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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ackval0012 | Structured version Visualization version GIF version |
Description: The Ackermann function at (0,0), (0,1), (0,2). (Contributed by AV, 2-May-2024.) |
Ref | Expression |
---|---|
ackval0012 | ⊢ 〈((Ack‘0)‘0), ((Ack‘0)‘1), ((Ack‘0)‘2)〉 = 〈1, 2, 3〉 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ackval0 46667 | . 2 ⊢ (Ack‘0) = (𝑛 ∈ ℕ0 ↦ (𝑛 + 1)) | |
2 | oveq1 7358 | . . . . 5 ⊢ (𝑛 = 0 → (𝑛 + 1) = (0 + 1)) | |
3 | 0p1e1 12233 | . . . . 5 ⊢ (0 + 1) = 1 | |
4 | 2, 3 | eqtrdi 2793 | . . . 4 ⊢ (𝑛 = 0 → (𝑛 + 1) = 1) |
5 | 0nn0 12386 | . . . . 5 ⊢ 0 ∈ ℕ0 | |
6 | 5 | a1i 11 | . . . 4 ⊢ ((Ack‘0) = (𝑛 ∈ ℕ0 ↦ (𝑛 + 1)) → 0 ∈ ℕ0) |
7 | 1nn0 12387 | . . . . 5 ⊢ 1 ∈ ℕ0 | |
8 | 7 | a1i 11 | . . . 4 ⊢ ((Ack‘0) = (𝑛 ∈ ℕ0 ↦ (𝑛 + 1)) → 1 ∈ ℕ0) |
9 | 1, 4, 6, 8 | fvmptd3 6968 | . . 3 ⊢ ((Ack‘0) = (𝑛 ∈ ℕ0 ↦ (𝑛 + 1)) → ((Ack‘0)‘0) = 1) |
10 | oveq1 7358 | . . . . 5 ⊢ (𝑛 = 1 → (𝑛 + 1) = (1 + 1)) | |
11 | 1p1e2 12236 | . . . . 5 ⊢ (1 + 1) = 2 | |
12 | 10, 11 | eqtrdi 2793 | . . . 4 ⊢ (𝑛 = 1 → (𝑛 + 1) = 2) |
13 | 2nn0 12388 | . . . . 5 ⊢ 2 ∈ ℕ0 | |
14 | 13 | a1i 11 | . . . 4 ⊢ ((Ack‘0) = (𝑛 ∈ ℕ0 ↦ (𝑛 + 1)) → 2 ∈ ℕ0) |
15 | 1, 12, 8, 14 | fvmptd3 6968 | . . 3 ⊢ ((Ack‘0) = (𝑛 ∈ ℕ0 ↦ (𝑛 + 1)) → ((Ack‘0)‘1) = 2) |
16 | oveq1 7358 | . . . . 5 ⊢ (𝑛 = 2 → (𝑛 + 1) = (2 + 1)) | |
17 | 2p1e3 12253 | . . . . 5 ⊢ (2 + 1) = 3 | |
18 | 16, 17 | eqtrdi 2793 | . . . 4 ⊢ (𝑛 = 2 → (𝑛 + 1) = 3) |
19 | 3nn0 12389 | . . . . 5 ⊢ 3 ∈ ℕ0 | |
20 | 19 | a1i 11 | . . . 4 ⊢ ((Ack‘0) = (𝑛 ∈ ℕ0 ↦ (𝑛 + 1)) → 3 ∈ ℕ0) |
21 | 1, 18, 14, 20 | fvmptd3 6968 | . . 3 ⊢ ((Ack‘0) = (𝑛 ∈ ℕ0 ↦ (𝑛 + 1)) → ((Ack‘0)‘2) = 3) |
22 | 9, 15, 21 | oteq123d 4843 | . 2 ⊢ ((Ack‘0) = (𝑛 ∈ ℕ0 ↦ (𝑛 + 1)) → 〈((Ack‘0)‘0), ((Ack‘0)‘1), ((Ack‘0)‘2)〉 = 〈1, 2, 3〉) |
23 | 1, 22 | ax-mp 5 | 1 ⊢ 〈((Ack‘0)‘0), ((Ack‘0)‘1), ((Ack‘0)‘2)〉 = 〈1, 2, 3〉 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 ∈ wcel 2106 〈cotp 4592 ↦ cmpt 5186 ‘cfv 6493 (class class class)co 7351 0cc0 11009 1c1 11010 + caddc 11012 2c2 12166 3c3 12167 ℕ0cn0 12371 Ackcack 46645 |
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 2708 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-ot 4593 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7307 df-ov 7354 df-oprab 7355 df-mpo 7356 df-om 7795 df-2nd 7914 df-frecs 8204 df-wrecs 8235 df-recs 8309 df-rdg 8348 df-er 8606 df-en 8842 df-dom 8843 df-sdom 8844 df-pnf 11149 df-mnf 11150 df-xr 11151 df-ltxr 11152 df-le 11153 df-sub 11345 df-neg 11346 df-nn 12112 df-2 12174 df-3 12175 df-n0 12372 df-z 12458 df-uz 12722 df-seq 13861 df-ack 46647 |
This theorem is referenced by: (None) |
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