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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ackval2012 | Structured version Visualization version GIF version |
Description: The Ackermann function at (2,0), (2,1), (2,2). (Contributed by AV, 4-May-2024.) |
Ref | Expression |
---|---|
ackval2012 | ⊢ 〈((Ack‘2)‘0), ((Ack‘2)‘1), ((Ack‘2)‘2)〉 = 〈3, 5, 7〉 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ackval2 48532 | . 2 ⊢ (Ack‘2) = (𝑛 ∈ ℕ0 ↦ ((2 · 𝑛) + 3)) | |
2 | oveq2 7439 | . . . . . 6 ⊢ (𝑛 = 0 → (2 · 𝑛) = (2 · 0)) | |
3 | 2 | oveq1d 7446 | . . . . 5 ⊢ (𝑛 = 0 → ((2 · 𝑛) + 3) = ((2 · 0) + 3)) |
4 | 2t0e0 12433 | . . . . . . 7 ⊢ (2 · 0) = 0 | |
5 | 4 | oveq1i 7441 | . . . . . 6 ⊢ ((2 · 0) + 3) = (0 + 3) |
6 | 3cn 12345 | . . . . . . 7 ⊢ 3 ∈ ℂ | |
7 | 6 | addlidi 11447 | . . . . . 6 ⊢ (0 + 3) = 3 |
8 | 5, 7 | eqtri 2763 | . . . . 5 ⊢ ((2 · 0) + 3) = 3 |
9 | 3, 8 | eqtrdi 2791 | . . . 4 ⊢ (𝑛 = 0 → ((2 · 𝑛) + 3) = 3) |
10 | 0nn0 12539 | . . . . 5 ⊢ 0 ∈ ℕ0 | |
11 | 10 | a1i 11 | . . . 4 ⊢ ((Ack‘2) = (𝑛 ∈ ℕ0 ↦ ((2 · 𝑛) + 3)) → 0 ∈ ℕ0) |
12 | 3nn0 12542 | . . . . 5 ⊢ 3 ∈ ℕ0 | |
13 | 12 | a1i 11 | . . . 4 ⊢ ((Ack‘2) = (𝑛 ∈ ℕ0 ↦ ((2 · 𝑛) + 3)) → 3 ∈ ℕ0) |
14 | 1, 9, 11, 13 | fvmptd3 7039 | . . 3 ⊢ ((Ack‘2) = (𝑛 ∈ ℕ0 ↦ ((2 · 𝑛) + 3)) → ((Ack‘2)‘0) = 3) |
15 | oveq2 7439 | . . . . . 6 ⊢ (𝑛 = 1 → (2 · 𝑛) = (2 · 1)) | |
16 | 15 | oveq1d 7446 | . . . . 5 ⊢ (𝑛 = 1 → ((2 · 𝑛) + 3) = ((2 · 1) + 3)) |
17 | 2t1e2 12427 | . . . . . . 7 ⊢ (2 · 1) = 2 | |
18 | 17 | oveq1i 7441 | . . . . . 6 ⊢ ((2 · 1) + 3) = (2 + 3) |
19 | 2cn 12339 | . . . . . . 7 ⊢ 2 ∈ ℂ | |
20 | 3p2e5 12415 | . . . . . . 7 ⊢ (3 + 2) = 5 | |
21 | 6, 19, 20 | addcomli 11451 | . . . . . 6 ⊢ (2 + 3) = 5 |
22 | 18, 21 | eqtri 2763 | . . . . 5 ⊢ ((2 · 1) + 3) = 5 |
23 | 16, 22 | eqtrdi 2791 | . . . 4 ⊢ (𝑛 = 1 → ((2 · 𝑛) + 3) = 5) |
24 | 1nn0 12540 | . . . . 5 ⊢ 1 ∈ ℕ0 | |
25 | 24 | a1i 11 | . . . 4 ⊢ ((Ack‘2) = (𝑛 ∈ ℕ0 ↦ ((2 · 𝑛) + 3)) → 1 ∈ ℕ0) |
26 | 5nn0 12544 | . . . . 5 ⊢ 5 ∈ ℕ0 | |
27 | 26 | a1i 11 | . . . 4 ⊢ ((Ack‘2) = (𝑛 ∈ ℕ0 ↦ ((2 · 𝑛) + 3)) → 5 ∈ ℕ0) |
28 | 1, 23, 25, 27 | fvmptd3 7039 | . . 3 ⊢ ((Ack‘2) = (𝑛 ∈ ℕ0 ↦ ((2 · 𝑛) + 3)) → ((Ack‘2)‘1) = 5) |
29 | oveq2 7439 | . . . . . 6 ⊢ (𝑛 = 2 → (2 · 𝑛) = (2 · 2)) | |
30 | 29 | oveq1d 7446 | . . . . 5 ⊢ (𝑛 = 2 → ((2 · 𝑛) + 3) = ((2 · 2) + 3)) |
31 | 2t2e4 12428 | . . . . . . 7 ⊢ (2 · 2) = 4 | |
32 | 31 | oveq1i 7441 | . . . . . 6 ⊢ ((2 · 2) + 3) = (4 + 3) |
33 | 4p3e7 12418 | . . . . . 6 ⊢ (4 + 3) = 7 | |
34 | 32, 33 | eqtri 2763 | . . . . 5 ⊢ ((2 · 2) + 3) = 7 |
35 | 30, 34 | eqtrdi 2791 | . . . 4 ⊢ (𝑛 = 2 → ((2 · 𝑛) + 3) = 7) |
36 | 2nn0 12541 | . . . . 5 ⊢ 2 ∈ ℕ0 | |
37 | 36 | a1i 11 | . . . 4 ⊢ ((Ack‘2) = (𝑛 ∈ ℕ0 ↦ ((2 · 𝑛) + 3)) → 2 ∈ ℕ0) |
38 | 7nn0 12546 | . . . . 5 ⊢ 7 ∈ ℕ0 | |
39 | 38 | a1i 11 | . . . 4 ⊢ ((Ack‘2) = (𝑛 ∈ ℕ0 ↦ ((2 · 𝑛) + 3)) → 7 ∈ ℕ0) |
40 | 1, 35, 37, 39 | fvmptd3 7039 | . . 3 ⊢ ((Ack‘2) = (𝑛 ∈ ℕ0 ↦ ((2 · 𝑛) + 3)) → ((Ack‘2)‘2) = 7) |
41 | 14, 28, 40 | oteq123d 4893 | . 2 ⊢ ((Ack‘2) = (𝑛 ∈ ℕ0 ↦ ((2 · 𝑛) + 3)) → 〈((Ack‘2)‘0), ((Ack‘2)‘1), ((Ack‘2)‘2)〉 = 〈3, 5, 7〉) |
42 | 1, 41 | ax-mp 5 | 1 ⊢ 〈((Ack‘2)‘0), ((Ack‘2)‘1), ((Ack‘2)‘2)〉 = 〈3, 5, 7〉 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2106 〈cotp 4639 ↦ cmpt 5231 ‘cfv 6563 (class class class)co 7431 0cc0 11153 1c1 11154 + caddc 11156 · cmul 11158 2c2 12319 3c3 12320 4c4 12321 5c5 12322 7c7 12324 ℕ0cn0 12524 Ackcack 48508 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-ot 4640 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-n0 12525 df-z 12612 df-uz 12877 df-seq 14040 df-itco 48509 df-ack 48510 |
This theorem is referenced by: (None) |
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