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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 12509 | . . 3 β’ 0 β β0 | |
| 2 | ackvalsuc1 47675 | . . 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 12356 | . . . . . 6 β’ (0 + 1) = 1 | |
| 5 | 4 | a1i 11 | . . . . 5 β’ (π β β0 β (0 + 1) = 1) |
| 6 | 5 | fveq2d 6895 | . . . 4 β’ (π β β0 β ((IterCompβ(Ackβπ))β(0 + 1)) = ((IterCompβ(Ackβπ))β1)) |
| 7 | ackfnnn0 47681 | . . . . . 6 β’ (π β β0 β (Ackβπ) Fn β0) | |
| 8 | fnfun 6648 | . . . . . 6 β’ ((Ackβπ) Fn β0 β Fun (Ackβπ)) | |
| 9 | funrel 6564 | . . . . . 6 β’ (Fun (Ackβπ) β Rel (Ackβπ)) | |
| 10 | 7, 8, 9 | 3syl 18 | . . . . 5 β’ (π β β0 β Rel (Ackβπ)) |
| 11 | fvex 6904 | . . . . 5 β’ (Ackβπ) β V | |
| 12 | itcoval1 47659 | . . . . 5 β’ ((Rel (Ackβπ) β§ (Ackβπ) β V) β ((IterCompβ(Ackβπ))β1) = (Ackβπ)) | |
| 13 | 10, 11, 12 | sylancl 585 | . . . 4 β’ (π β β0 β ((IterCompβ(Ackβπ))β1) = (Ackβπ)) |
| 14 | 6, 13 | eqtrd 2767 | . . 3 β’ (π β β0 β ((IterCompβ(Ackβπ))β(0 + 1)) = (Ackβπ)) |
| 15 | 14 | fveq1d 6893 | . 2 β’ (π β β0 β (((IterCompβ(Ackβπ))β(0 + 1))β1) = ((Ackβπ)β1)) |
| 16 | 3, 15 | eqtrd 2767 | 1 β’ (π β β0 β ((Ackβ(π + 1))β0) = ((Ackβπ)β1)) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 = wceq 1534 β wcel 2099 Vcvv 3469 Rel wrel 5677 Fun wfun 6536 Fn wfn 6537 βcfv 6542 (class class class)co 7414 0cc0 11130 1c1 11131 + caddc 11133 β0cn0 12494 IterCompcitco 47653 Ackcack 47654 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-inf2 9656 ax-cnex 11186 ax-resscn 11187 ax-1cn 11188 ax-icn 11189 ax-addcl 11190 ax-addrcl 11191 ax-mulcl 11192 ax-mulrcl 11193 ax-mulcom 11194 ax-addass 11195 ax-mulass 11196 ax-distr 11197 ax-i2m1 11198 ax-1ne0 11199 ax-1rid 11200 ax-rnegex 11201 ax-rrecex 11202 ax-cnre 11203 ax-pre-lttri 11204 ax-pre-lttrn 11205 ax-pre-ltadd 11206 ax-pre-mulgt0 11207 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-er 8718 df-en 8956 df-dom 8957 df-sdom 8958 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11468 df-neg 11469 df-nn 12235 df-n0 12495 df-z 12581 df-uz 12845 df-seq 13991 df-itco 47655 df-ack 47656 |
| This theorem is referenced by: ackval40 47689 ackval50 47694 |
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