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Theorem ackval1 46367
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.
StepHypRef Expression
1 1e0p1 12572 . . 3 1 = (0 + 1)
21fveq2i 6822 . 2 (Ack‘1) = (Ack‘(0 + 1))
3 0nn0 12341 . . 3 0 ∈ ℕ0
4 ackvalsuc1mpt 46364 . . 3 (0 ∈ ℕ0 → (Ack‘(0 + 1)) = (𝑛 ∈ ℕ0 ↦ (((IterComp‘(Ack‘0))‘(𝑛 + 1))‘1)))
53, 4ax-mp 5 . 2 (Ack‘(0 + 1)) = (𝑛 ∈ ℕ0 ↦ (((IterComp‘(Ack‘0))‘(𝑛 + 1))‘1))
6 peano2nn0 12366 . . . . . . 7 (𝑛 ∈ ℕ0 → (𝑛 + 1) ∈ ℕ0)
7 1nn0 12342 . . . . . . 7 1 ∈ ℕ0
8 ackval0 46366 . . . . . . . 8 (Ack‘0) = (𝑖 ∈ ℕ0 ↦ (𝑖 + 1))
98itcovalpc 46358 . . . . . . 7 (((𝑛 + 1) ∈ ℕ0 ∧ 1 ∈ ℕ0) → ((IterComp‘(Ack‘0))‘(𝑛 + 1)) = (𝑖 ∈ ℕ0 ↦ (𝑖 + (1 · (𝑛 + 1)))))
106, 7, 9sylancl 586 . . . . . 6 (𝑛 ∈ ℕ0 → ((IterComp‘(Ack‘0))‘(𝑛 + 1)) = (𝑖 ∈ ℕ0 ↦ (𝑖 + (1 · (𝑛 + 1)))))
11 nn0cn 12336 . . . . . . . . . 10 ((𝑛 + 1) ∈ ℕ0 → (𝑛 + 1) ∈ ℂ)
126, 11syl 17 . . . . . . . . 9 (𝑛 ∈ ℕ0 → (𝑛 + 1) ∈ ℂ)
1312mulid2d 11086 . . . . . . . 8 (𝑛 ∈ ℕ0 → (1 · (𝑛 + 1)) = (𝑛 + 1))
1413oveq2d 7345 . . . . . . 7 (𝑛 ∈ ℕ0 → (𝑖 + (1 · (𝑛 + 1))) = (𝑖 + (𝑛 + 1)))
1514mpteq2dv 5191 . . . . . 6 (𝑛 ∈ ℕ0 → (𝑖 ∈ ℕ0 ↦ (𝑖 + (1 · (𝑛 + 1)))) = (𝑖 ∈ ℕ0 ↦ (𝑖 + (𝑛 + 1))))
1610, 15eqtrd 2776 . . . . 5 (𝑛 ∈ ℕ0 → ((IterComp‘(Ack‘0))‘(𝑛 + 1)) = (𝑖 ∈ ℕ0 ↦ (𝑖 + (𝑛 + 1))))
1716fveq1d 6821 . . . 4 (𝑛 ∈ ℕ0 → (((IterComp‘(Ack‘0))‘(𝑛 + 1))‘1) = ((𝑖 ∈ ℕ0 ↦ (𝑖 + (𝑛 + 1)))‘1))
18 eqidd 2737 . . . . 5 (𝑛 ∈ ℕ0 → (𝑖 ∈ ℕ0 ↦ (𝑖 + (𝑛 + 1))) = (𝑖 ∈ ℕ0 ↦ (𝑖 + (𝑛 + 1))))
19 oveq1 7336 . . . . . 6 (𝑖 = 1 → (𝑖 + (𝑛 + 1)) = (1 + (𝑛 + 1)))
2019adantl 482 . . . . 5 ((𝑛 ∈ ℕ0𝑖 = 1) → (𝑖 + (𝑛 + 1)) = (1 + (𝑛 + 1)))
217a1i 11 . . . . 5 (𝑛 ∈ ℕ0 → 1 ∈ ℕ0)
22 ovexd 7364 . . . . 5 (𝑛 ∈ ℕ0 → (1 + (𝑛 + 1)) ∈ V)
2318, 20, 21, 22fvmptd 6932 . . . 4 (𝑛 ∈ ℕ0 → ((𝑖 ∈ ℕ0 ↦ (𝑖 + (𝑛 + 1)))‘1) = (1 + (𝑛 + 1)))
24 1cnd 11063 . . . . . 6 (𝑛 ∈ ℕ0 → 1 ∈ ℂ)
25 nn0cn 12336 . . . . . . 7 (𝑛 ∈ ℕ0𝑛 ∈ ℂ)
26 peano2cn 11240 . . . . . . 7 (𝑛 ∈ ℂ → (𝑛 + 1) ∈ ℂ)
2725, 26syl 17 . . . . . 6 (𝑛 ∈ ℕ0 → (𝑛 + 1) ∈ ℂ)
2824, 27addcomd 11270 . . . . 5 (𝑛 ∈ ℕ0 → (1 + (𝑛 + 1)) = ((𝑛 + 1) + 1))
2925, 24, 24addassd 11090 . . . . 5 (𝑛 ∈ ℕ0 → ((𝑛 + 1) + 1) = (𝑛 + (1 + 1)))
30 1p1e2 12191 . . . . . . 7 (1 + 1) = 2
3130oveq2i 7340 . . . . . 6 (𝑛 + (1 + 1)) = (𝑛 + 2)
3231a1i 11 . . . . 5 (𝑛 ∈ ℕ0 → (𝑛 + (1 + 1)) = (𝑛 + 2))
