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Mirrors > Home > MPE Home > Th. List > revcl | Structured version Visualization version GIF version |
Description: The reverse of a word is a word. (Contributed by Stefan O'Rear, 26-Aug-2015.) |
Ref | Expression |
---|---|
revcl | β’ (π β Word π΄ β (reverseβπ) β Word π΄) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | revval 14655 | . 2 β’ (π β Word π΄ β (reverseβπ) = (π₯ β (0..^(β―βπ)) β¦ (πβ(((β―βπ) β 1) β π₯)))) | |
2 | wrdf 14414 | . . . . . 6 β’ (π β Word π΄ β π:(0..^(β―βπ))βΆπ΄) | |
3 | 2 | adantr 482 | . . . . 5 β’ ((π β Word π΄ β§ π₯ β (0..^(β―βπ))) β π:(0..^(β―βπ))βΆπ΄) |
4 | simpr 486 | . . . . . . . 8 β’ ((π β Word π΄ β§ π₯ β (0..^(β―βπ))) β π₯ β (0..^(β―βπ))) | |
5 | lencl 14428 | . . . . . . . . . . 11 β’ (π β Word π΄ β (β―βπ) β β0) | |
6 | 5 | adantr 482 | . . . . . . . . . 10 β’ ((π β Word π΄ β§ π₯ β (0..^(β―βπ))) β (β―βπ) β β0) |
7 | 6 | nn0zd 12532 | . . . . . . . . 9 β’ ((π β Word π΄ β§ π₯ β (0..^(β―βπ))) β (β―βπ) β β€) |
8 | fzoval 13580 | . . . . . . . . 9 β’ ((β―βπ) β β€ β (0..^(β―βπ)) = (0...((β―βπ) β 1))) | |
9 | 7, 8 | syl 17 | . . . . . . . 8 β’ ((π β Word π΄ β§ π₯ β (0..^(β―βπ))) β (0..^(β―βπ)) = (0...((β―βπ) β 1))) |
10 | 4, 9 | eleqtrd 2840 | . . . . . . 7 β’ ((π β Word π΄ β§ π₯ β (0..^(β―βπ))) β π₯ β (0...((β―βπ) β 1))) |
11 | fznn0sub2 13555 | . . . . . . 7 β’ (π₯ β (0...((β―βπ) β 1)) β (((β―βπ) β 1) β π₯) β (0...((β―βπ) β 1))) | |
12 | 10, 11 | syl 17 | . . . . . 6 β’ ((π β Word π΄ β§ π₯ β (0..^(β―βπ))) β (((β―βπ) β 1) β π₯) β (0...((β―βπ) β 1))) |
13 | 12, 9 | eleqtrrd 2841 | . . . . 5 β’ ((π β Word π΄ β§ π₯ β (0..^(β―βπ))) β (((β―βπ) β 1) β π₯) β (0..^(β―βπ))) |
14 | 3, 13 | ffvelcdmd 7041 | . . . 4 β’ ((π β Word π΄ β§ π₯ β (0..^(β―βπ))) β (πβ(((β―βπ) β 1) β π₯)) β π΄) |
15 | 14 | fmpttd 7068 | . . 3 β’ (π β Word π΄ β (π₯ β (0..^(β―βπ)) β¦ (πβ(((β―βπ) β 1) β π₯))):(0..^(β―βπ))βΆπ΄) |
16 | iswrdi 14413 | . . 3 β’ ((π₯ β (0..^(β―βπ)) β¦ (πβ(((β―βπ) β 1) β π₯))):(0..^(β―βπ))βΆπ΄ β (π₯ β (0..^(β―βπ)) β¦ (πβ(((β―βπ) β 1) β π₯))) β Word π΄) | |
17 | 15, 16 | syl 17 | . 2 β’ (π β Word π΄ β (π₯ β (0..^(β―βπ)) β¦ (πβ(((β―βπ) β 1) β π₯))) β Word π΄) |
18 | 1, 17 | eqeltrd 2838 | 1 β’ (π β Word π΄ β (reverseβπ) β Word π΄) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 β¦ cmpt 5193 βΆwf 6497 βcfv 6501 (class class class)co 7362 0cc0 11058 1c1 11059 β cmin 11392 β0cn0 12420 β€cz 12506 ...cfz 13431 ..^cfzo 13574 β―chash 14237 Word cword 14409 reversecreverse 14653 |
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 5247 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-1st 7926 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-z 12507 df-uz 12771 df-fz 13432 df-fzo 13575 df-hash 14238 df-word 14410 df-reverse 14654 |
This theorem is referenced by: revs1 14660 revccat 14661 revrev 14662 revco 14730 gsumwrev 19154 psgnuni 19288 efginvrel2 19516 efginvrel1 19517 frgp0 19549 frgpinv 19553 revpfxsfxrev 33749 swrdrevpfx 33750 revwlk 33758 |
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