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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ackval41a | Structured version Visualization version GIF version |
Description: The Ackermann function at (4,1). (Contributed by AV, 9-May-2024.) |
Ref | Expression |
---|---|
ackval41a | ⊢ ((Ack‘4)‘1) = ((2↑;16) − 3) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-4 11724 | . . . 4 ⊢ 4 = (3 + 1) | |
2 | 1 | fveq2i 6654 | . . 3 ⊢ (Ack‘4) = (Ack‘(3 + 1)) |
3 | 1e0p1 12164 | . . 3 ⊢ 1 = (0 + 1) | |
4 | 2, 3 | fveq12i 6657 | . 2 ⊢ ((Ack‘4)‘1) = ((Ack‘(3 + 1))‘(0 + 1)) |
5 | 3nn0 11937 | . . . 4 ⊢ 3 ∈ ℕ0 | |
6 | 0nn0 11934 | . . . 4 ⊢ 0 ∈ ℕ0 | |
7 | ackvalsucsucval 45452 | . . . 4 ⊢ ((3 ∈ ℕ0 ∧ 0 ∈ ℕ0) → ((Ack‘(3 + 1))‘(0 + 1)) = ((Ack‘3)‘((Ack‘(3 + 1))‘0))) | |
8 | 5, 6, 7 | mp2an 692 | . . 3 ⊢ ((Ack‘(3 + 1))‘(0 + 1)) = ((Ack‘3)‘((Ack‘(3 + 1))‘0)) |
9 | 3p1e4 11804 | . . . . . . . 8 ⊢ (3 + 1) = 4 | |
10 | 9 | fveq2i 6654 | . . . . . . 7 ⊢ (Ack‘(3 + 1)) = (Ack‘4) |
11 | 10 | fveq1i 6652 | . . . . . 6 ⊢ ((Ack‘(3 + 1))‘0) = ((Ack‘4)‘0) |
12 | ackval40 45457 | . . . . . 6 ⊢ ((Ack‘4)‘0) = ;13 | |
13 | 11, 12 | eqtri 2782 | . . . . 5 ⊢ ((Ack‘(3 + 1))‘0) = ;13 |
14 | 13 | fveq2i 6654 | . . . 4 ⊢ ((Ack‘3)‘((Ack‘(3 + 1))‘0)) = ((Ack‘3)‘;13) |
15 | 1nn0 11935 | . . . . . 6 ⊢ 1 ∈ ℕ0 | |
16 | 15, 5 | deccl 12137 | . . . . 5 ⊢ ;13 ∈ ℕ0 |
17 | oveq1 7150 | . . . . . . . . 9 ⊢ (𝑛 = ;13 → (𝑛 + 3) = (;13 + 3)) | |
18 | 17 | oveq2d 7159 | . . . . . . . 8 ⊢ (𝑛 = ;13 → (2↑(𝑛 + 3)) = (2↑(;13 + 3))) |
19 | 18 | oveq1d 7158 | . . . . . . 7 ⊢ (𝑛 = ;13 → ((2↑(𝑛 + 3)) − 3) = ((2↑(;13 + 3)) − 3)) |
20 | eqid 2759 | . . . . . . . . . 10 ⊢ ;13 = ;13 | |
21 | 3p3e6 11811 | . . . . . . . . . 10 ⊢ (3 + 3) = 6 | |
22 | 15, 5, 5, 20, 21 | decaddi 12182 | . . . . . . . . 9 ⊢ (;13 + 3) = ;16 |
23 | 22 | oveq2i 7154 | . . . . . . . 8 ⊢ (2↑(;13 + 3)) = (2↑;16) |
24 | 23 | oveq1i 7153 | . . . . . . 7 ⊢ ((2↑(;13 + 3)) − 3) = ((2↑;16) − 3) |
25 | 19, 24 | eqtrdi 2810 | . . . . . 6 ⊢ (𝑛 = ;13 → ((2↑(𝑛 + 3)) − 3) = ((2↑;16) − 3)) |
26 | ackval3 45447 | . . . . . 6 ⊢ (Ack‘3) = (𝑛 ∈ ℕ0 ↦ ((2↑(𝑛 + 3)) − 3)) | |
27 | ovex 7176 | . . . . . 6 ⊢ ((2↑;16) − 3) ∈ V | |
28 | 25, 26, 27 | fvmpt 6752 | . . . . 5 ⊢ (;13 ∈ ℕ0 → ((Ack‘3)‘;13) = ((2↑;16) − 3)) |
29 | 16, 28 | ax-mp 5 | . . . 4 ⊢ ((Ack‘3)‘;13) = ((2↑;16) − 3) |
30 | 14, 29 | eqtri 2782 | . . 3 ⊢ ((Ack‘3)‘((Ack‘(3 + 1))‘0)) = ((2↑;16) − 3) |
31 | 8, 30 | eqtri 2782 | . 2 ⊢ ((Ack‘(3 + 1))‘(0 + 1)) = ((2↑;16) − 3) |
32 | 4, 31 | eqtri 2782 | 1 ⊢ ((Ack‘4)‘1) = ((2↑;16) − 3) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1539 ∈ wcel 2112 ‘cfv 6328 (class class class)co 7143 0cc0 10560 1c1 10561 + caddc 10563 − cmin 10893 2c2 11714 3c3 11715 4c4 11716 6c6 11718 ℕ0cn0 11919 ;cdc 12122 ↑cexp 13464 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-inf2 9122 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-4 11724 df-5 11725 df-6 11726 df-7 11727 df-8 11728 df-9 11729 df-n0 11920 df-z 12006 df-dec 12123 df-uz 12268 df-seq 13404 df-exp 13465 df-itco 45423 df-ack 45424 |
This theorem is referenced by: ackval41 45459 ackval42 45460 |
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