Theorem ackval1 48415

Description: The Ackermann function at 1. (Contributed by AV, 4-May-2024.)

Assertion

Ref	Expression
ackval1	⊢ (Ack‘1) = (𝑛 ∈ ℕ0 ↦ (𝑛 + 2))

Proof of Theorem ackval1

Dummy variable 𝑖 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		1e0p1 12800	. . 3 ⊢ 1 = (0 + 1)
2	1	fveq2i 6923	. 2 ⊢ (Ack‘1) = (Ack‘(0 + 1))
3		0nn0 12568	. . 3 ⊢ 0 ∈ ℕ0
4		ackvalsuc1mpt 48412	. . 3 ⊢ (0 ∈ ℕ0 → (Ack‘(0 + 1)) = (𝑛 ∈ ℕ0 ↦ (((IterComp‘(Ack‘0))‘(𝑛 + 1))‘1)))
5	3, 4	ax-mp 5	. 2 ⊢ (Ack‘(0 + 1)) = (𝑛 ∈ ℕ0 ↦ (((IterComp‘(Ack‘0))‘(𝑛 + 1))‘1))
6		peano2nn0 12593	. . . . . . 7 ⊢ (𝑛 ∈ ℕ0 → (𝑛 + 1) ∈ ℕ0)
7		1nn0 12569	. . . . . . 7 ⊢ 1 ∈ ℕ0
8		ackval0 48414	. . . . . . . 8 ⊢ (Ack‘0) = (𝑖 ∈ ℕ0 ↦ (𝑖 + 1))
9	8	itcovalpc 48406	. . . . . . 7 ⊢ (((𝑛 + 1) ∈ ℕ0 ∧ 1 ∈ ℕ0) → ((IterComp‘(Ack‘0))‘(𝑛 + 1)) = (𝑖 ∈ ℕ0 ↦ (𝑖 + (1 · (𝑛 + 1)))))
10	6, 7, 9	sylancl 585	. . . . . 6 ⊢ (𝑛 ∈ ℕ0 → ((IterComp‘(Ack‘0))‘(𝑛 + 1)) = (𝑖 ∈ ℕ0 ↦ (𝑖 + (1 · (𝑛 + 1)))))
11		nn0cn 12563	. . . . . . . . . 10 ⊢ ((𝑛 + 1) ∈ ℕ0 → (𝑛 + 1) ∈ ℂ)
12	6, 11	syl 17	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ0 → (𝑛 + 1) ∈ ℂ)
13	12	mullidd 11308	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ0 → (1 · (𝑛 + 1)) = (𝑛 + 1))
14	13	oveq2d 7464	. . . . . . 7 ⊢ (𝑛 ∈ ℕ0 → (𝑖 + (1 · (𝑛 + 1))) = (𝑖 + (𝑛 + 1)))
15	14	mpteq2dv 5268	. . . . . 6 ⊢ (𝑛 ∈ ℕ0 → (𝑖 ∈ ℕ0 ↦ (𝑖 + (1 · (𝑛 + 1)))) = (𝑖 ∈ ℕ0 ↦ (𝑖 + (𝑛 + 1))))
16	10, 15	eqtrd 2780	. . . . 5 ⊢ (𝑛 ∈ ℕ0 → ((IterComp‘(Ack‘0))‘(𝑛 + 1)) = (𝑖 ∈ ℕ0 ↦ (𝑖 + (𝑛 + 1))))
17	16	fveq1d 6922	. . . 4 ⊢ (𝑛 ∈ ℕ0 → (((IterComp‘(Ack‘0))‘(𝑛 + 1))‘1) = ((𝑖 ∈ ℕ0 ↦ (𝑖 + (𝑛 + 1)))‘1))
18		eqidd 2741	. . . . 5 ⊢ (𝑛 ∈ ℕ0 → (𝑖 ∈ ℕ0 ↦ (𝑖 + (𝑛 + 1))) = (𝑖 ∈ ℕ0 ↦ (𝑖 + (𝑛 + 1))))
19		oveq1 7455	. . . . . 6 ⊢ (𝑖 = 1 → (𝑖 + (𝑛 + 1)) = (1 + (𝑛 + 1)))
