Theorem itcovalpc 48661

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 6858	. . . 4 ⊢ (𝑥 = 0 → ((IterComp‘𝐹)‘𝑥) = ((IterComp‘𝐹)‘0))
2		oveq2 7395	. . . . . 6 ⊢ (𝑥 = 0 → (𝐶 · 𝑥) = (𝐶 · 0))
3	2	oveq2d 7403	. . . . 5 ⊢ (𝑥 = 0 → (𝑛 + (𝐶 · 𝑥)) = (𝑛 + (𝐶 · 0)))
4	3	mpteq2dv 5201	. . . 4 ⊢ (𝑥 = 0 → (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 𝑥))) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 0))))
5	1, 4	eqeq12d 2745	. . 3 ⊢ (𝑥 = 0 → (((IterComp‘𝐹)‘𝑥) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 𝑥))) ↔ ((IterComp‘𝐹)‘0) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 0)))))
6		fveq2 6858	. . . 4 ⊢ (𝑥 = 𝑦 → ((IterComp‘𝐹)‘𝑥) = ((IterComp‘𝐹)‘𝑦))
7		oveq2 7395	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝐶 · 𝑥) = (𝐶 · 𝑦))
8	7	oveq2d 7403	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑛 + (𝐶 · 𝑥)) = (𝑛 + (𝐶 · 𝑦)))
9	8	mpteq2dv 5201	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 𝑥))) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 𝑦))))
10	6, 9	eqeq12d 2745	. . 3 ⊢ (𝑥 = 𝑦 → (((IterComp‘𝐹)‘𝑥) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 𝑥))) ↔ ((IterComp‘𝐹)‘𝑦) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 𝑦)))))
11		fveq2 6858	. . . 4 ⊢ (𝑥 = (𝑦 + 1) → ((IterComp‘𝐹)‘𝑥) = ((IterComp‘𝐹)‘(𝑦 + 1)))
12		oveq2 7395	. . . . . 6 ⊢ (𝑥 = (𝑦 + 1) → (𝐶 · 𝑥) = (𝐶 · (𝑦 + 1)))
13	12	oveq2d 7403	. . . . 5 ⊢ (𝑥 = (𝑦 + 1) → (𝑛 + (𝐶 · 𝑥)) = (𝑛 + (𝐶 · (𝑦 + 1))))
14	13	mpteq2dv 5201	. . . 4 ⊢ (𝑥 = (𝑦 + 1) → (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 𝑥))) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · (𝑦 + 1)))))
15	11, 14	eqeq12d 2745	. . 3 ⊢ (𝑥 = (𝑦 + 1) → (((IterComp‘𝐹)‘𝑥) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 𝑥))) ↔ ((IterComp‘𝐹)‘(𝑦 + 1)) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · (𝑦 + 1))))))
16		fveq2 6858	. . . 4 ⊢ (𝑥 = 𝐼 → ((IterComp‘𝐹)‘𝑥) = ((IterComp‘𝐹)‘𝐼))
17		oveq2 7395	. . . . . 6 ⊢ (𝑥 = 𝐼 → (𝐶 · 𝑥) = (𝐶 · 𝐼))
18	17	oveq2d 7403	. . . . 5 ⊢ (𝑥 = 𝐼 → (𝑛 + (𝐶 · 𝑥)) = (𝑛 + (𝐶 · 𝐼)))
19	18	mpteq2dv 5201	. . . 4 ⊢ (𝑥 = 𝐼 → (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 𝑥))) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 𝐼))))
20	16, 19	eqeq12d 2745	. . 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 48659	. . 3 ⊢ (𝐶 ∈ ℕ0 → ((IterComp‘𝐹)‘0) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 0))))
23	21	itcovalpclem2 48660	. . . . 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 12631	. 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 1540   ∈ wcel 2109   ↦ cmpt 5188  ‘cfv 6511  (class class class)co 7387  0cc0 11068  1c1 11069   + caddc 11071   · cmul 11073  ℕ₀cn0 12442  IterCompcitco 48646

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-inf2 9594  ax-cnex 11124  ax-resscn 11125  ax-1cn 11126  ax-icn 11127  ax-addcl 11128  ax-addrcl 11129  ax-mulcl 11130  ax-mulrcl 11131  ax-mulcom 11132  ax-addass 11133  ax-mulass 11134  ax-distr 11135  ax-i2m1 11136  ax-1ne0 11137  ax-1rid 11138  ax-rnegex 11139  ax-rrecex 11140  ax-cnre 11141  ax-pre-lttri 11142  ax-pre-lttrn 11143  ax-pre-ltadd 11144  ax-pre-mulgt0 11145

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-er 8671  df-en 8919  df-dom 8920  df-sdom 8921  df-pnf 11210  df-mnf 11211  df-xr 11212  df-ltxr 11213  df-le 11214  df-sub 11407  df-neg 11408  df-nn 12187  df-n0 12443  df-z 12530  df-uz 12794  df-seq 13967  df-itco 48648

This theorem is referenced by:  ackval1  48670  ackval2  48671