Theorem itcovalpc 48703

Description: The value of the function that returns the n-th iterate of the "plus a constant" function with regard to composition. (Contributed by AV, 4-May-2024.)

Hypothesis

Ref	Expression
itcovalpc.f	⊢ 𝐹 = (𝑛 ∈ ℕ0 ↦ (𝑛 + 𝐶))

Assertion

Ref	Expression
itcovalpc	⊢ ((𝐼 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) → ((IterComp‘𝐹)‘𝐼) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 𝐼))))

Distinct variable groups:   𝐶,𝑛   𝑛,𝐼

Allowed substitution hint:   𝐹(𝑛)

Proof of Theorem itcovalpc

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

Step	Hyp	Ref	Expression
1		fveq2 6822	. . . 4 ⊢ (𝑥 = 0 → ((IterComp‘𝐹)‘𝑥) = ((IterComp‘𝐹)‘0))
2		oveq2 7354	. . . . . 6 ⊢ (𝑥 = 0 → (𝐶 · 𝑥) = (𝐶 · 0))
3	2	oveq2d 7362	. . . . 5 ⊢ (𝑥 = 0 → (𝑛 + (𝐶 · 𝑥)) = (𝑛 + (𝐶 · 0)))
4	3	mpteq2dv 5185	. . . 4 ⊢ (𝑥 = 0 → (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 𝑥))) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 0))))
5	1, 4	eqeq12d 2747	. . 3 ⊢ (𝑥 = 0 → (((IterComp‘𝐹)‘𝑥) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 𝑥))) ↔ ((IterComp‘𝐹)‘0) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 0)))))
6		fveq2 6822	. . . 4 ⊢ (𝑥 = 𝑦 → ((IterComp‘𝐹)‘𝑥) = ((IterComp‘𝐹)‘𝑦))
7		oveq2 7354	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝐶 · 𝑥) = (𝐶 · 𝑦))
8	7	oveq2d 7362	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑛 + (𝐶 · 𝑥)) = (𝑛 + (𝐶 · 𝑦)))
9	8	mpteq2dv 5185	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 𝑥))) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 𝑦))))
10	6, 9	eqeq12d 2747	. . 3 ⊢ (𝑥 = 𝑦 → (((IterComp‘𝐹)‘𝑥) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 𝑥))) ↔ ((IterComp‘𝐹)‘𝑦) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 𝑦)))))
11		fveq2 6822	. . . 4 ⊢ (𝑥 = (𝑦 + 1) → ((IterComp‘𝐹)‘𝑥) = ((IterComp‘𝐹)‘(𝑦 + 1)))
12		oveq2 7354	. . . . . 6 ⊢ (𝑥 = (𝑦 + 1) → (𝐶 · 𝑥) = (𝐶 · (𝑦 + 1)))
13	12	oveq2d 7362	. . . . 5 ⊢ (𝑥 = (𝑦 + 1) → (𝑛 + (𝐶 · 𝑥)) = (𝑛 + (𝐶 · (𝑦 + 1))))
14	13	mpteq2dv 5185	. . . 4 ⊢ (𝑥 = (𝑦 + 1) → (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 𝑥))) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · (𝑦 + 1)))))
15	11, 14	eqeq12d 2747	. . 3 ⊢ (𝑥 = (𝑦 + 1) → (((IterComp‘𝐹)‘𝑥) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 𝑥))) ↔ ((IterComp‘𝐹)‘(𝑦 + 1)) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · (𝑦 + 1))))))
16		fveq2 6822	. . . 4 ⊢ (𝑥 = 𝐼 → ((IterComp‘𝐹)‘𝑥) = ((IterComp‘𝐹)‘𝐼))
17		oveq2 7354	. . . . . 6 ⊢ (𝑥 = 𝐼 → (𝐶 · 𝑥) = (𝐶 · 𝐼))
18	17	oveq2d 7362	. . . . 5 ⊢ (𝑥 = 𝐼 → (𝑛 + (𝐶 · 𝑥)) = (𝑛 + (𝐶 · 𝐼)))
19	18	mpteq2dv 5185	. . . 4 ⊢ (𝑥 = 𝐼 → (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 𝑥))) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 𝐼))))
20	16, 19	eqeq12d 2747	. . 3 ⊢ (𝑥 = 𝐼 → (((IterComp‘𝐹)‘𝑥) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 𝑥))) ↔ ((IterComp‘𝐹)‘𝐼) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 𝐼)))))
21		itcovalpc.f	. . . 4 ⊢ 𝐹 = (𝑛 ∈ ℕ0 ↦ (𝑛 + 𝐶))
22	21	itcovalpclem1 48701	. . 3 ⊢ (𝐶 ∈ ℕ0 → ((IterComp‘𝐹)‘0) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 0))))
23	21	itcovalpclem2 48702	. . . . 5 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) → (((IterComp‘𝐹)‘𝑦) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 𝑦))) → ((IterComp‘𝐹)‘(𝑦 + 1)) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · (𝑦 + 1))))))
24	23	ancoms 458	. . . 4 ⊢ ((𝐶 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) → (((IterComp‘𝐹)‘𝑦) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 𝑦))) → ((IterComp‘𝐹)‘(𝑦 + 1)) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · (𝑦 + 1))))))
25	24	imp 406	. . 3 ⊢ (((𝐶 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ ((IterComp‘𝐹)‘𝑦) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 𝑦)))) → ((IterComp‘𝐹)‘(𝑦 + 1)) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · (𝑦 + 1)))))
26	5, 10, 15, 20, 22, 25	nn0indd 12567	. 2 ⊢ ((𝐶 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0) → ((IterComp‘𝐹)‘𝐼) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 𝐼))))
27	26	ancoms 458	1 ⊢ ((𝐼 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) → ((IterComp‘𝐹)‘𝐼) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 𝐼))))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111   ↦ cmpt 5172  ‘cfv 6481  (class class class)co 7346  0cc0 11003  1c1 11004   + caddc 11006   · cmul 11008  ℕ₀cn0 12378  IterCompcitco 48688

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 11059  ax-resscn 11060  ax-1cn 11061  ax-icn 11062  ax-addcl 11063  ax-addrcl 11064  ax-mulcl 11065  ax-mulrcl 11066  ax-mulcom 11067  ax-addass 11068  ax-mulass 11069  ax-distr 11070  ax-i2m1 11071  ax-1ne0 11072  ax-1rid 11073  ax-rnegex 11074  ax-rrecex 11075  ax-cnre 11076  ax-pre-lttri 11077  ax-pre-lttrn 11078  ax-pre-ltadd 11079  ax-pre-mulgt0 11080

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 11145  df-mnf 11146  df-xr 11147  df-ltxr 11148  df-le 11149  df-sub 11343  df-neg 11344  df-nn 12123  df-n0 12379  df-z 12466  df-uz 12730  df-seq 13906  df-itco 48690

This theorem is referenced by:  ackval1  48712  ackval2  48713