20	19	adantl 481	. . . . 5 ⊢ ((𝑛 ∈ ℕ0 ∧ 𝑖 = 1) → (𝑖 + (𝑛 + 1)) = (1 + (𝑛 + 1)))
21	7	a1i 11	. . . . 5 ⊢ (𝑛 ∈ ℕ0 → 1 ∈ ℕ0)
22		ovexd 7483	. . . . 5 ⊢ (𝑛 ∈ ℕ0 → (1 + (𝑛 + 1)) ∈ V)
23	18, 20, 21, 22	fvmptd 7036	. . . 4 ⊢ (𝑛 ∈ ℕ0 → ((𝑖 ∈ ℕ0 ↦ (𝑖 + (𝑛 + 1)))‘1) = (1 + (𝑛 + 1)))
24		1cnd 11285	. . . . . 6 ⊢ (𝑛 ∈ ℕ0 → 1 ∈ ℂ)
25		nn0cn 12563	. . . . . . 7 ⊢ (𝑛 ∈ ℕ0 → 𝑛 ∈ ℂ)
26		peano2cn 11462	. . . . . . 7 ⊢ (𝑛 ∈ ℂ → (𝑛 + 1) ∈ ℂ)
27	25, 26	syl 17	. . . . . 6 ⊢ (𝑛 ∈ ℕ0 → (𝑛 + 1) ∈ ℂ)
28	24, 27	addcomd 11492	. . . . 5 ⊢ (𝑛 ∈ ℕ0 → (1 + (𝑛 + 1)) = ((𝑛 + 1) + 1))
29	25, 24, 24	addassd 11312	. . . . 5 ⊢ (𝑛 ∈ ℕ0 → ((𝑛 + 1) + 1) = (𝑛 + (1 + 1)))
30		1p1e2 12418	. . . . . . 7 ⊢ (1 + 1) = 2
31	30	oveq2i 7459	. . . . . 6 ⊢ (𝑛 + (1 + 1)) = (𝑛 + 2)
32	31	a1i 11	. . . . 5 ⊢ (𝑛 ∈ ℕ0 → (𝑛 + (1 + 1)) = (𝑛 + 2))
33	28, 29, 32	3eqtrd 2784	. . . 4 ⊢ (𝑛 ∈ ℕ0 → (1 + (𝑛 + 1)) = (𝑛 + 2))
34	17, 23, 33	3eqtrd 2784	. . 3 ⊢ (𝑛 ∈ ℕ0 → (((IterComp‘(Ack‘0))‘(𝑛 + 1))‘1) = (𝑛 + 2))
35	34	mpteq2ia 5269	. 2 ⊢ (𝑛 ∈ ℕ0 ↦ (((IterComp‘(Ack‘0))‘(𝑛 + 1))‘1)) = (𝑛 ∈ ℕ0 ↦ (𝑛 + 2))
36	2, 5, 35	3eqtri 2772	1 ⊢ (Ack‘1) = (𝑛 ∈ ℕ0 ↦ (𝑛 + 2))

Syntax hints:   = wceq 1537   ∈ wcel 2108  Vcvv 3488   ↦ cmpt 5249  ‘cfv 6573  (class class class)co 7448  ℂcc 11182  0cc0 11184  1c1 11185   + caddc 11187   · cmul 11189  2c2 12348  ℕ₀cn0 12553  IterCompcitco 48391  Ackcack 48392

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-inf2 9710  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-er 8763  df-en 9004  df-dom 9005  df-sdom 9006  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-nn 12294  df-2 12356  df-n0 12554  df-z 12640  df-uz 12904  df-seq 14053  df-itco 48393  df-ack 48394

This theorem is referenced by:  ackval2  48416  ackval1012  48424