Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ackval0val | Structured version Visualization version GIF version |
Description: The Ackermann function at 0 (for the first argument). This is the first equation of PΓ©ter's definition of the Ackermann function. (Contributed by AV, 4-May-2024.) |
Ref | Expression |
---|---|
ackval0val | β’ (π β β0 β ((Ackβ0)βπ) = (π + 1)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ackval0 46366 | . . 3 β’ (Ackβ0) = (π β β0 β¦ (π + 1)) | |
2 | 1 | a1i 11 | . 2 β’ (π β β0 β (Ackβ0) = (π β β0 β¦ (π + 1))) |
3 | oveq1 7336 | . . 3 β’ (π = π β (π + 1) = (π + 1)) | |
4 | 3 | adantl 482 | . 2 β’ ((π β β0 β§ π = π) β (π + 1) = (π + 1)) |
5 | id 22 | . 2 β’ (π β β0 β π β β0) | |
6 | peano2nn0 12366 | . 2 β’ (π β β0 β (π + 1) β β0) | |
7 | 2, 4, 5, 6 | fvmptd 6932 | 1 β’ (π β β0 β ((Ackβ0)βπ) = (π + 1)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1540 β wcel 2105 β¦ cmpt 5172 βcfv 6473 (class class class)co 7329 0cc0 10964 1c1 10965 + caddc 10967 β0cn0 12326 Ackcack 46344 |
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 2707 ax-rep 5226 ax-sep 5240 ax-nul 5247 ax-pow 5305 ax-pr 5369 ax-un 7642 ax-cnex 11020 ax-resscn 11021 ax-1cn 11022 ax-icn 11023 ax-addcl 11024 ax-addrcl 11025 ax-mulcl 11026 ax-mulrcl 11027 ax-mulcom 11028 ax-addass 11029 ax-mulass 11030 ax-distr 11031 ax-i2m1 11032 ax-1ne0 11033 ax-1rid 11034 ax-rnegex 11035 ax-rrecex 11036 ax-cnre 11037 ax-pre-lttri 11038 ax-pre-lttrn 11039 ax-pre-ltadd 11040 ax-pre-mulgt0 11041 |
This theorem depends on definitions: df-bi 206 df-an 397 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 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3350 df-rab 3404 df-v 3443 df-sbc 3727 df-csb 3843 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3916 df-nul 4269 df-if 4473 df-pw 4548 df-sn 4573 df-pr 4575 df-op 4579 df-uni 4852 df-iun 4940 df-br 5090 df-opab 5152 df-mpt 5173 df-tr 5207 df-id 5512 df-eprel 5518 df-po 5526 df-so 5527 df-fr 5569 df-we 5571 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6232 df-ord 6299 df-on 6300 df-lim 6301 df-suc 6302 df-iota 6425 df-fun 6475 df-fn 6476 df-f 6477 df-f1 6478 df-fo 6479 df-f1o 6480 df-fv 6481 df-riota 7286 df-ov 7332 df-oprab 7333 df-mpo 7334 df-om 7773 df-2nd 7892 df-frecs 8159 df-wrecs 8190 df-recs 8264 df-rdg 8303 df-er 8561 df-en 8797 df-dom 8798 df-sdom 8799 df-pnf 11104 df-mnf 11105 df-xr 11106 df-ltxr 11107 df-le 11108 df-sub 11300 df-neg 11301 df-nn 12067 df-n0 12327 df-z 12413 df-uz 12676 df-seq 13815 df-ack 46346 |
This theorem is referenced by: (None) |
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