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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 12223 | . . . 4 ⊢ 4 = (3 + 1) | |
2 | 1 | fveq2i 6846 | . . 3 ⊢ (Ack‘4) = (Ack‘(3 + 1)) |
3 | 1e0p1 12665 | . . 3 ⊢ 1 = (0 + 1) | |
4 | 2, 3 | fveq12i 6849 | . 2 ⊢ ((Ack‘4)‘1) = ((Ack‘(3 + 1))‘(0 + 1)) |
5 | 3nn0 12436 | . . . 4 ⊢ 3 ∈ ℕ0 | |
6 | 0nn0 12433 | . . . 4 ⊢ 0 ∈ ℕ0 | |
7 | ackvalsucsucval 46860 | . . . 4 ⊢ ((3 ∈ ℕ0 ∧ 0 ∈ ℕ0) → ((Ack‘(3 + 1))‘(0 + 1)) = ((Ack‘3)‘((Ack‘(3 + 1))‘0))) | |
8 | 5, 6, 7 | mp2an 691 | . . 3 ⊢ ((Ack‘(3 + 1))‘(0 + 1)) = ((Ack‘3)‘((Ack‘(3 + 1))‘0)) |
9 | 3p1e4 12303 | . . . . . . . 8 ⊢ (3 + 1) = 4 | |
10 | 9 | fveq2i 6846 | . . . . . . 7 ⊢ (Ack‘(3 + 1)) = (Ack‘4) |
11 | 10 | fveq1i 6844 | . . . . . 6 ⊢ ((Ack‘(3 + 1))‘0) = ((Ack‘4)‘0) |
12 | ackval40 46865 | . . . . . 6 ⊢ ((Ack‘4)‘0) = ;13 | |
13 | 11, 12 | eqtri 2761 | . . . . 5 ⊢ ((Ack‘(3 + 1))‘0) = ;13 |
14 | 13 | fveq2i 6846 | . . . 4 ⊢ ((Ack‘3)‘((Ack‘(3 + 1))‘0)) = ((Ack‘3)‘;13) |
15 | 1nn0 12434 | . . . . . 6 ⊢ 1 ∈ ℕ0 | |
16 | 15, 5 | deccl 12638 | . . . . 5 ⊢ ;13 ∈ ℕ0 |
17 | oveq1 7365 | . . . . . . . . 9 ⊢ (𝑛 = ;13 → (𝑛 + 3) = (;13 + 3)) | |
18 | 17 | oveq2d 7374 | . . . . . . . 8 ⊢ (𝑛 = ;13 → (2↑(𝑛 + 3)) = (2↑(;13 + 3))) |
19 | 18 | oveq1d 7373 | . . . . . . 7 ⊢ (𝑛 = ;13 → ((2↑(𝑛 + 3)) − 3) = ((2↑(;13 + 3)) − 3)) |
20 | eqid 2733 | . . . . . . . . . 10 ⊢ ;13 = ;13 | |
21 | 3p3e6 12310 | . . . . . . . . . 10 ⊢ (3 + 3) = 6 | |
22 | 15, 5, 5, 20, 21 | decaddi 12683 | . . . . . . . . 9 ⊢ (;13 + 3) = ;16 |
23 | 22 | oveq2i 7369 | . . . . . . . 8 ⊢ (2↑(;13 + 3)) = (2↑;16) |
24 | 23 | oveq1i 7368 | . . . . . . 7 ⊢ ((2↑(;13 + 3)) − 3) = ((2↑;16) − 3) |
25 | 19, 24 | eqtrdi 2789 | . . . . . 6 ⊢ (𝑛 = ;13 → ((2↑(𝑛 + 3)) − 3) = ((2↑;16) − 3)) |
26 | ackval3 46855 | . . . . . 6 ⊢ (Ack‘3) = (𝑛 ∈ ℕ0 ↦ ((2↑(𝑛 + 3)) − 3)) | |
27 | ovex 7391 | . . . . . 6 ⊢ ((2↑;16) − 3) ∈ V | |
28 | 25, 26, 27 | fvmpt 6949 | . . . . 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 2761 | . . 3 ⊢ ((Ack‘3)‘((Ack‘(3 + 1))‘0)) = ((2↑;16) − 3) |
31 | 8, 30 | eqtri 2761 | . 2 ⊢ ((Ack‘(3 + 1))‘(0 + 1)) = ((2↑;16) − 3) |
32 | 4, 31 | eqtri 2761 | 1 ⊢ ((Ack‘4)‘1) = ((2↑;16) − 3) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∈ wcel 2107 ‘cfv 6497 (class class class)co 7358 0cc0 11056 1c1 11057 + caddc 11059 − cmin 11390 2c2 12213 3c3 12214 4c4 12215 6c6 12217 ℕ0cn0 12418 ;cdc 12623 ↑cexp 13973 Ackcack 46830 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-inf2 9582 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-ot 4596 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-er 8651 df-en 8887 df-dom 8888 df-sdom 8889 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-nn 12159 df-2 12221 df-3 12222 df-4 12223 df-5 12224 df-6 12225 df-7 12226 df-8 12227 df-9 12228 df-n0 12419 df-z 12505 df-dec 12624 df-uz 12769 df-seq 13913 df-exp 13974 df-itco 46831 df-ack 46832 |
This theorem is referenced by: ackval41 46867 ackval42 46868 |
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