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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 45444 | . 2 ⊢ (Ack‘0) = (𝑛 ∈ ℕ0 ↦ (𝑛 + 1)) | |
2 | oveq1 7150 | . . . . 5 ⊢ (𝑛 = 0 → (𝑛 + 1) = (0 + 1)) | |
3 | 0p1e1 11781 | . . . . 5 ⊢ (0 + 1) = 1 | |
4 | 2, 3 | eqtrdi 2810 | . . . 4 ⊢ (𝑛 = 0 → (𝑛 + 1) = 1) |
5 | 0nn0 11934 | . . . . 5 ⊢ 0 ∈ ℕ0 | |
6 | 5 | a1i 11 | . . . 4 ⊢ ((Ack‘0) = (𝑛 ∈ ℕ0 ↦ (𝑛 + 1)) → 0 ∈ ℕ0) |
7 | 1nn0 11935 | . . . . 5 ⊢ 1 ∈ ℕ0 | |
8 | 7 | a1i 11 | . . . 4 ⊢ ((Ack‘0) = (𝑛 ∈ ℕ0 ↦ (𝑛 + 1)) → 1 ∈ ℕ0) |
9 | 1, 4, 6, 8 | fvmptd3 6775 | . . 3 ⊢ ((Ack‘0) = (𝑛 ∈ ℕ0 ↦ (𝑛 + 1)) → ((Ack‘0)‘0) = 1) |
10 | oveq1 7150 | . . . . 5 ⊢ (𝑛 = 1 → (𝑛 + 1) = (1 + 1)) | |
11 | 1p1e2 11784 | . . . . 5 ⊢ (1 + 1) = 2 | |
12 | 10, 11 | eqtrdi 2810 | . . . 4 ⊢ (𝑛 = 1 → (𝑛 + 1) = 2) |
13 | 2nn0 11936 | . . . . 5 ⊢ 2 ∈ ℕ0 | |
14 | 13 | a1i 11 | . . . 4 ⊢ ((Ack‘0) = (𝑛 ∈ ℕ0 ↦ (𝑛 + 1)) → 2 ∈ ℕ0) |
15 | 1, 12, 8, 14 | fvmptd3 6775 | . . 3 ⊢ ((Ack‘0) = (𝑛 ∈ ℕ0 ↦ (𝑛 + 1)) → ((Ack‘0)‘1) = 2) |
16 | oveq1 7150 | . . . . 5 ⊢ (𝑛 = 2 → (𝑛 + 1) = (2 + 1)) | |
17 | 2p1e3 11801 | . . . . 5 ⊢ (2 + 1) = 3 | |
18 | 16, 17 | eqtrdi 2810 | . . . 4 ⊢ (𝑛 = 2 → (𝑛 + 1) = 3) |
19 | 3nn0 11937 | . . . . 5 ⊢ 3 ∈ ℕ0 | |
20 | 19 | a1i 11 | . . . 4 ⊢ ((Ack‘0) = (𝑛 ∈ ℕ0 ↦ (𝑛 + 1)) → 3 ∈ ℕ0) |
21 | 1, 18, 14, 20 | fvmptd3 6775 | . . 3 ⊢ ((Ack‘0) = (𝑛 ∈ ℕ0 ↦ (𝑛 + 1)) → ((Ack‘0)‘2) = 3) |
22 | 9, 15, 21 | oteq123d 4771 | . 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 1539 ∈ wcel 2112 〈cotp 4523 ↦ cmpt 5105 ‘cfv 6328 (class class class)co 7143 0cc0 10560 1c1 10561 + caddc 10563 2c2 11714 3c3 11715 ℕ0cn0 11919 Ackcack 45422 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2159 ax-12 2176 ax-ext 2730 ax-rep 5149 ax-sep 5162 ax-nul 5169 ax-pow 5227 ax-pr 5291 ax-un 7452 ax-cnex 10616 ax-resscn 10617 ax-1cn 10618 ax-icn 10619 ax-addcl 10620 ax-addrcl 10621 ax-mulcl 10622 ax-mulrcl 10623 ax-mulcom 10624 ax-addass 10625 ax-mulass 10626 ax-distr 10627 ax-i2m1 10628 ax-1ne0 10629 ax-1rid 10630 ax-rnegex 10631 ax-rrecex 10632 ax-cnre 10633 ax-pre-lttri 10634 ax-pre-lttrn 10635 ax-pre-ltadd 10636 ax-pre-mulgt0 10637 |
This theorem depends on definitions: df-bi 210 df-an 401 df-or 846 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2071 df-mo 2558 df-eu 2589 df-clab 2737 df-cleq 2751 df-clel 2831 df-nfc 2899 df-ne 2950 df-nel 3054 df-ral 3073 df-rex 3074 df-reu 3075 df-rab 3077 df-v 3409 df-sbc 3694 df-csb 3802 df-dif 3857 df-un 3859 df-in 3861 df-ss 3871 df-pss 3873 df-nul 4222 df-if 4414 df-pw 4489 df-sn 4516 df-pr 4518 df-tp 4520 df-op 4522 df-ot 4524 df-uni 4792 df-iun 4878 df-br 5026 df-opab 5088 df-mpt 5106 df-tr 5132 df-id 5423 df-eprel 5428 df-po 5436 df-so 5437 df-fr 5476 df-we 5478 df-xp 5523 df-rel 5524 df-cnv 5525 df-co 5526 df-dm 5527 df-rn 5528 df-res 5529 df-ima 5530 df-pred 6119 df-ord 6165 df-on 6166 df-lim 6167 df-suc 6168 df-iota 6287 df-fun 6330 df-fn 6331 df-f 6332 df-f1 6333 df-fo 6334 df-f1o 6335 df-fv 6336 df-riota 7101 df-ov 7146 df-oprab 7147 df-mpo 7148 df-om 7573 df-2nd 7687 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-er 8292 df-en 8521 df-dom 8522 df-sdom 8523 df-pnf 10700 df-mnf 10701 df-xr 10702 df-ltxr 10703 df-le 10704 df-sub 10895 df-neg 10896 df-nn 11660 df-2 11722 df-3 11723 df-n0 11920 df-z 12006 df-uz 12268 df-seq 13404 df-ack 45424 |
This theorem is referenced by: (None) |
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