| Mathbox for Alexander van der Vekens |
< Previous
Next >
Nearby theorems |
||
| 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 12331 | . . . 4 ⊢ 4 = (3 + 1) | |
| 2 | 1 | fveq2i 6909 | . . 3 ⊢ (Ack‘4) = (Ack‘(3 + 1)) |
| 3 | 1e0p1 12775 | . . 3 ⊢ 1 = (0 + 1) | |
| 4 | 2, 3 | fveq12i 6912 | . 2 ⊢ ((Ack‘4)‘1) = ((Ack‘(3 + 1))‘(0 + 1)) |
| 5 | 3nn0 12544 | . . . 4 ⊢ 3 ∈ ℕ0 | |
| 6 | 0nn0 12541 | . . . 4 ⊢ 0 ∈ ℕ0 | |
| 7 | ackvalsucsucval 48609 | . . . 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 12411 | . . . . . . . 8 ⊢ (3 + 1) = 4 | |
| 10 | 9 | fveq2i 6909 | . . . . . . 7 ⊢ (Ack‘(3 + 1)) = (Ack‘4) |
| 11 | 10 | fveq1i 6907 | . . . . . 6 ⊢ ((Ack‘(3 + 1))‘0) = ((Ack‘4)‘0) |
| 12 | ackval40 48614 | . . . . . 6 ⊢ ((Ack‘4)‘0) = ;13 | |
| 13 | 11, 12 | eqtri 2765 | . . . . 5 ⊢ ((Ack‘(3 + 1))‘0) = ;13 |
| 14 | 13 | fveq2i 6909 | . . . 4 ⊢ ((Ack‘3)‘((Ack‘(3 + 1))‘0)) = ((Ack‘3)‘;13) |
| 15 | 1nn0 12542 | . . . . . 6 ⊢ 1 ∈ ℕ0 | |
| 16 | 15, 5 | deccl 12748 | . . . . 5 ⊢ ;13 ∈ ℕ0 |
| 17 | oveq1 7438 | . . . . . . . . 9 ⊢ (𝑛 = ;13 → (𝑛 + 3) = (;13 + 3)) | |
| 18 | 17 | oveq2d 7447 | . . . . . . . 8 ⊢ (𝑛 = ;13 → (2↑(𝑛 + 3)) = (2↑(;13 + 3))) |
| 19 | 18 | oveq1d 7446 | . . . . . . 7 ⊢ (𝑛 = ;13 → ((2↑(𝑛 + 3)) − 3) = ((2↑(;13 + 3)) − 3)) |
| 20 | eqid 2737 | . . . . . . . . . 10 ⊢ ;13 = ;13 | |
| 21 | 3p3e6 12418 | . . . . . . . . . 10 ⊢ (3 + 3) = 6 | |
| 22 | 15, 5, 5, 20, 21 | decaddi 12793 | . . . . . . . . 9 ⊢ (;13 + 3) = ;16 |
| 23 | 22 | oveq2i 7442 | . . . . . . . 8 ⊢ (2↑(;13 + 3)) = (2↑;16) |
| 24 | 23 | oveq1i 7441 | . . . . . . 7 ⊢ ((2↑(;13 + 3)) − 3) = ((2↑;16) − 3) |
| 25 | 19, 24 | eqtrdi 2793 | . . . . . 6 ⊢ (𝑛 = ;13 → ((2↑(𝑛 + 3)) − 3) = ((2↑;16) − 3)) |
| 26 | ackval3 48604 | . . . . . 6 ⊢ (Ack‘3) = (𝑛 ∈ ℕ0 ↦ ((2↑(𝑛 + 3)) − 3)) | |
| 27 | ovex 7464 | . . . . . 6 ⊢ ((2↑;16) − 3) ∈ V | |
| 28 | 25, 26, 27 | fvmpt 7016 | . . . . 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 2765 | . . 3 ⊢ ((Ack‘3)‘((Ack‘(3 + 1))‘0)) = ((2↑;16) − 3) |
| 31 | 8, 30 | eqtri 2765 | . 2 ⊢ ((Ack‘(3 + 1))‘(0 + 1)) = ((2↑;16) − 3) |
| 32 | 4, 31 | eqtri 2765 | 1 ⊢ ((Ack‘4)‘1) = ((2↑;16) − 3) |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 ∈ wcel 2108 ‘cfv 6561 (class class class)co 7431 0cc0 11155 1c1 11156 + caddc 11158 − cmin 11492 2c2 12321 3c3 12322 4c4 12323 6c6 12325 ℕ0cn0 12526 ;cdc 12733 ↑cexp 14102 Ackcack 48579 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-ot 4635 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-seq 14043 df-exp 14103 df-itco 48580 df-ack 48581 |
| This theorem is referenced by: ackval41 48616 ackval42 48617 |
| Copyright terms: Public domain | W3C validator |