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Mathbox for Alexander van der Vekens |
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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 12284 | . . . 4 ⊢ 4 = (3 + 1) | |
2 | 1 | fveq2i 6894 | . . 3 ⊢ (Ack‘4) = (Ack‘(3 + 1)) |
3 | 2 | fveq1i 6892 | . 2 ⊢ ((Ack‘4)‘0) = ((Ack‘(3 + 1))‘0) |
4 | 3nn0 12497 | . . 3 ⊢ 3 ∈ ℕ0 | |
5 | ackvalsuc0val 47535 | . . 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 47540 | . . 3 ⊢ 〈((Ack‘3)‘0), ((Ack‘3)‘1), ((Ack‘3)‘2)〉 = 〈5, ;13, ;29〉 | |
8 | fvex 6904 | . . . . 5 ⊢ ((Ack‘3)‘0) ∈ V | |
9 | fvex 6904 | . . . . 5 ⊢ ((Ack‘3)‘1) ∈ V | |
10 | fvex 6904 | . . . . 5 ⊢ ((Ack‘3)‘2) ∈ V | |
11 | 8, 9, 10 | otth 5484 | . . . 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 2763 | 1 ⊢ ((Ack‘4)‘0) = ;13 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1540 ∈ wcel 2105 〈cotp 4636 ‘cfv 6543 (class class class)co 7412 0cc0 11116 1c1 11117 + caddc 11119 2c2 12274 3c3 12275 4c4 12276 5c5 12277 9c9 12281 ℕ0cn0 12479 ;cdc 12684 Ackcack 47506 |
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 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-inf2 9642 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 |
This theorem depends on definitions: df-bi 206 df-an 396 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 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-ot 4637 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-er 8709 df-en 8946 df-dom 8947 df-sdom 8948 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-nn 12220 df-2 12282 df-3 12283 df-4 12284 df-5 12285 df-6 12286 df-7 12287 df-8 12288 df-9 12289 df-n0 12480 df-z 12566 df-dec 12685 df-uz 12830 df-seq 13974 df-exp 14035 df-itco 47507 df-ack 47508 |
This theorem is referenced by: ackval41a 47542 |
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