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Mirrors > Home > MPE Home > Th. List > pfxfvlsw | Structured version Visualization version GIF version |
Description: The last symbol in a nonempty prefix of a word. (Contributed by Alexander van der Vekens, 24-Jun-2018.) (Revised by AV, 3-May-2020.) |
Ref | Expression |
---|---|
pfxfvlsw | β’ ((π β Word π β§ πΏ β (1...(β―βπ))) β (lastSβ(π prefix πΏ)) = (πβ(πΏ β 1))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pfxcl 14633 | . . . 4 β’ (π β Word π β (π prefix πΏ) β Word π) | |
2 | 1 | adantr 480 | . . 3 β’ ((π β Word π β§ πΏ β (1...(β―βπ))) β (π prefix πΏ) β Word π) |
3 | lsw 14520 | . . 3 β’ ((π prefix πΏ) β Word π β (lastSβ(π prefix πΏ)) = ((π prefix πΏ)β((β―β(π prefix πΏ)) β 1))) | |
4 | 2, 3 | syl 17 | . 2 β’ ((π β Word π β§ πΏ β (1...(β―βπ))) β (lastSβ(π prefix πΏ)) = ((π prefix πΏ)β((β―β(π prefix πΏ)) β 1))) |
5 | fz1ssfz0 13603 | . . . . 5 β’ (1...(β―βπ)) β (0...(β―βπ)) | |
6 | 5 | sseli 3973 | . . . 4 β’ (πΏ β (1...(β―βπ)) β πΏ β (0...(β―βπ))) |
7 | pfxlen 14639 | . . . 4 β’ ((π β Word π β§ πΏ β (0...(β―βπ))) β (β―β(π prefix πΏ)) = πΏ) | |
8 | 6, 7 | sylan2 592 | . . 3 β’ ((π β Word π β§ πΏ β (1...(β―βπ))) β (β―β(π prefix πΏ)) = πΏ) |
9 | 8 | fvoveq1d 7427 | . 2 β’ ((π β Word π β§ πΏ β (1...(β―βπ))) β ((π prefix πΏ)β((β―β(π prefix πΏ)) β 1)) = ((π prefix πΏ)β(πΏ β 1))) |
10 | simpl 482 | . . 3 β’ ((π β Word π β§ πΏ β (1...(β―βπ))) β π β Word π) | |
11 | 6 | adantl 481 | . . 3 β’ ((π β Word π β§ πΏ β (1...(β―βπ))) β πΏ β (0...(β―βπ))) |
12 | elfznn 13536 | . . . . 5 β’ (πΏ β (1...(β―βπ)) β πΏ β β) | |
13 | fzo0end 13730 | . . . . 5 β’ (πΏ β β β (πΏ β 1) β (0..^πΏ)) | |
14 | 12, 13 | syl 17 | . . . 4 β’ (πΏ β (1...(β―βπ)) β (πΏ β 1) β (0..^πΏ)) |
15 | 14 | adantl 481 | . . 3 β’ ((π β Word π β§ πΏ β (1...(β―βπ))) β (πΏ β 1) β (0..^πΏ)) |
16 | pfxfv 14638 | . . 3 β’ ((π β Word π β§ πΏ β (0...(β―βπ)) β§ (πΏ β 1) β (0..^πΏ)) β ((π prefix πΏ)β(πΏ β 1)) = (πβ(πΏ β 1))) | |
17 | 10, 11, 15, 16 | syl3anc 1368 | . 2 β’ ((π β Word π β§ πΏ β (1...(β―βπ))) β ((π prefix πΏ)β(πΏ β 1)) = (πβ(πΏ β 1))) |
18 | 4, 9, 17 | 3eqtrd 2770 | 1 β’ ((π β Word π β§ πΏ β (1...(β―βπ))) β (lastSβ(π prefix πΏ)) = (πβ(πΏ β 1))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1533 β wcel 2098 βcfv 6537 (class class class)co 7405 0cc0 11112 1c1 11113 β cmin 11448 βcn 12216 ...cfz 13490 ..^cfzo 13633 β―chash 14295 Word cword 14470 lastSclsw 14518 prefix cpfx 14626 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-n0 12477 df-z 12563 df-uz 12827 df-fz 13491 df-fzo 13634 df-hash 14296 df-word 14471 df-lsw 14519 df-substr 14597 df-pfx 14627 |
This theorem is referenced by: pfxtrcfvl 14653 wwlksnredwwlkn 29658 wwlksnextproplem2 29673 clwwlkinwwlk 29802 clwwlkf 29809 numclwlk2lem2f 30139 |
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