp‘𝐹)‘𝑥) = ((IterComp‘𝐹)‘𝐼))
23		oveq2 7283	. . . . . . . 8 ⊢ (𝑥 = 𝐼 → (2↑𝑥) = (2↑𝐼))
24	23	oveq2d 7291	. . . . . . 7 ⊢ (𝑥 = 𝐼 → ((𝑛 + 𝐶) · (2↑𝑥)) = ((𝑛 + 𝐶) · (2↑𝐼)))
25	24	oveq1d 7290	. . . . . 6 ⊢ (𝑥 = 𝐼 → (((𝑛 + 𝐶) · (2↑𝑥)) − 𝐶) = (((𝑛 + 𝐶) · (2↑𝐼)) − 𝐶))
26	25	mpteq2dv 5176	. . . . 5 ⊢ (𝑥 = 𝐼 → (𝑛 ∈ ℕ0 ↦ (((𝑛 + 𝐶) · (2↑𝑥)) − 𝐶)) = (𝑛 ∈ ℕ0 ↦ (((𝑛 + 𝐶) · (2↑𝐼)) − 𝐶)))
27	22, 26	eqeq12d 2754	. . . 4 ⊢ (𝑥 = 𝐼 → (((IterComp‘𝐹)‘𝑥) = (𝑛 ∈ ℕ0 ↦ (((𝑛 + 𝐶) · (2↑𝑥)) − 𝐶)) ↔ ((IterComp‘𝐹)‘𝐼) = (𝑛 ∈ ℕ0 ↦ (((𝑛 + 𝐶) · (2↑𝐼)) − 𝐶))))
28	27	imbi2d 341	. . 3 ⊢ (𝑥 = 𝐼 → ((𝐶 ∈ ℕ0 → ((IterComp‘𝐹)‘𝑥) = (𝑛 ∈ ℕ0 ↦ (((𝑛 + 𝐶) · (2↑𝑥)) − 𝐶))) ↔ (𝐶 ∈ ℕ0 → ((IterComp‘𝐹)‘𝐼) = (𝑛 ∈ ℕ0 ↦ (((𝑛 + 𝐶) · (2↑𝐼)) − 𝐶)))))
29		itcovalt2.f	. . . 4 ⊢ 𝐹 = (𝑛 ∈ ℕ0 ↦ ((2 · 𝑛) + 𝐶))
30	29	itcovalt2lem1 46021	. . 3 ⊢ (𝐶 ∈ ℕ0 → ((IterComp‘𝐹)‘0) = (𝑛 ∈ ℕ0 ↦ (((𝑛 + 𝐶) · (2↑0)) − 𝐶)))
31		pm2.27 42	. . . . . . 7 ⊢ (𝐶 ∈ ℕ0 → ((𝐶 ∈ ℕ0 → ((IterComp‘𝐹)‘𝑦) = (𝑛 ∈ ℕ0 ↦ (((𝑛 + 𝐶) · (2↑𝑦)) − 𝐶))) → ((IterComp‘𝐹)‘𝑦) = (𝑛 ∈ ℕ0 ↦ (((𝑛 + 𝐶) · (2↑𝑦)) − 𝐶))))
32	31	adantl 482	. . . . . 6 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) → ((𝐶 ∈ ℕ0 → ((IterComp‘𝐹)‘𝑦) = (𝑛 ∈ ℕ0 ↦ (((𝑛 + 𝐶) · (2↑𝑦)) − 𝐶))) → ((IterComp‘𝐹)‘𝑦) = (𝑛 ∈ ℕ0 ↦ (((𝑛 + 𝐶) · (2↑𝑦)) − 𝐶))))
33	29	itcovalt2lem2 46022	. . . . . 6 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) → (((IterComp‘𝐹)‘𝑦) = (𝑛 ∈ ℕ0 ↦ (((𝑛 + 𝐶) · (2↑𝑦)) − 𝐶)) → ((IterComp‘𝐹)‘(𝑦 + 1)) = (𝑛 ∈ ℕ0 ↦ (((𝑛 + 𝐶) · (2↑(𝑦 + 1))) − 𝐶))))
34	32, 33	syld 47	. . . . 5 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) → ((𝐶 ∈ ℕ0 → ((IterComp‘𝐹)‘𝑦) = (𝑛 ∈ ℕ0 ↦ (((𝑛 + 𝐶) · (2↑𝑦)) − 𝐶))) → ((IterComp‘𝐹)‘(𝑦 + 1)) = (𝑛 ∈ ℕ0 ↦ (((𝑛 + 𝐶) · (2↑(𝑦 + 1))) − 𝐶))))
35	34	ex 413	. . . 4 ⊢ (𝑦 ∈ ℕ0 → (𝐶 ∈ ℕ0 → ((𝐶 ∈ ℕ0 → ((IterComp‘𝐹)‘𝑦) = (𝑛 ∈ ℕ0 ↦ (((𝑛 + 𝐶) · (2↑𝑦)) − 𝐶))) → ((IterComp‘𝐹)‘(𝑦 + 1)) = (𝑛 ∈ ℕ0 ↦ (((𝑛 + 𝐶) · (2↑(𝑦 + 1))) − 𝐶)))))
36	35	com23 86	. . 3 ⊢ (𝑦 ∈ ℕ0 → ((𝐶 ∈ ℕ0 → ((IterComp‘𝐹)‘𝑦) = (𝑛 ∈ ℕ0 ↦ (((𝑛 + 𝐶) · (2↑𝑦)) − 𝐶))) → (𝐶 ∈ ℕ0 → ((IterComp‘𝐹)‘(𝑦 + 1)) = (𝑛 ∈ ℕ0 ↦ (((𝑛 + 𝐶) · (2↑(𝑦 + 1))) − 𝐶)))))
37	7, 14, 21, 28, 30, 36	nn0ind 12415	. 2 ⊢ (𝐼 ∈ ℕ0 → (𝐶 ∈ ℕ0 → ((IterComp‘𝐹)‘𝐼) = (𝑛 ∈ ℕ0 ↦ (((𝑛 + 𝐶) · (2↑𝐼)) − 𝐶))))
38	37	imp 407	1 ⊢ ((𝐼 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) → ((IterComp‘𝐹)‘𝐼) = (𝑛 ∈ ℕ0 ↦ (((𝑛 + 𝐶) · (2↑𝐼)) − 𝐶)))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539   ∈ wcel 2106   ↦ cmpt 5157  ‘cfv 6433  (class class class)co 7275  0cc0 10871  1c1 10872   + caddc 10874   · cmul 10876   − cmin 11205  2c2 12028  ℕ₀cn0 12233  ↑cexp 13782  IterCompcitco 46003

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-inf2 9399  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-er 8498  df-en 8734  df-dom 8735  df-sdom 8736  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-nn 11974  df-2 12036  df-n0 12234  df-z 12320  df-uz 12583  df-seq 13722  df-exp 13783  df-itco 46005

This theorem is referenced by:  ackval3  46029