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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ackval1012 | Structured version Visualization version GIF version |
Description: The Ackermann function at (1,0), (1,1), (1,2). (Contributed by AV, 4-May-2024.) |
Ref | Expression |
---|---|
ackval1012 | ⊢ 〈((Ack‘1)‘0), ((Ack‘1)‘1), ((Ack‘1)‘2)〉 = 〈2, 3, 4〉 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ackval1 48334 | . 2 ⊢ (Ack‘1) = (𝑛 ∈ ℕ0 ↦ (𝑛 + 2)) | |
2 | oveq1 7452 | . . . . 5 ⊢ (𝑛 = 0 → (𝑛 + 2) = (0 + 2)) | |
3 | 2cn 12364 | . . . . . 6 ⊢ 2 ∈ ℂ | |
4 | 3 | addlidi 11474 | . . . . 5 ⊢ (0 + 2) = 2 |
5 | 2, 4 | eqtrdi 2790 | . . . 4 ⊢ (𝑛 = 0 → (𝑛 + 2) = 2) |
6 | 0nn0 12564 | . . . . 5 ⊢ 0 ∈ ℕ0 | |
7 | 6 | a1i 11 | . . . 4 ⊢ ((Ack‘1) = (𝑛 ∈ ℕ0 ↦ (𝑛 + 2)) → 0 ∈ ℕ0) |
8 | 2nn0 12566 | . . . . 5 ⊢ 2 ∈ ℕ0 | |
9 | 8 | a1i 11 | . . . 4 ⊢ ((Ack‘1) = (𝑛 ∈ ℕ0 ↦ (𝑛 + 2)) → 2 ∈ ℕ0) |
10 | 1, 5, 7, 9 | fvmptd3 7050 | . . 3 ⊢ ((Ack‘1) = (𝑛 ∈ ℕ0 ↦ (𝑛 + 2)) → ((Ack‘1)‘0) = 2) |
11 | oveq1 7452 | . . . . 5 ⊢ (𝑛 = 1 → (𝑛 + 2) = (1 + 2)) | |
12 | 1p2e3 12432 | . . . . 5 ⊢ (1 + 2) = 3 | |
13 | 11, 12 | eqtrdi 2790 | . . . 4 ⊢ (𝑛 = 1 → (𝑛 + 2) = 3) |
14 | 1nn0 12565 | . . . . 5 ⊢ 1 ∈ ℕ0 | |
15 | 14 | a1i 11 | . . . 4 ⊢ ((Ack‘1) = (𝑛 ∈ ℕ0 ↦ (𝑛 + 2)) → 1 ∈ ℕ0) |
16 | 3nn0 12567 | . . . . 5 ⊢ 3 ∈ ℕ0 | |
17 | 16 | a1i 11 | . . . 4 ⊢ ((Ack‘1) = (𝑛 ∈ ℕ0 ↦ (𝑛 + 2)) → 3 ∈ ℕ0) |
18 | 1, 13, 15, 17 | fvmptd3 7050 | . . 3 ⊢ ((Ack‘1) = (𝑛 ∈ ℕ0 ↦ (𝑛 + 2)) → ((Ack‘1)‘1) = 3) |
19 | oveq1 7452 | . . . . 5 ⊢ (𝑛 = 2 → (𝑛 + 2) = (2 + 2)) | |
20 | 2p2e4 12424 | . . . . 5 ⊢ (2 + 2) = 4 | |
21 | 19, 20 | eqtrdi 2790 | . . . 4 ⊢ (𝑛 = 2 → (𝑛 + 2) = 4) |
22 | 4nn0 12568 | . . . . 5 ⊢ 4 ∈ ℕ0 | |
23 | 22 | a1i 11 | . . . 4 ⊢ ((Ack‘1) = (𝑛 ∈ ℕ0 ↦ (𝑛 + 2)) → 4 ∈ ℕ0) |
24 | 1, 21, 9, 23 | fvmptd3 7050 | . . 3 ⊢ ((Ack‘1) = (𝑛 ∈ ℕ0 ↦ (𝑛 + 2)) → ((Ack‘1)‘2) = 4) |
25 | 10, 18, 24 | oteq123d 4912 | . 2 ⊢ ((Ack‘1) = (𝑛 ∈ ℕ0 ↦ (𝑛 + 2)) → 〈((Ack‘1)‘0), ((Ack‘1)‘1), ((Ack‘1)‘2)〉 = 〈2, 3, 4〉) |
26 | 1, 25 | ax-mp 5 | 1 ⊢ 〈((Ack‘1)‘0), ((Ack‘1)‘1), ((Ack‘1)‘2)〉 = 〈2, 3, 4〉 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2103 〈cotp 4656 ↦ cmpt 5252 ‘cfv 6572 (class class class)co 7445 0cc0 11180 1c1 11181 + caddc 11183 2c2 12344 3c3 12345 4c4 12346 ℕ0cn0 12549 Ackcack 48311 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2105 ax-9 2113 ax-10 2136 ax-11 2153 ax-12 2173 ax-ext 2705 ax-rep 5306 ax-sep 5320 ax-nul 5327 ax-pow 5386 ax-pr 5450 ax-un 7766 ax-inf2 9706 ax-cnex 11236 ax-resscn 11237 ax-1cn 11238 ax-icn 11239 ax-addcl 11240 ax-addrcl 11241 ax-mulcl 11242 ax-mulrcl 11243 ax-mulcom 11244 ax-addass 11245 ax-mulass 11246 ax-distr 11247 ax-i2m1 11248 ax-1ne0 11249 ax-1rid 11250 ax-rnegex 11251 ax-rrecex 11252 ax-cnre 11253 ax-pre-lttri 11254 ax-pre-lttrn 11255 ax-pre-ltadd 11256 ax-pre-mulgt0 11257 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2890 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-reu 3384 df-rab 3439 df-v 3484 df-sbc 3799 df-csb 3916 df-dif 3973 df-un 3975 df-in 3977 df-ss 3987 df-pss 3990 df-nul 4348 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-ot 4657 df-uni 4932 df-iun 5021 df-br 5170 df-opab 5232 df-mpt 5253 df-tr 5287 df-id 5597 df-eprel 5603 df-po 5611 df-so 5612 df-fr 5654 df-we 5656 df-xp 5705 df-rel 5706 df-cnv 5707 df-co 5708 df-dm 5709 df-rn 5710 df-res 5711 df-ima 5712 df-pred 6331 df-ord 6397 df-on 6398 df-lim 6399 df-suc 6400 df-iota 6524 df-fun 6574 df-fn 6575 df-f 6576 df-f1 6577 df-fo 6578 df-f1o 6579 df-fv 6580 df-riota 7401 df-ov 7448 df-oprab 7449 df-mpo 7450 df-om 7900 df-2nd 8027 df-frecs 8318 df-wrecs 8349 df-recs 8423 df-rdg 8462 df-er 8759 df-en 9000 df-dom 9001 df-sdom 9002 df-pnf 11322 df-mnf 11323 df-xr 11324 df-ltxr 11325 df-le 11326 df-sub 11518 df-neg 11519 df-nn 12290 df-2 12352 df-3 12353 df-4 12354 df-n0 12550 df-z 12636 df-uz 12900 df-seq 14049 df-itco 48312 df-ack 48313 |
This theorem is referenced by: (None) |
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