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Mirrors > Home > MPE Home > Th. List > revfv | Structured version Visualization version GIF version |
Description: Reverse of a word at a point. (Contributed by Stefan O'Rear, 26-Aug-2015.) |
Ref | Expression |
---|---|
revfv | β’ ((π β Word π΄ β§ π β (0..^(β―βπ))) β ((reverseβπ)βπ) = (πβ(((β―βπ) β 1) β π))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | revval 14655 | . . 3 β’ (π β Word π΄ β (reverseβπ) = (π₯ β (0..^(β―βπ)) β¦ (πβ(((β―βπ) β 1) β π₯)))) | |
2 | 1 | fveq1d 6849 | . 2 β’ (π β Word π΄ β ((reverseβπ)βπ) = ((π₯ β (0..^(β―βπ)) β¦ (πβ(((β―βπ) β 1) β π₯)))βπ)) |
3 | oveq2 7370 | . . . 4 β’ (π₯ = π β (((β―βπ) β 1) β π₯) = (((β―βπ) β 1) β π)) | |
4 | 3 | fveq2d 6851 | . . 3 β’ (π₯ = π β (πβ(((β―βπ) β 1) β π₯)) = (πβ(((β―βπ) β 1) β π))) |
5 | eqid 2737 | . . 3 β’ (π₯ β (0..^(β―βπ)) β¦ (πβ(((β―βπ) β 1) β π₯))) = (π₯ β (0..^(β―βπ)) β¦ (πβ(((β―βπ) β 1) β π₯))) | |
6 | fvex 6860 | . . 3 β’ (πβ(((β―βπ) β 1) β π)) β V | |
7 | 4, 5, 6 | fvmpt 6953 | . 2 β’ (π β (0..^(β―βπ)) β ((π₯ β (0..^(β―βπ)) β¦ (πβ(((β―βπ) β 1) β π₯)))βπ) = (πβ(((β―βπ) β 1) β π))) |
8 | 2, 7 | sylan9eq 2797 | 1 β’ ((π β Word π΄ β§ π β (0..^(β―βπ))) β ((reverseβπ)βπ) = (πβ(((β―βπ) β 1) β π))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 β¦ cmpt 5193 βcfv 6501 (class class class)co 7362 0cc0 11058 1c1 11059 β cmin 11392 ..^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-pr 5389 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 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-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-nul 4288 df-if 4492 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-id 5536 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-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-ov 7365 df-reverse 14654 |
This theorem is referenced by: revs1 14660 revccat 14661 revrev 14662 revco 14730 revpfxsfxrev 33749 revwlk 33758 |
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