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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 12660 | . . 3 ⊢ 1 = (0 + 1) | |
2 | 1 | fveq2i 6845 | . 2 ⊢ (Ack‘1) = (Ack‘(0 + 1)) |
3 | 0nn0 12428 | . . 3 ⊢ 0 ∈ ℕ0 | |
4 | ackvalsuc1mpt 46754 | . . 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 12453 | . . . . . . 7 ⊢ (𝑛 ∈ ℕ0 → (𝑛 + 1) ∈ ℕ0) | |
7 | 1nn0 12429 | . . . . . . 7 ⊢ 1 ∈ ℕ0 | |
8 | ackval0 46756 | . . . . . . . 8 ⊢ (Ack‘0) = (𝑖 ∈ ℕ0 ↦ (𝑖 + 1)) | |
9 | 8 | itcovalpc 46748 | . . . . . . 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 12423 | . . . . . . . . . 10 ⊢ ((𝑛 + 1) ∈ ℕ0 → (𝑛 + 1) ∈ ℂ) | |
12 | 6, 11 | syl 17 | . . . . . . . . 9 ⊢ (𝑛 ∈ ℕ0 → (𝑛 + 1) ∈ ℂ) |
13 | 12 | mulid2d 11173 | . . . . . . . 8 ⊢ (𝑛 ∈ ℕ0 → (1 · (𝑛 + 1)) = (𝑛 + 1)) |
14 | 13 | oveq2d 7373 | . . . . . . 7 ⊢ (𝑛 ∈ ℕ0 → (𝑖 + (1 · (𝑛 + 1))) = (𝑖 + (𝑛 + 1))) |
15 | 14 | mpteq2dv 5207 | . . . . . 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 6844 | . . . 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 7364 | . . . . . 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 7392 | . . . . 5 ⊢ (𝑛 ∈ ℕ0 → (1 + (𝑛 + 1)) ∈ V) | |
23 | 18, 20, 21, 22 | fvmptd 6955 | . . . 4 ⊢ (𝑛 ∈ ℕ0 → ((𝑖 ∈ ℕ0 ↦ (𝑖 + (𝑛 + 1)))‘1) = (1 + (𝑛 + 1))) |
24 | 1cnd 11150 | . . . . . 6 ⊢ (𝑛 ∈ ℕ0 → 1 ∈ ℂ) | |
25 | nn0cn 12423 | . . . . . . 7 ⊢ (𝑛 ∈ ℕ0 → 𝑛 ∈ ℂ) | |
26 | peano2cn 11327 | . . . . . . 7 ⊢ (𝑛 ∈ ℂ → (𝑛 + 1) ∈ ℂ) | |
27 | 25, 26 | syl 17 | . . . . . 6 ⊢ (𝑛 ∈ ℕ0 → (𝑛 + 1) ∈ ℂ) |
28 | 24, 27 | addcomd 11357 | . . . . 5 ⊢ (𝑛 ∈ ℕ0 → (1 + (𝑛 + 1)) = ((𝑛 + 1) + 1)) |
29 | 25, 24, 24 | addassd 11177 | . . . . 5 ⊢ (𝑛 ∈ ℕ0 → ((𝑛 + 1) + 1) = (𝑛 + (1 + 1))) |
30 | 1p1e2 12278 | . . . . . . 7 ⊢ (1 + 1) = 2 | |
31 | 30 | oveq2i 7368 | . . . . . 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 5208 | . 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 1541 ∈ wcel 2106 Vcvv 3445 ↦ cmpt 5188 ‘cfv 6496 (class class class)co 7357 ℂcc 11049 0cc0 11051 1c1 11052 + caddc 11054 · cmul 11056 2c2 12208 ℕ0cn0 12413 IterCompcitco 46733 Ackcack 46734 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-inf2 9577 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-om 7803 df-2nd 7922 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-er 8648 df-en 8884 df-dom 8885 df-sdom 8886 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-nn 12154 df-2 12216 df-n0 12414 df-z 12500 df-uz 12764 df-seq 13907 df-itco 46735 df-ack 46736 |
This theorem is referenced by: ackval2 46758 ackval1012 46766 |
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