Theorem itcovalpc 49397

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 6881	. . . 4 ⊢ (𝑥 = 0 → ((IterComp‘𝐹)‘𝑥) = ((IterComp‘𝐹)‘0))
2		oveq2 7418	. . . . . 6 ⊢ (𝑥 = 0 → (𝐶 · 𝑥) = (𝐶 · 0))
3	2	oveq2d 7426	. . . . 5 ⊢ (𝑥 = 0 → (𝑛 + (𝐶 · 𝑥)) = (𝑛 + (𝐶 · 0)))
4	3	mpteq2dv 5204	. . . 4 ⊢ (𝑥 = 0 → (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 𝑥))) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 0))))
5	1, 4	eqeq12d 2777	. . 3 ⊢ (𝑥 = 0 → (((IterComp‘𝐹)‘𝑥) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 𝑥))) ↔ ((IterComp‘𝐹)‘0) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 0)))))
6		fveq2 6881	. . . 4 ⊢ (𝑥 = 𝑦 → ((IterComp‘𝐹)‘𝑥) = ((IterComp‘𝐹)‘𝑦))
7		oveq2 7418	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝐶 · 𝑥) = (𝐶 · 𝑦))
8	7	oveq2d 7426	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑛 + (𝐶 · 𝑥)) = (𝑛 + (𝐶 · 𝑦)))
9	8	mpteq2dv 5204	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 𝑥))) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 𝑦))))
10	6, 9	eqeq12d 2777	. . 3 ⊢ (𝑥 = 𝑦 → (((IterComp‘𝐹)‘𝑥) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 𝑥))) ↔ ((IterComp‘𝐹)‘𝑦) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 𝑦)))))
11		fveq2 6881	. . . 4 ⊢ (𝑥 = (𝑦 + 1) → ((IterComp‘𝐹)‘𝑥) = ((IterComp‘𝐹)‘(𝑦 + 1)))
12		oveq2 7418	. . . . . 6 ⊢ (𝑥 = (𝑦 + 1) → (𝐶 · 𝑥) = (𝐶 · (𝑦 + 1)))
13	12	oveq2d 7426	. . . . 5 ⊢ (𝑥 = (𝑦 + 1) → (𝑛 + (𝐶 · 𝑥)) = (𝑛 + (𝐶 · (𝑦 + 1))))
14	13	mpteq2dv 5204	. . . 4 ⊢ (𝑥 = (𝑦 + 1) → (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 𝑥))) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · (𝑦 + 1)))))
15	11, 14	eqeq12d 2777	. . 3 ⊢ (𝑥 = (𝑦 + 1) → (((IterComp‘𝐹)‘𝑥) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 𝑥))) ↔ ((IterComp‘𝐹)‘(𝑦 + 1)) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · (𝑦 + 1))))))
16		fveq2 6881	. . . 4 ⊢ (𝑥 = 𝐼 → ((IterComp‘𝐹)‘𝑥) = ((IterComp‘𝐹)‘𝐼))
17		oveq2 7418	. . . . . 6 ⊢ (𝑥 = 𝐼 → (𝐶 · 𝑥) = (𝐶 · 𝐼))
18	17	oveq2d 7426	. . . . 5 ⊢ (𝑥 = 𝐼 → (𝑛 + (𝐶 · 𝑥)) = (𝑛 + (𝐶 · 𝐼)))
19	18	mpteq2dv 5204	. . . 4 ⊢ (𝑥 = 𝐼 → (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 𝑥))) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 𝐼))))
20	16, 19	eqeq12d 2777	. . 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 49395	. . 3 ⊢ (𝐶 ∈ ℕ0 → ((IterComp‘𝐹)‘0) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 0))))
23	21	itcovalpclem2 49396	. . . . 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 463	. . . 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 411	. . 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 12692	. 2 ⊢ ((𝐶 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0) → ((IterComp‘𝐹)‘𝐼) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 𝐼))))
27	26	ancoms 463	1 ⊢ ((𝐼 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) → ((IterComp‘𝐹)‘𝐼) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 𝐼))))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1568   ∈ wcel 2141   ↦ cmpt 5191  ‘cfv 6536  (class class class)co 7410  0cc0 11099  1c1 11100   + caddc 11102   · cmul 11104  ℕ₀cn0 12503  IterCompcitco 49382

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-inf2 9609  ax-cnex 11155  ax-resscn 11156  ax-1cn 11157  ax-icn 11158  ax-addcl 11159  ax-addrcl 11160  ax-mulcl 11161  ax-mulrcl 11162  ax-mulcom 11163  ax-addass 11164  ax-mulass 11165  ax-distr 11166  ax-i2m1 11167  ax-1ne0 11168  ax-1rid 11169  ax-rnegex 11170  ax-rrecex 11171  ax-cnre 11172  ax-pre-lttri 11173  ax-pre-lttrn 11174  ax-pre-ltadd 11175  ax-pre-mulgt0 11176

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7862  df-2nd 7986  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-er 8693  df-en 8943  df-dom 8944  df-sdom 8945  df-pnf 11244  df-mnf 11245  df-xr 11246  df-ltxr 11247  df-le 11248  df-sub 11442  df-neg 11443  df-nn 12233  df-n0 12504  df-z 12591  df-uz 12862  df-seq 14037  df-itco 49384

This theorem is referenced by:  ackval1  49406  ackval2  49407