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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ackval1 | Structured version Visualization version GIF version |
Description: The Ackermann function at 1. (Contributed by AV, 4-May-2024.) |
Ref | Expression |
---|---|
ackval1 | ⊢ (Ack‘1) = (𝑛 ∈ ℕ0 ↦ (𝑛 + 2)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1e0p1 12572 | . . 3 ⊢ 1 = (0 + 1) | |
2 | 1 | fveq2i 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))) | |
5 | 3, 4 | ax-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)) | |
9 | 8 | itcovalpc 46358 | . . . . . . 7 ⊢ (((𝑛 + 1) ∈ ℕ0 ∧ 1 ∈ ℕ0) → ((IterComp‘(Ack‘0))‘(𝑛 + 1)) = (𝑖 ∈ ℕ0 ↦ (𝑖 + (1 · (𝑛 + 1))))) |
10 | 6, 7, 9 | sylancl 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) | |
12 | 6, 11 | syl 17 | . . . . . . . . 9 ⊢ (𝑛 ∈ ℕ0 → (𝑛 + 1) ∈ ℂ) |
13 | 12 | mulid2d 11086 | . . . . . . . 8 ⊢ (𝑛 ∈ ℕ0 → (1 · (𝑛 + 1)) = (𝑛 + 1)) |
14 | 13 | oveq2d 7345 | . . . . . . 7 ⊢ (𝑛 ∈ ℕ0 → (𝑖 + (1 · (𝑛 + 1))) = (𝑖 + (𝑛 + 1))) |
15 | 14 | mpteq2dv 5191 | . . . . . 6 ⊢ (𝑛 ∈ ℕ0 → (𝑖 ∈ ℕ0 ↦ (𝑖 + (1 · (𝑛 + 1)))) = (𝑖 ∈ ℕ0 ↦ (𝑖 + (𝑛 + 1)))) |
16 | 10, 15 | eqtrd 2776 | . . . . 5 ⊢ (𝑛 ∈ ℕ0 → ((IterComp‘(Ack‘0))‘(𝑛 + 1)) = (𝑖 ∈ ℕ0 ↦ (𝑖 + (𝑛 + 1)))) |
17 | 16 | fveq1d 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))) | |
20 | 19 | adantl 482 | . . . . 5 ⊢ ((𝑛 ∈ ℕ0 ∧ 𝑖 = 1) → (𝑖 + (𝑛 + 1)) = (1 + (𝑛 + 1))) |
21 | 7 | a1i 11 | . . . . 5 ⊢ (𝑛 ∈ ℕ0 → 1 ∈ ℕ0) |
22 | ovexd 7364 | . . . . 5 ⊢ (𝑛 ∈ ℕ0 → (1 + (𝑛 + 1)) ∈ V) | |
23 | 18, 20, 21, 22 | fvmptd 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 ∈ ℕ0 → 𝑛 ∈ ℂ) | |
26 | peano2cn 11240 | . . . . . . 7 ⊢ (𝑛 ∈ ℂ → (𝑛 + 1) ∈ ℂ) | |
27 | 25, 26 | syl 17 | . . . . . 6 ⊢ (𝑛 ∈ ℕ0 → (𝑛 + 1) ∈ ℂ) |
28 | 24, 27 | addcomd 11270 | . . . . 5 ⊢ (𝑛 ∈ ℕ0 → (1 + (𝑛 + 1)) = ((𝑛 + 1) + 1)) |
29 | 25, 24, 24 | addassd 11090 | . . . . 5 ⊢ (𝑛 ∈ ℕ0 → ((𝑛 + 1) + 1) = (𝑛 + (1 + 1))) |
30 | 1p1e2 12191 | . . . . . . 7 ⊢ (1 + 1) = 2 | |
31 | 30 | oveq2i 7340 | . . . . . 6 ⊢ (𝑛 + (1 + 1)) = (𝑛 + 2) |
32 | 31 | a1i 11 | . . . . 5 ⊢ (𝑛 ∈ ℕ0 → (𝑛 + (1 + 1)) = (𝑛 + 2)) |
33 | 28, 29, 32 | 3eqtrd 2780 | . . . 4 ⊢ (𝑛 ∈ ℕ0 → (1 + (𝑛 + 1)) = (𝑛 + 2)) |
34 | 17, 23, 33 | 3eqtrd 2780 | . . 3 ⊢ (𝑛 ∈ ℕ0 → (((IterComp‘(Ack‘0))‘(𝑛 + 1))‘1) = (𝑛 + 2)) |
35 | 34 | mpteq2ia 5192 | . 2 ⊢ (𝑛 ∈ ℕ0 ↦ (((IterComp‘(Ack‘0))‘(𝑛 + 1))‘1)) = (𝑛 ∈ ℕ0 ↦ (𝑛 + 2)) |
36 | 2, 5, 35 | 3eqtri 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 ℂcc 10962 0cc0 10964 1c1 10965 + caddc 10967 · cmul 10969 2c2 12121 ℕ0cn0 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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