Mathbox for Alexander van der Vekens |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > ackval40 | Structured version Visualization version GIF version |
Description: The Ackermann function at (4,0). (Contributed by AV, 9-May-2024.) |
Ref | Expression |
---|---|
ackval40 | ⊢ ((Ack‘4)‘0) = ;13 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-4 12131 | . . . 4 ⊢ 4 = (3 + 1) | |
2 | 1 | fveq2i 6822 | . . 3 ⊢ (Ack‘4) = (Ack‘(3 + 1)) |
3 | 2 | fveq1i 6820 | . 2 ⊢ ((Ack‘4)‘0) = ((Ack‘(3 + 1))‘0) |
4 | 3nn0 12344 | . . 3 ⊢ 3 ∈ ℕ0 | |
5 | ackvalsuc0val 46373 | . . 3 ⊢ (3 ∈ ℕ0 → ((Ack‘(3 + 1))‘0) = ((Ack‘3)‘1)) | |
6 | 4, 5 | ax-mp 5 | . 2 ⊢ ((Ack‘(3 + 1))‘0) = ((Ack‘3)‘1) |
7 | ackval3012 46378 | . . 3 ⊢ ⟨((Ack‘3)‘0), ((Ack‘3)‘1), ((Ack‘3)‘2)⟩ = ⟨5, ;13, ;29⟩ | |
8 | fvex 6832 | . . . . 5 ⊢ ((Ack‘3)‘0) ∈ V | |
9 | fvex 6832 | . . . . 5 ⊢ ((Ack‘3)‘1) ∈ V | |
10 | fvex 6832 | . . . . 5 ⊢ ((Ack‘3)‘2) ∈ V | |
11 | 8, 9, 10 | otth 5423 | . . . 4 ⊢ (⟨((Ack‘3)‘0), ((Ack‘3)‘1), ((Ack‘3)‘2)⟩ = ⟨5, ;13, ;29⟩ ↔ (((Ack‘3)‘0) = 5 ∧ ((Ack‘3)‘1) = ;13 ∧ ((Ack‘3)‘2) = ;29)) |
12 | 11 | simp2bi 1145 | . . 3 ⊢ (⟨((Ack‘3)‘0), ((Ack‘3)‘1), ((Ack‘3)‘2)⟩ = ⟨5, ;13, ;29⟩ → ((Ack‘3)‘1) = ;13) |
13 | 7, 12 | ax-mp 5 | . 2 ⊢ ((Ack‘3)‘1) = ;13 |
14 | 3, 6, 13 | 3eqtri 2768 | 1 ⊢ ((Ack‘4)‘0) = ;13 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1540 ∈ wcel 2105 ⟨cotp 4580 ‘cfv 6473 (class class class)co 7329 0cc0 10964 1c1 10965 + caddc 10967 2c2 12121 3c3 12122 4c4 12123 5c5 12124 9c9 12128 ℕ0cn0 12326 ;cdc 12530 Ackcack 46344 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-rep 5226 ax-sep 5240 ax-nul 5247 ax-pow 5305 ax-pr 5369 ax-un 7642 ax-inf2 9490 ax-cnex 11020 ax-resscn 11021 ax-1cn 11022 ax-icn 11023 ax-addcl 11024 ax-addrcl 11025 ax-mulcl 11026 ax-mulrcl 11027 ax-mulcom 11028 ax-addass 11029 ax-mulass 11030 ax-distr 11031 ax-i2m1 11032 ax-1ne0 11033 ax-1rid 11034 ax-rnegex 11035 ax-rrecex 11036 ax-cnre 11037 ax-pre-lttri 11038 ax-pre-lttrn 11039 ax-pre-ltadd 11040 ax-pre-mulgt0 11041 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3350 df-rab 3404 df-v 3443 df-sbc 3727 df-csb 3843 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3916 df-nul 4269 df-if 4473 df-pw 4548 df-sn 4573 df-pr 4575 df-op 4579 df-ot 4581 df-uni 4852 df-iun 4940 df-br 5090 df-opab 5152 df-mpt 5173 df-tr 5207 df-id 5512 df-eprel 5518 df-po 5526 df-so 5527 df-fr 5569 df-we 5571 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6232 df-ord 6299 df-on 6300 df-lim 6301 df-suc 6302 df-iota 6425 df-fun 6475 df-fn 6476 df-f 6477 df-f1 6478 df-fo 6479 df-f1o 6480 df-fv 6481 df-riota 7286 df-ov 7332 df-oprab 7333 df-mpo 7334 df-om 7773 df-2nd 7892 df-frecs 8159 df-wrecs 8190 df-recs 8264 df-rdg 8303 df-er 8561 df-en 8797 df-dom 8798 df-sdom 8799 df-pnf 11104 df-mnf 11105 df-xr 11106 df-ltxr 11107 df-le 11108 df-sub 11300 df-neg 11301 df-nn 12067 df-2 12129 df-3 12130 df-4 12131 df-5 12132 df-6 12133 df-7 12134 df-8 12135 df-9 12136 df-n0 12327 df-z 12413 df-dec 12531 df-uz 12676 df-seq 13815 df-exp 13876 df-itco 46345 df-ack 46346 |
This theorem is referenced by: ackval41a 46380 |
Copyright terms: Public domain | W3C validator |