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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 12512 | . . 3 β’ 0 β β0 | |
| 2 | ackvalsuc1 47860 | . . 3 β’ ((π β β0 β§ 0 β β0) β ((Ackβ(π + 1))β0) = (((IterCompβ(Ackβπ))β(0 + 1))β1)) | |
| 3 | 1, 2 | mpan2 689 | . 2 β’ (π β β0 β ((Ackβ(π + 1))β0) = (((IterCompβ(Ackβπ))β(0 + 1))β1)) |
| 4 | 0p1e1 12359 | . . . . . 6 β’ (0 + 1) = 1 | |
| 5 | 4 | a1i 11 | . . . . 5 β’ (π β β0 β (0 + 1) = 1) |
| 6 | 5 | fveq2d 6894 | . . . 4 β’ (π β β0 β ((IterCompβ(Ackβπ))β(0 + 1)) = ((IterCompβ(Ackβπ))β1)) |
| 7 | ackfnnn0 47866 | . . . . . 6 β’ (π β β0 β (Ackβπ) Fn β0) | |
| 8 | fnfun 6649 | . . . . . 6 β’ ((Ackβπ) Fn β0 β Fun (Ackβπ)) | |
| 9 | funrel 6565 | . . . . . 6 β’ (Fun (Ackβπ) β Rel (Ackβπ)) | |
| 10 | 7, 8, 9 | 3syl 18 | . . . . 5 β’ (π β β0 β Rel (Ackβπ)) |
| 11 | fvex 6903 | . . . . 5 β’ (Ackβπ) β V | |
| 12 | itcoval1 47844 | . . . . 5 β’ ((Rel (Ackβπ) β§ (Ackβπ) β V) β ((IterCompβ(Ackβπ))β1) = (Ackβπ)) | |
| 13 | 10, 11, 12 | sylancl 584 | . . . 4 β’ (π β β0 β ((IterCompβ(Ackβπ))β1) = (Ackβπ)) |
| 14 | 6, 13 | eqtrd 2765 | . . 3 β’ (π β β0 β ((IterCompβ(Ackβπ))β(0 + 1)) = (Ackβπ)) |
| 15 | 14 | fveq1d 6892 | . 2 β’ (π β β0 β (((IterCompβ(Ackβπ))β(0 + 1))β1) = ((Ackβπ)β1)) |
| 16 | 3, 15 | eqtrd 2765 | 1 β’ (π β β0 β ((Ackβ(π + 1))β0) = ((Ackβπ)β1)) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 = wceq 1533 β wcel 2098 Vcvv 3463 Rel wrel 5678 Fun wfun 6537 Fn wfn 6538 βcfv 6543 (class class class)co 7413 0cc0 11133 1c1 11134 + caddc 11136 β0cn0 12497 IterCompcitco 47838 Ackcack 47839 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-inf2 9659 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210 |
| This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3961 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-iun 4994 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 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 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7866 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-er 8718 df-en 8958 df-dom 8959 df-sdom 8960 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-nn 12238 df-n0 12498 df-z 12584 df-uz 12848 df-seq 13994 df-itco 47840 df-ack 47841 |
| This theorem is referenced by: ackval40 47874 ackval50 47879 |
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