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Mathbox for Alexander van der Vekens |
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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 12429 | . . 3 β’ 0 β β0 | |
2 | ackvalsuc1 46772 | . . 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 12276 | . . . . . 6 β’ (0 + 1) = 1 | |
5 | 4 | a1i 11 | . . . . 5 β’ (π β β0 β (0 + 1) = 1) |
6 | 5 | fveq2d 6847 | . . . 4 β’ (π β β0 β ((IterCompβ(Ackβπ))β(0 + 1)) = ((IterCompβ(Ackβπ))β1)) |
7 | ackfnnn0 46778 | . . . . . 6 β’ (π β β0 β (Ackβπ) Fn β0) | |
8 | fnfun 6603 | . . . . . 6 β’ ((Ackβπ) Fn β0 β Fun (Ackβπ)) | |
9 | funrel 6519 | . . . . . 6 β’ (Fun (Ackβπ) β Rel (Ackβπ)) | |
10 | 7, 8, 9 | 3syl 18 | . . . . 5 β’ (π β β0 β Rel (Ackβπ)) |
11 | fvex 6856 | . . . . 5 β’ (Ackβπ) β V | |
12 | itcoval1 46756 | . . . . 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 2777 | . . 3 β’ (π β β0 β ((IterCompβ(Ackβπ))β(0 + 1)) = (Ackβπ)) |
15 | 14 | fveq1d 6845 | . 2 β’ (π β β0 β (((IterCompβ(Ackβπ))β(0 + 1))β1) = ((Ackβπ)β1)) |
16 | 3, 15 | eqtrd 2777 | 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 3446 Rel wrel 5639 Fun wfun 6491 Fn wfn 6492 βcfv 6497 (class class class)co 7358 0cc0 11052 1c1 11053 + caddc 11055 β0cn0 12414 IterCompcitco 46750 Ackcack 46751 |
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 2708 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-inf2 9578 ax-cnex 11108 ax-resscn 11109 ax-1cn 11110 ax-icn 11111 ax-addcl 11112 ax-addrcl 11113 ax-mulcl 11114 ax-mulrcl 11115 ax-mulcom 11116 ax-addass 11117 ax-mulass 11118 ax-distr 11119 ax-i2m1 11120 ax-1ne0 11121 ax-1rid 11122 ax-rnegex 11123 ax-rrecex 11124 ax-cnre 11125 ax-pre-lttri 11126 ax-pre-lttrn 11127 ax-pre-ltadd 11128 ax-pre-mulgt0 11129 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-er 8649 df-en 8885 df-dom 8886 df-sdom 8887 df-pnf 11192 df-mnf 11193 df-xr 11194 df-ltxr 11195 df-le 11196 df-sub 11388 df-neg 11389 df-nn 12155 df-n0 12415 df-z 12501 df-uz 12765 df-seq 13908 df-itco 46752 df-ack 46753 |
This theorem is referenced by: ackval40 46786 ackval50 46791 |
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