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Mirrors > Home > MPE Home > Th. List > hashgval2 | Structured version Visualization version GIF version |
Description: A short expression for the πΊ function of hashgf1o 13883. (Contributed by Mario Carneiro, 24-Jan-2015.) |
Ref | Expression |
---|---|
hashgval2 | β’ (β― βΎ Ο) = (rec((π₯ β V β¦ (π₯ + 1)), 0) βΎ Ο) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hashresfn 14247 | . . 3 β’ (β― βΎ Ο) Fn Ο | |
2 | frfnom 8386 | . . 3 β’ (rec((π₯ β V β¦ (π₯ + 1)), 0) βΎ Ο) Fn Ο | |
3 | eqfnfv 6987 | . . 3 β’ (((β― βΎ Ο) Fn Ο β§ (rec((π₯ β V β¦ (π₯ + 1)), 0) βΎ Ο) Fn Ο) β ((β― βΎ Ο) = (rec((π₯ β V β¦ (π₯ + 1)), 0) βΎ Ο) β βπ¦ β Ο ((β― βΎ Ο)βπ¦) = ((rec((π₯ β V β¦ (π₯ + 1)), 0) βΎ Ο)βπ¦))) | |
4 | 1, 2, 3 | mp2an 691 | . 2 β’ ((β― βΎ Ο) = (rec((π₯ β V β¦ (π₯ + 1)), 0) βΎ Ο) β βπ¦ β Ο ((β― βΎ Ο)βπ¦) = ((rec((π₯ β V β¦ (π₯ + 1)), 0) βΎ Ο)βπ¦)) |
5 | fvres 6866 | . . 3 β’ (π¦ β Ο β ((β― βΎ Ο)βπ¦) = (β―βπ¦)) | |
6 | nnfi 9118 | . . . 4 β’ (π¦ β Ο β π¦ β Fin) | |
7 | eqid 2737 | . . . . 5 β’ (rec((π₯ β V β¦ (π₯ + 1)), 0) βΎ Ο) = (rec((π₯ β V β¦ (π₯ + 1)), 0) βΎ Ο) | |
8 | 7 | hashgval 14240 | . . . 4 β’ (π¦ β Fin β ((rec((π₯ β V β¦ (π₯ + 1)), 0) βΎ Ο)β(cardβπ¦)) = (β―βπ¦)) |
9 | 6, 8 | syl 17 | . . 3 β’ (π¦ β Ο β ((rec((π₯ β V β¦ (π₯ + 1)), 0) βΎ Ο)β(cardβπ¦)) = (β―βπ¦)) |
10 | cardnn 9906 | . . . 4 β’ (π¦ β Ο β (cardβπ¦) = π¦) | |
11 | 10 | fveq2d 6851 | . . 3 β’ (π¦ β Ο β ((rec((π₯ β V β¦ (π₯ + 1)), 0) βΎ Ο)β(cardβπ¦)) = ((rec((π₯ β V β¦ (π₯ + 1)), 0) βΎ Ο)βπ¦)) |
12 | 5, 9, 11 | 3eqtr2d 2783 | . 2 β’ (π¦ β Ο β ((β― βΎ Ο)βπ¦) = ((rec((π₯ β V β¦ (π₯ + 1)), 0) βΎ Ο)βπ¦)) |
13 | 4, 12 | mprgbir 3072 | 1 β’ (β― βΎ Ο) = (rec((π₯ β V β¦ (π₯ + 1)), 0) βΎ Ο) |
Colors of variables: wff setvar class |
Syntax hints: β wb 205 = wceq 1542 β wcel 2107 βwral 3065 Vcvv 3448 β¦ cmpt 5193 βΎ cres 5640 Fn wfn 6496 βcfv 6501 (class class class)co 7362 Οcom 7807 reccrdg 8360 Fincfn 8890 cardccrd 9878 0cc0 11058 1c1 11059 + caddc 11061 β―chash 14237 |
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-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-cnex 11114 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 ax-pre-mulgt0 11135 |
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 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-int 4913 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-1o 8417 df-er 8655 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-card 9882 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-nn 12161 df-n0 12421 df-xnn0 12493 df-z 12507 df-uz 12771 df-hash 14238 |
This theorem is referenced by: ackbijnn 15720 ltbwe 21461 |
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