Theorem itcovalt2 48715

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 6822	. . . . 5 ⊢ (𝑥 = 0 → ((IterComp‘𝐹)‘𝑥) = ((IterComp‘𝐹)‘0))
2		oveq2 7354	. . . . . . . 8 ⊢ (𝑥 = 0 → (2↑𝑥) = (2↑0))
3	2	oveq2d 7362	. . . . . . 7 ⊢ (𝑥 = 0 → ((𝑛 + 𝐶) · (2↑𝑥)) = ((𝑛 + 𝐶) · (2↑0)))
4	3	oveq1d 7361	. . . . . 6 ⊢ (𝑥 = 0 → (((𝑛 + 𝐶) · (2↑𝑥)) − 𝐶) = (((𝑛 + 𝐶) · (2↑0)) − 𝐶))
5	4	mpteq2dv 5185	. . . . 5 ⊢ (𝑥 = 0 → (𝑛 ∈ ℕ0 ↦ (((𝑛 + 𝐶) · (2↑𝑥)) − 𝐶)) = (𝑛 ∈ ℕ0 ↦ (((𝑛 + 𝐶) · (2↑0)) − 𝐶)))
6	1, 5	eqeq12d 2747	. . . 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 340	. . 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 6822	. . . . 5 ⊢ (𝑥 = 𝑦 → ((IterComp‘𝐹)‘𝑥) = ((IterComp‘𝐹)‘𝑦))
9		oveq2 7354	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (2↑𝑥) = (2↑𝑦))
10	9	oveq2d 7362	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ((𝑛 + 𝐶) · (2↑𝑥)) = ((𝑛 + 𝐶) · (2↑𝑦)))
11	10	oveq1d 7361	. . . . . 6 ⊢ (𝑥 = 𝑦 → (((𝑛 + 𝐶) · (2↑𝑥)) − 𝐶) = (((𝑛 + 𝐶) · (2↑𝑦)) − 𝐶))
12	11	mpteq2dv 5185	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑛 ∈ ℕ0 ↦ (((𝑛 + 𝐶) · (2↑𝑥)) − 𝐶)) = (𝑛 ∈ ℕ0 ↦ (((𝑛 + 𝐶) · (2↑𝑦)) − 𝐶)))
13	8, 12	eqeq12d 2747	. . . 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 340	. . 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 6822	. . . . 5 ⊢ (𝑥 = (𝑦 + 1) → ((IterComp‘𝐹)‘𝑥) = ((IterComp‘𝐹)‘(𝑦 + 1)))
16		oveq2 7354	. . . . . . . 8 ⊢ (𝑥 = (𝑦 + 1) → (2↑𝑥) = (2↑(𝑦 + 1)))
17	16	oveq2d 7362	. . . . . . 7 ⊢ (𝑥 = (𝑦 + 1) → ((𝑛 + 𝐶) · (2↑𝑥)) = ((𝑛 + 𝐶) · (2↑(𝑦 + 1))))
18	17	oveq1d 7361	. . . . . 6 ⊢ (𝑥 = (𝑦 + 1) → (((𝑛 + 𝐶) · (2↑𝑥)) − 𝐶) = (((𝑛 + 𝐶) · (2↑(𝑦 + 1))) − 𝐶))
19	18	mpteq2dv 5185	. . . . 5 ⊢ (𝑥 = (𝑦 + 1) → (𝑛 ∈ ℕ0 ↦ (((𝑛 + 𝐶) · (2↑𝑥)) − 𝐶)) = (𝑛 ∈ ℕ0 ↦ (((𝑛 + 𝐶) · (2↑(𝑦 + 1))) − 𝐶)))
20	15, 19	eqeq12d 2747	. . . 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 340	. . 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 6822	. . . . 5 ⊢ (𝑥 = 𝐼 → ((IterComp‘𝐹)‘𝑥) = ((IterComp‘𝐹)‘𝐼))
23		oveq2 7354	. . . . . . . 8 ⊢ (𝑥 = 𝐼 → (2↑𝑥) = (2↑𝐼))
24	23	oveq2d 7362	. . . . . . 7 ⊢ (𝑥 = 𝐼 → ((𝑛 + 𝐶) · (2↑𝑥)) = ((𝑛 + 𝐶) · (2↑𝐼)))
25	24	oveq1d 7361	. . . . . 6 ⊢ (𝑥 = 𝐼 → (((𝑛 + 𝐶) · (2↑𝑥)) − 𝐶) = (((𝑛 + 𝐶) · (2↑𝐼)) − 𝐶))
26	25	mpteq2dv 5185	. . . . 5 ⊢ (𝑥 = 𝐼 → (𝑛 ∈ ℕ0 ↦ (((𝑛 + 𝐶) · (2↑𝑥)) − 𝐶)) = (𝑛 ∈ ℕ0 ↦ (((𝑛 + 𝐶) · (2↑𝐼)) − 𝐶)))
27	22, 26	eqeq12d 2747	. . . 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 340	. . 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 48713	. . 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 481	. . . . . 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 48714	. . . . . 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 412	. . . 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 12568	. 2 ⊢ (𝐼 ∈ ℕ0 → (𝐶 ∈ ℕ0 → ((IterComp‘𝐹)‘𝐼) = (𝑛 ∈ ℕ0 ↦ (((𝑛 + 𝐶) · (2↑𝐼)) − 𝐶))))
38	37	imp 406	1 ⊢ ((𝐼 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) → ((IterComp‘𝐹)‘𝐼) = (𝑛 ∈ ℕ0 ↦ (((𝑛 + 𝐶) · (2↑𝐼)) − 𝐶)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111   ↦ cmpt 5172  ‘cfv 6481  (class class class)co 7346  0cc0 11006  1c1 11007   + caddc 11009   · cmul 11011   − cmin 11344  2c2 12180  ℕ₀cn0 12381  ↑cexp 13968  IterCompcitco 48695

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5217  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668  ax-inf2 9531  ax-cnex 11062  ax-resscn 11063  ax-1cn 11064  ax-icn 11065  ax-addcl 11066  ax-addrcl 11067  ax-mulcl 11068  ax-mulrcl 11069  ax-mulcom 11070  ax-addass 11071  ax-mulass 11072  ax-distr 11073  ax-i2m1 11074  ax-1ne0 11075  ax-1rid 11076  ax-rnegex 11077  ax-rrecex 11078  ax-cnre 11079  ax-pre-lttri 11080  ax-pre-lttrn 11081  ax-pre-ltadd 11082  ax-pre-mulgt0 11083

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-iun 4943  df-br 5092  df-opab 5154  df-mpt 5173  df-tr 5199  df-id 5511  df-eprel 5516  df-po 5524  df-so 5525  df-fr 5569  df-we 5571  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7797  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-er 8622  df-en 8870  df-dom 8871  df-sdom 8872  df-pnf 11148  df-mnf 11149  df-xr 11150  df-ltxr 11151  df-le 11152  df-sub 11346  df-neg 11347  df-nn 12126  df-2 12188  df-n0 12382  df-z 12469  df-uz 12733  df-seq 13909  df-exp 13969  df-itco 48697

This theorem is referenced by:  ackval3  48721