Theorem itcovalt2 48839

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 6831	. . . . 5 ⊢ (𝑥 = 0 → ((IterComp‘𝐹)‘𝑥) = ((IterComp‘𝐹)‘0))
2		oveq2 7363	. . . . . . . 8 ⊢ (𝑥 = 0 → (2↑𝑥) = (2↑0))
3	2	oveq2d 7371	. . . . . . 7 ⊢ (𝑥 = 0 → ((𝑛 + 𝐶) · (2↑𝑥)) = ((𝑛 + 𝐶) · (2↑0)))
4	3	oveq1d 7370	. . . . . 6 ⊢ (𝑥 = 0 → (((𝑛 + 𝐶) · (2↑𝑥)) − 𝐶) = (((𝑛 + 𝐶) · (2↑0)) − 𝐶))
5	4	mpteq2dv 5189	. . . . 5 ⊢ (𝑥 = 0 → (𝑛 ∈ ℕ0 ↦ (((𝑛 + 𝐶) · (2↑𝑥)) − 𝐶)) = (𝑛 ∈ ℕ0 ↦ (((𝑛 + 𝐶) · (2↑0)) − 𝐶)))
6	1, 5	eqeq12d 2749	. . . 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 6831	. . . . 5 ⊢ (𝑥 = 𝑦 → ((IterComp‘𝐹)‘𝑥) = ((IterComp‘𝐹)‘𝑦))
9		oveq2 7363	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (2↑𝑥) = (2↑𝑦))
10	9	oveq2d 7371	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ((𝑛 + 𝐶) · (2↑𝑥)) = ((𝑛 + 𝐶) · (2↑𝑦)))
11	10	oveq1d 7370	. . . . . 6 ⊢ (𝑥 = 𝑦 → (((𝑛 + 𝐶) · (2↑𝑥)) − 𝐶) = (((𝑛 + 𝐶) · (2↑𝑦)) − 𝐶))
12	11	mpteq2dv 5189	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑛 ∈ ℕ0 ↦ (((𝑛 + 𝐶) · (2↑𝑥)) − 𝐶)) = (𝑛 ∈ ℕ0 ↦ (((𝑛 + 𝐶) · (2↑𝑦)) − 𝐶)))
13	8, 12	eqeq12d 2749	. . . 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 6831	. . . . 5 ⊢ (𝑥 = (𝑦 + 1) → ((IterComp‘𝐹)‘𝑥) = ((IterComp‘𝐹)‘(𝑦 + 1)))
16		oveq2 7363	. . . . . . . 8 ⊢ (𝑥 = (𝑦 + 1) → (2↑𝑥) = (2↑(𝑦 + 1)))
17	16	oveq2d 7371	. . . . . . 7 ⊢ (𝑥 = (𝑦 + 1) → ((𝑛 + 𝐶) · (2↑𝑥)) = ((𝑛 + 𝐶) · (2↑(𝑦 + 1))))
18	17	oveq1d 7370	. . . . . 6 ⊢ (𝑥 = (𝑦 + 1) → (((𝑛 + 𝐶) · (2↑𝑥)) − 𝐶) = (((𝑛 + 𝐶) · (2↑(𝑦 + 1))) − 𝐶))
19	18	mpteq2dv 5189	. . . . 5 ⊢ (𝑥 = (𝑦 + 1) → (𝑛 ∈ ℕ0 ↦ (((𝑛 + 𝐶) · (2↑𝑥)) − 𝐶)) = (𝑛 ∈ ℕ0 ↦ (((𝑛 + 𝐶) · (2↑(𝑦 + 1))) − 𝐶)))
20	15, 19	eqeq12d 2749	. . . 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 6831	. . . . 5 ⊢ (𝑥 = 𝐼 → ((IterComp‘𝐹)‘𝑥) = ((IterComp‘𝐹)‘𝐼))
23		oveq2 7363	. . . . . . . 8 ⊢ (𝑥 = 𝐼 → (2↑𝑥) = (2↑𝐼))
24	23	oveq2d 7371	. . . . . . 7 ⊢ (𝑥 = 𝐼 → ((𝑛 + 𝐶) · (2↑𝑥)) = ((𝑛 + 𝐶) · (2↑𝐼)))
25	24	oveq1d 7370	. . . . . 6 ⊢ (𝑥 = 𝐼 → (((𝑛 + 𝐶) · (2↑𝑥)) − 𝐶) = (((𝑛 + 𝐶) · (2↑𝐼)) − 𝐶))
26	25	mpteq2dv 5189	. . . . 5 ⊢ (𝑥 = 𝐼 → (𝑛 ∈ ℕ0 ↦ (((𝑛 + 𝐶) · (2↑𝑥)) − 𝐶)) = (𝑛 ∈ ℕ0 ↦ (((𝑛 + 𝐶) · (2↑𝐼)) − 𝐶)))
27	22, 26	eqeq12d 2749	. . . 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 48837	. . 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 48838	. . . . . 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 12578	. 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 2113   ↦ cmpt 5176  ‘cfv 6489  (class class class)co 7355  0cc0 11017  1c1 11018   + caddc 11020   · cmul 11022   − cmin 11355  2c2 12191  ℕ₀cn0 12392  ↑cexp 13975  IterCompcitco 48819

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 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7677  ax-inf2 9542  ax-cnex 11073  ax-resscn 11074  ax-1cn 11075  ax-icn 11076  ax-addcl 11077  ax-addrcl 11078  ax-mulcl 11079  ax-mulrcl 11080  ax-mulcom 11081  ax-addass 11082  ax-mulass 11083  ax-distr 11084  ax-i2m1 11085  ax-1ne0 11086  ax-1rid 11087  ax-rnegex 11088  ax-rrecex 11089  ax-cnre 11090  ax-pre-lttri 11091  ax-pre-lttrn 11092  ax-pre-ltadd 11093  ax-pre-mulgt0 11094

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 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-nel 3034  df-ral 3049  df-rex 3058  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-iun 4945  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6256  df-ord 6317  df-on 6318  df-lim 6319  df-suc 6320  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-riota 7312  df-ov 7358  df-oprab 7359  df-mpo 7360  df-om 7806  df-2nd 7931  df-frecs 8220  df-wrecs 8251  df-recs 8300  df-rdg 8338  df-er 8631  df-en 8880  df-dom 8881  df-sdom 8882  df-pnf 11159  df-mnf 11160  df-xr 11161  df-ltxr 11162  df-le 11163  df-sub 11357  df-neg 11358  df-nn 12137  df-2 12199  df-n0 12393  df-z 12480  df-uz 12743  df-seq 13916  df-exp 13976  df-itco 48821

This theorem is referenced by:  ackval3  48845