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Mirrors > Home > MPE Home > Th. List > pfxtrcfvl | Structured version Visualization version GIF version |
Description: The last symbol in a word truncated by one symbol. (Contributed by AV, 16-Jun-2018.) (Revised by AV, 5-May-2020.) |
Ref | Expression |
---|---|
pfxtrcfvl | β’ ((π β Word π β§ 2 β€ (β―βπ)) β (lastSβ(π prefix ((β―βπ) β 1))) = (πβ((β―βπ) β 2))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2z 12542 | . . . . . 6 β’ 2 β β€ | |
2 | 1 | a1i 11 | . . . . 5 β’ ((π β Word π β§ 2 β€ (β―βπ)) β 2 β β€) |
3 | lencl 14428 | . . . . . . 7 β’ (π β Word π β (β―βπ) β β0) | |
4 | 3 | nn0zd 12532 | . . . . . 6 β’ (π β Word π β (β―βπ) β β€) |
5 | 4 | adantr 482 | . . . . 5 β’ ((π β Word π β§ 2 β€ (β―βπ)) β (β―βπ) β β€) |
6 | simpr 486 | . . . . 5 β’ ((π β Word π β§ 2 β€ (β―βπ)) β 2 β€ (β―βπ)) | |
7 | eluz2 12776 | . . . . 5 β’ ((β―βπ) β (β€β₯β2) β (2 β β€ β§ (β―βπ) β β€ β§ 2 β€ (β―βπ))) | |
8 | 2, 5, 6, 7 | syl3anbrc 1344 | . . . 4 β’ ((π β Word π β§ 2 β€ (β―βπ)) β (β―βπ) β (β€β₯β2)) |
9 | ige2m1fz1 13537 | . . . 4 β’ ((β―βπ) β (β€β₯β2) β ((β―βπ) β 1) β (1...(β―βπ))) | |
10 | 8, 9 | syl 17 | . . 3 β’ ((π β Word π β§ 2 β€ (β―βπ)) β ((β―βπ) β 1) β (1...(β―βπ))) |
11 | pfxfvlsw 14590 | . . 3 β’ ((π β Word π β§ ((β―βπ) β 1) β (1...(β―βπ))) β (lastSβ(π prefix ((β―βπ) β 1))) = (πβ(((β―βπ) β 1) β 1))) | |
12 | 10, 11 | syldan 592 | . 2 β’ ((π β Word π β§ 2 β€ (β―βπ)) β (lastSβ(π prefix ((β―βπ) β 1))) = (πβ(((β―βπ) β 1) β 1))) |
13 | 3 | nn0cnd 12482 | . . . . 5 β’ (π β Word π β (β―βπ) β β) |
14 | sub1m1 12412 | . . . . 5 β’ ((β―βπ) β β β (((β―βπ) β 1) β 1) = ((β―βπ) β 2)) | |
15 | 13, 14 | syl 17 | . . . 4 β’ (π β Word π β (((β―βπ) β 1) β 1) = ((β―βπ) β 2)) |
16 | 15 | adantr 482 | . . 3 β’ ((π β Word π β§ 2 β€ (β―βπ)) β (((β―βπ) β 1) β 1) = ((β―βπ) β 2)) |
17 | 16 | fveq2d 6851 | . 2 β’ ((π β Word π β§ 2 β€ (β―βπ)) β (πβ(((β―βπ) β 1) β 1)) = (πβ((β―βπ) β 2))) |
18 | 12, 17 | eqtrd 2777 | 1 β’ ((π β Word π β§ 2 β€ (β―βπ)) β (lastSβ(π prefix ((β―βπ) β 1))) = (πβ((β―βπ) β 2))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 class class class wbr 5110 βcfv 6501 (class class class)co 7362 βcc 11056 1c1 11059 β€ cle 11197 β cmin 11392 2c2 12215 β€cz 12506 β€β₯cuz 12770 ...cfz 13431 β―chash 14237 Word cword 14409 lastSclsw 14457 prefix cpfx 14565 |
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-2 12223 df-n0 12421 df-z 12507 df-uz 12771 df-fz 13432 df-fzo 13575 df-hash 14238 df-word 14410 df-lsw 14458 df-substr 14536 df-pfx 14566 |
This theorem is referenced by: clwlkclwwlk 28988 pfxlsw2ccat 31848 |
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