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| Mirrors > Home > MPE Home > Th. List > Mathboxes > ackvalsuc0val | Structured version Visualization version GIF version | ||
| Description: The Ackermann function at a successor (of the first argument). This is the second equation of PΓ©ter's definition of the Ackermann function. (Contributed by AV, 4-May-2024.) |
| Ref | Expression |
|---|---|
| ackvalsuc0val | β’ (π β β0 β ((Ackβ(π + 1))β0) = ((Ackβπ)β1)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | 0nn0 12487 | . . 3 β’ 0 β β0 | |
| 2 | ackvalsuc1 47365 | . . 3 β’ ((π β β0 β§ 0 β β0) β ((Ackβ(π + 1))β0) = (((IterCompβ(Ackβπ))β(0 + 1))β1)) | |
| 3 | 1, 2 | mpan2 690 | . 2 β’ (π β β0 β ((Ackβ(π + 1))β0) = (((IterCompβ(Ackβπ))β(0 + 1))β1)) |
| 4 | 0p1e1 12334 | . . . . . 6 β’ (0 + 1) = 1 | |
| 5 | 4 | a1i 11 | . . . . 5 β’ (π β β0 β (0 + 1) = 1) |
| 6 | 5 | fveq2d 6896 | . . . 4 β’ (π β β0 β ((IterCompβ(Ackβπ))β(0 + 1)) = ((IterCompβ(Ackβπ))β1)) |
| 7 | ackfnnn0 47371 | . . . . . 6 β’ (π β β0 β (Ackβπ) Fn β0) | |
| 8 | fnfun 6650 | . . . . . 6 β’ ((Ackβπ) Fn β0 β Fun (Ackβπ)) | |
| 9 | funrel 6566 | . . . . . 6 β’ (Fun (Ackβπ) β Rel (Ackβπ)) | |
| 10 | 7, 8, 9 | 3syl 18 | . . . . 5 β’ (π β β0 β Rel (Ackβπ)) |
| 11 | fvex 6905 | . . . . 5 β’ (Ackβπ) β V | |
| 12 | itcoval1 47349 | . . . . 5 β’ ((Rel (Ackβπ) β§ (Ackβπ) β V) β ((IterCompβ(Ackβπ))β1) = (Ackβπ)) | |
| 13 | 10, 11, 12 | sylancl 587 | . . . 4 β’ (π β β0 β ((IterCompβ(Ackβπ))β1) = (Ackβπ)) |
| 14 | 6, 13 | eqtrd 2773 | . . 3 β’ (π β β0 β ((IterCompβ(Ackβπ))β(0 + 1)) = (Ackβπ)) |
| 15 | 14 | fveq1d 6894 | . 2 β’ (π β β0 β (((IterCompβ(Ackβπ))β(0 + 1))β1) = ((Ackβπ)β1)) |
| 16 | 3, 15 | eqtrd 2773 | 1 β’ (π β β0 β ((Ackβ(π + 1))β0) = ((Ackβπ)β1)) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 = wceq 1542 β wcel 2107 Vcvv 3475 Rel wrel 5682 Fun wfun 6538 Fn wfn 6539 βcfv 6544 (class class class)co 7409 0cc0 11110 1c1 11111 + caddc 11113 β0cn0 12472 IterCompcitco 47343 Ackcack 47344 |
| 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 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-inf2 9636 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 |
| 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 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-er 8703 df-en 8940 df-dom 8941 df-sdom 8942 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-nn 12213 df-n0 12473 df-z 12559 df-uz 12823 df-seq 13967 df-itco 47345 df-ack 47346 |
| This theorem is referenced by: ackval40 47379 ackval50 47384 |
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