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Mirrors > Home > MPE Home > Th. List > lsw | Structured version Visualization version GIF version |
Description: Extract the last symbol of a word. May be not meaningful for other sets which are not words. (Contributed by Alexander van der Vekens, 18-Mar-2018.) |
Ref | Expression |
---|---|
lsw | β’ (π β π β (lastSβπ) = (πβ((β―βπ) β 1))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3493 | . 2 β’ (π β π β π β V) | |
2 | fvex 6905 | . 2 β’ (πβ((β―βπ) β 1)) β V | |
3 | id 22 | . . . 4 β’ (π€ = π β π€ = π) | |
4 | fveq2 6892 | . . . . 5 β’ (π€ = π β (β―βπ€) = (β―βπ)) | |
5 | 4 | oveq1d 7424 | . . . 4 β’ (π€ = π β ((β―βπ€) β 1) = ((β―βπ) β 1)) |
6 | 3, 5 | fveq12d 6899 | . . 3 β’ (π€ = π β (π€β((β―βπ€) β 1)) = (πβ((β―βπ) β 1))) |
7 | df-lsw 14513 | . . 3 β’ lastS = (π€ β V β¦ (π€β((β―βπ€) β 1))) | |
8 | 6, 7 | fvmptg 6997 | . 2 β’ ((π β V β§ (πβ((β―βπ) β 1)) β V) β (lastSβπ) = (πβ((β―βπ) β 1))) |
9 | 1, 2, 8 | sylancl 587 | 1 β’ (π β π β (lastSβπ) = (πβ((β―βπ) β 1))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1542 β wcel 2107 Vcvv 3475 βcfv 6544 (class class class)co 7409 1c1 11111 β cmin 11444 β―chash 14290 lastSclsw 14512 |
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 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428 |
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 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-iota 6496 df-fun 6546 df-fv 6552 df-ov 7412 df-lsw 14513 |
This theorem is referenced by: lsw0 14515 lsw1 14517 lswcl 14518 ccatval1lsw 14534 lswccatn0lsw 14541 swrdlsw 14617 pfxfvlsw 14645 repswlsw 14732 lswcshw 14765 lswco 14790 lsws2 14855 lsws3 14856 lsws4 14857 wrdl2exs2 14897 swrd2lsw 14903 psgnunilem5 19362 wlkonwlk1l 28920 wwlknlsw 29101 wwlksnext 29147 wwlksnredwwlkn 29149 wwlksnextproplem2 29164 clwlkclwwlklem2a1 29245 clwlkclwwlklem2a3 29247 clwlkclwwlklem2a4 29250 clwlkclwwlklem2 29253 clwwisshclwwslem 29267 clwwlknlbonbgr1 29292 clwwlkn2 29297 clwwlkel 29299 clwwlkf 29300 clwwlkwwlksb 29307 clwwlknonex2lem2 29361 2clwwlk2clwwlklem 29599 numclwwlk1lem2f1 29610 pfxlsw2ccat 32116 iwrdsplit 33386 signsvtn0 33581 signstfveq0 33588 lswn0 46112 |
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