Theorem itcovalpc 45645

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 6706	. . . 4 ⊢ (𝑥 = 0 → ((IterComp‘𝐹)‘𝑥) = ((IterComp‘𝐹)‘0))
2		oveq2 7210	. . . . . 6 ⊢ (𝑥 = 0 → (𝐶 · 𝑥) = (𝐶 · 0))
3	2	oveq2d 7218	. . . . 5 ⊢ (𝑥 = 0 → (𝑛 + (𝐶 · 𝑥)) = (𝑛 + (𝐶 · 0)))
4	3	mpteq2dv 5140	. . . 4 ⊢ (𝑥 = 0 → (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 𝑥))) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 0))))
5	1, 4	eqeq12d 2750	. . 3 ⊢ (𝑥 = 0 → (((IterComp‘𝐹)‘𝑥) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 𝑥))) ↔ ((IterComp‘𝐹)‘0) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 0)))))
6		fveq2 6706	. . . 4 ⊢ (𝑥 = 𝑦 → ((IterComp‘𝐹)‘𝑥) = ((IterComp‘𝐹)‘𝑦))
7		oveq2 7210	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝐶 · 𝑥) = (𝐶 · 𝑦))
8	7	oveq2d 7218	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑛 + (𝐶 · 𝑥)) = (𝑛 + (𝐶 · 𝑦)))
9	8	mpteq2dv 5140	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 𝑥))) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 𝑦))))
10	6, 9	eqeq12d 2750	. . 3 ⊢ (𝑥 = 𝑦 → (((IterComp‘𝐹)‘𝑥) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 𝑥))) ↔ ((IterComp‘𝐹)‘𝑦) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 𝑦)))))
11		fveq2 6706	. . . 4 ⊢ (𝑥 = (𝑦 + 1) → ((IterComp‘𝐹)‘𝑥) = ((IterComp‘𝐹)‘(𝑦 + 1)))
12		oveq2 7210	. . . . . 6 ⊢ (𝑥 = (𝑦 + 1) → (𝐶 · 𝑥) = (𝐶 · (𝑦 + 1)))
13	12	oveq2d 7218	. . . . 5 ⊢ (𝑥 = (𝑦 + 1) → (𝑛 + (𝐶 · 𝑥)) = (𝑛 + (𝐶 · (𝑦 + 1))))
14	13	mpteq2dv 5140	. . . 4 ⊢ (𝑥 = (𝑦 + 1) → (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 𝑥))) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · (𝑦 + 1)))))
15	11, 14	eqeq12d 2750	. . 3 ⊢ (𝑥 = (𝑦 + 1) → (((IterComp‘𝐹)‘𝑥) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 𝑥))) ↔ ((IterComp‘𝐹)‘(𝑦 + 1)) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · (𝑦 + 1))))))
16		fveq2 6706	. . . 4 ⊢ (𝑥 = 𝐼 → ((IterComp‘𝐹)‘𝑥) = ((IterComp‘𝐹)‘𝐼))
17		oveq2 7210	. . . . . 6 ⊢ (𝑥 = 𝐼 → (𝐶 · 𝑥) = (𝐶 · 𝐼))
18	17	oveq2d 7218	. . . . 5 ⊢ (𝑥 = 𝐼 → (𝑛 + (𝐶 · 𝑥)) = (𝑛 + (𝐶 · 𝐼)))
19	18	mpteq2dv 5140	. . . 4 ⊢ (𝑥 = 𝐼 → (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 𝑥))) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 𝐼))))
20	16, 19	eqeq12d 2750	. . 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 45643	. . 3 ⊢ (𝐶 ∈ ℕ0 → ((IterComp‘𝐹)‘0) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 0))))
23	21	itcovalpclem2 45644	. . . . 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 462	. . . 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 410	. . 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 12257	. 2 ⊢ ((𝐶 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0) → ((IterComp‘𝐹)‘𝐼) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 𝐼))))
27	26	ancoms 462	1 ⊢ ((𝐼 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) → ((IterComp‘𝐹)‘𝐼) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 𝐼))))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1543   ∈ wcel 2110   ↦ cmpt 5124  ‘cfv 6369  (class class class)co 7202  0cc0 10712  1c1 10713   + caddc 10715   · cmul 10717  ℕ₀cn0 12073  IterCompcitco 45630

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2706  ax-rep 5168  ax-sep 5181  ax-nul 5188  ax-pow 5247  ax-pr 5311  ax-un 7512  ax-inf2 9245  ax-cnex 10768  ax-resscn 10769  ax-1cn 10770  ax-icn 10771  ax-addcl 10772  ax-addrcl 10773  ax-mulcl 10774  ax-mulrcl 10775  ax-mulcom 10776  ax-addass 10777  ax-mulass 10778  ax-distr 10779  ax-i2m1 10780  ax-1ne0 10781  ax-1rid 10782  ax-rnegex 10783  ax-rrecex 10784  ax-cnre 10785  ax-pre-lttri 10786  ax-pre-lttrn 10787  ax-pre-ltadd 10788  ax-pre-mulgt0 10789

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2537  df-eu 2566  df-clab 2713  df-cleq 2726  df-clel 2812  df-nfc 2882  df-ne 2936  df-nel 3040  df-ral 3059  df-rex 3060  df-reu 3061  df-rab 3063  df-v 3403  df-sbc 3688  df-csb 3803  df-dif 3860  df-un 3862  df-in 3864  df-ss 3874  df-pss 3876  df-nul 4228  df-if 4430  df-pw 4505  df-sn 4532  df-pr 4534  df-tp 4536  df-op 4538  df-uni 4810  df-iun 4896  df-br 5044  df-opab 5106  df-mpt 5125  df-tr 5151  df-id 5444  df-eprel 5449  df-po 5457  df-so 5458  df-fr 5498  df-we 5500  df-xp 5546  df-rel 5547  df-cnv 5548  df-co 5549  df-dm 5550  df-rn 5551  df-res 5552  df-ima 5553  df-pred 6149  df-ord 6205  df-on 6206  df-lim 6207  df-suc 6208  df-iota 6327  df-fun 6371  df-fn 6372  df-f 6373  df-f1 6374  df-fo 6375  df-f1o 6376  df-fv 6377  df-riota 7159  df-ov 7205  df-oprab 7206  df-mpo 7207  df-om 7634  df-2nd 7751  df-wrecs 8036  df-recs 8097  df-rdg 8135  df-er 8380  df-en 8616  df-dom 8617  df-sdom 8618  df-pnf 10852  df-mnf 10853  df-xr 10854  df-ltxr 10855  df-le 10856  df-sub 11047  df-neg 11048  df-nn 11814  df-n0 12074  df-z 12160  df-uz 12422  df-seq 13558  df-itco 45632

This theorem is referenced by:  ackval1  45654  ackval2  45655