3328, 29, 323eqtrd 2780 . . . 4 (𝑛 ∈ ℕ0 → (1 + (𝑛 + 1)) = (𝑛 + 2))
3417, 23, 333eqtrd 2780 . . 3 (𝑛 ∈ ℕ0 → (((IterComp‘(Ack‘0))‘(𝑛 + 1))‘1) = (𝑛 + 2))
3534mpteq2ia 5192 . 2 (𝑛 ∈ ℕ0 ↦ (((IterComp‘(Ack‘0))‘(𝑛 + 1))‘1)) = (𝑛 ∈ ℕ0 ↦ (𝑛 + 2))
362, 5, 353eqtri 2768 1 (Ack‘1) = (𝑛 ∈ ℕ0 ↦ (𝑛 + 2))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1540  wcel 2105  Vcvv 3441  cmpt 5172  cfv 6473  (class class class)co 7329  cc 10962  0cc0 10964  1c1 10965   + caddc 10967   · cmul 10969  2c2 12121  0cn0 12326  IterCompcitco 46343  Ackcack 46344
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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-rep 5226  ax-sep 5240  ax-nul 5247  ax-pow 5305  ax-pr 5369  ax-un 7642  ax-inf2 9490  ax-cnex 11020  ax-resscn 11021  ax-1cn 11022  ax-icn 11023  ax-addcl 11024  ax-addrcl 11025  ax-mulcl 11026  ax-mulrcl 11027  ax-mulcom 11028  ax-addass 11029  ax-mulass 11030  ax-distr 11031  ax-i2m1 11032  ax-1ne0 11033  ax-1rid 11034  ax-rnegex 11035  ax-rrecex 11036  ax-cnre 11037  ax-pre-lttri 11038  ax-pre-lttrn 11039  ax-pre-ltadd 11040  ax-pre-mulgt0 11041
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3350  df-rab 3404  df-v 3443  df-sbc 3727  df-csb 3843  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3916  df-nul 4269  df-if 4473  df-pw 4548  df-sn 4573  df-pr 4575  df-op 4579  df-uni 4852  df-iun 4940  df-br 5090  df-opab 5152  df-mpt 5173  df-tr 5207  df-id 5512  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5569  df-we 5571  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6232  df-ord 6299  df-on 6300  df-lim 6301  df-suc 6302  df-iota 6425  df-fun 6475  df-fn 6476  df-f 6477  df-f1 6478  df-fo 6479  df-f1o 6480  df-fv 6481  df-riota 7286  df-ov 7332  df-oprab 7333  df-mpo 7334  df-om 7773  df-2nd 7892  df-frecs 8159  df-wrecs 8190  df-recs 8264  df-rdg 8303  df-er 8561  df-en 8797  df-dom 8798  df-sdom 8799  df-pnf 11104  df-mnf 11105  df-xr 11106  df-ltxr 11107  df-le 11108  df-sub 11300  df-neg 11301  df-nn 12067  df-2 12129  df-n0 12327  df-z 12413  df-uz 12676  df-seq 13815  df-itco 46345  df-ack 46346
This theorem is referenced by:  ackval2  46368  ackval1012  46376
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