Theorem itcovalt2 48688

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 6817	. . . . 5 ⊢ (𝑥 = 0 → ((IterComp‘𝐹)‘𝑥) = ((IterComp‘𝐹)‘0))
2		oveq2 7349	. . . . . . . 8 ⊢ (𝑥 = 0 → (2↑𝑥) = (2↑0))
3	2	oveq2d 7357	. . . . . . 7 ⊢ (𝑥 = 0 → ((𝑛 + 𝐶) · (2↑𝑥)) = ((𝑛 + 𝐶) · (2↑0)))
4	3	oveq1d 7356	. . . . . 6 ⊢ (𝑥 = 0 → (((𝑛 + 𝐶) · (2↑𝑥)) − 𝐶) = (((𝑛 + 𝐶) · (2↑0)) − 𝐶))
5	4	mpteq2dv 5183	. . . . 5 ⊢ (𝑥 = 0 → (𝑛 ∈ ℕ0 ↦ (((𝑛 + 𝐶) · (2↑𝑥)) − 𝐶)) = (𝑛 ∈ ℕ0 ↦ (((𝑛 + 𝐶) · (2↑0)) − 𝐶)))
6	1, 5	eqeq12d 2746	. . . 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 6817	. . . . 5 ⊢ (𝑥 = 𝑦 → ((IterComp‘𝐹)‘𝑥) = ((IterComp‘𝐹)‘𝑦))
9		oveq2 7349	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (2↑𝑥) = (2↑𝑦))
10	9	oveq2d 7357	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ((𝑛 + 𝐶) · (2↑𝑥)) = ((𝑛 + 𝐶) · (2↑𝑦)))
11	10	oveq1d 7356	. . . . . 6 ⊢ (𝑥 = 𝑦 → (((𝑛 + 𝐶) · (2↑𝑥)) − 𝐶) = (((𝑛 + 𝐶) · (2↑𝑦)) − 𝐶))
12	11	mpteq2dv 5183	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑛 ∈ ℕ0 ↦ (((𝑛 + 𝐶) · (2↑𝑥)) − 𝐶)) = (𝑛 ∈ ℕ0 ↦ (((𝑛 + 𝐶) · (2↑𝑦)) − 𝐶)))
13	8, 12	eqeq12d 2746	. . . 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 6817	. . . . 5 ⊢ (𝑥 = (𝑦 + 1) → ((IterComp‘𝐹)‘𝑥) = ((IterComp‘𝐹)‘(𝑦 + 1)))
16		oveq2 7349	. . . . . . . 8 ⊢ (𝑥 = (𝑦 + 1) → (2↑𝑥) = (2↑(𝑦 + 1)))
17	16	oveq2d 7357	. . . . . . 7 ⊢ (𝑥 = (𝑦 + 1) → ((𝑛 + 𝐶) · (2↑𝑥)) = ((𝑛 + 𝐶) · (2↑(𝑦 + 1))))
18	17	oveq1d 7356	. . . . . 6 ⊢ (𝑥 = (𝑦 + 1) → (((𝑛 + 𝐶) · (2↑𝑥)) − 𝐶) = (((𝑛 + 𝐶) · (2↑(𝑦 + 1))) − 𝐶))
19	18	mpteq2dv 5183	. . . . 5 ⊢ (𝑥 = (𝑦 + 1) → (𝑛 ∈ ℕ0 ↦ (((𝑛 + 𝐶) · (2↑𝑥)) − 𝐶)) = (𝑛 ∈ ℕ0 ↦ (((𝑛 + 𝐶) · (2↑(𝑦 + 1))) − 𝐶)))
20	15, 19	eqeq12d 2746	. . . 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 6817	. . . . 5 ⊢ (𝑥 = 𝐼 → ((IterComp‘𝐹)‘𝑥) = ((IterComp‘𝐹)‘𝐼))
23		oveq2 7349	. . . . . . . 8 ⊢ (𝑥 = 𝐼 → (2↑𝑥) = (2↑𝐼))
24	23	oveq2d 7357	. . . . . . 7 ⊢ (𝑥 = 𝐼 → ((𝑛 + 𝐶) · (2↑𝑥)) = ((𝑛 + 𝐶) · (2↑𝐼)))
25	24	oveq1d 7356	. . . . . 6 ⊢ (𝑥 = 𝐼 → (((𝑛 + 𝐶) · (2↑𝑥)) − 𝐶) = (((𝑛 + 𝐶) · (2↑𝐼)) − 𝐶))
26	25	mpteq2dv 5183	. . . . 5 ⊢ (𝑥 = 𝐼 → (𝑛 ∈ ℕ0 ↦ (((𝑛 + 𝐶) · (2↑𝑥)) − 𝐶)) = (𝑛 ∈ ℕ0 ↦ (((𝑛 + 𝐶) · (2↑𝐼)) − 𝐶)))
27	22, 26	eqeq12d 2746	. . . 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 48686	. . 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 48687	. . . . . 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 12560	. 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 2110   ↦ cmpt 5170  ‘cfv 6477  (class class class)co 7341  0cc0 10998  1c1 10999   + caddc 11001   · cmul 11003   − cmin 11336  2c2 12172  ℕ₀cn0 12373  ↑cexp 13960  IterCompcitco 48668

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 2112  ax-9 2120  ax-10 2143  ax-11 2159  ax-12 2179  ax-ext 2702  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7663  ax-inf2 9526  ax-cnex 11054  ax-resscn 11055  ax-1cn 11056  ax-icn 11057  ax-addcl 11058  ax-addrcl 11059  ax-mulcl 11060  ax-mulrcl 11061  ax-mulcom 11062  ax-addass 11063  ax-mulass 11064  ax-distr 11065  ax-i2m1 11066  ax-1ne0 11067  ax-1rid 11068  ax-rnegex 11069  ax-rrecex 11070  ax-cnre 11071  ax-pre-lttri 11072  ax-pre-lttrn 11073  ax-pre-ltadd 11074  ax-pre-mulgt0 11075

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 2067  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-reu 3345  df-rab 3394  df-v 3436  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-pss 3920  df-nul 4282  df-if 4474  df-pw 4550  df-sn 4575  df-pr 4577  df-op 4581  df-uni 4858  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6244  df-ord 6305  df-on 6306  df-lim 6307  df-suc 6308  df-iota 6433  df-fun 6479  df-fn 6480  df-f 6481  df-f1 6482  df-fo 6483  df-f1o 6484  df-fv 6485  df-riota 7298  df-ov 7344  df-oprab 7345  df-mpo 7346  df-om 7792  df-2nd 7917  df-frecs 8206  df-wrecs 8237  df-recs 8286  df-rdg 8324  df-er 8617  df-en 8865  df-dom 8866  df-sdom 8867  df-pnf 11140  df-mnf 11141  df-xr 11142  df-ltxr 11143  df-le 11144  df-sub 11338  df-neg 11339  df-nn 12118  df-2 12180  df-n0 12374  df-z 12461  df-uz 12725  df-seq 13901  df-exp 13961  df-itco 48670

This theorem is referenced by:  ackval3  48694