![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > lswccats1fst | Structured version Visualization version GIF version |
Description: The last symbol of a nonempty word concatenated with its first symbol is the first symbol. (Contributed by AV, 28-Jun-2018.) (Proof shortened by AV, 1-May-2020.) |
Ref | Expression |
---|---|
lswccats1fst | β’ ((π β Word π β§ 1 β€ (β―βπ)) β (lastSβ(π ++ β¨β(πβ0)ββ©)) = ((π ++ β¨β(πβ0)ββ©)β0)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wrdsymb1 14530 | . . 3 β’ ((π β Word π β§ 1 β€ (β―βπ)) β (πβ0) β π) | |
2 | lswccats1 14611 | . . 3 β’ ((π β Word π β§ (πβ0) β π) β (lastSβ(π ++ β¨β(πβ0)ββ©)) = (πβ0)) | |
3 | 1, 2 | syldan 589 | . 2 β’ ((π β Word π β§ 1 β€ (β―βπ)) β (lastSβ(π ++ β¨β(πβ0)ββ©)) = (πβ0)) |
4 | simpl 481 | . . 3 β’ ((π β Word π β§ 1 β€ (β―βπ)) β π β Word π) | |
5 | 1 | s1cld 14580 | . . 3 β’ ((π β Word π β§ 1 β€ (β―βπ)) β β¨β(πβ0)ββ© β Word π) |
6 | lencl 14510 | . . . . 5 β’ (π β Word π β (β―βπ) β β0) | |
7 | elnnnn0c 12542 | . . . . . 6 β’ ((β―βπ) β β β ((β―βπ) β β0 β§ 1 β€ (β―βπ))) | |
8 | 7 | biimpri 227 | . . . . 5 β’ (((β―βπ) β β0 β§ 1 β€ (β―βπ)) β (β―βπ) β β) |
9 | 6, 8 | sylan 578 | . . . 4 β’ ((π β Word π β§ 1 β€ (β―βπ)) β (β―βπ) β β) |
10 | lbfzo0 13699 | . . . 4 β’ (0 β (0..^(β―βπ)) β (β―βπ) β β) | |
11 | 9, 10 | sylibr 233 | . . 3 β’ ((π β Word π β§ 1 β€ (β―βπ)) β 0 β (0..^(β―βπ))) |
12 | ccatval1 14554 | . . 3 β’ ((π β Word π β§ β¨β(πβ0)ββ© β Word π β§ 0 β (0..^(β―βπ))) β ((π ++ β¨β(πβ0)ββ©)β0) = (πβ0)) | |
13 | 4, 5, 11, 12 | syl3anc 1368 | . 2 β’ ((π β Word π β§ 1 β€ (β―βπ)) β ((π ++ β¨β(πβ0)ββ©)β0) = (πβ0)) |
14 | 3, 13 | eqtr4d 2768 | 1 β’ ((π β Word π β§ 1 β€ (β―βπ)) β (lastSβ(π ++ β¨β(πβ0)ββ©)) = ((π ++ β¨β(πβ0)ββ©)β0)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 = wceq 1533 β wcel 2098 class class class wbr 5144 βcfv 6543 (class class class)co 7413 0cc0 11133 1c1 11134 β€ cle 11274 βcn 12237 β0cn0 12497 ..^cfzo 13654 β―chash 14316 Word cword 14491 lastSclsw 14539 ++ cconcat 14547 β¨βcs1 14572 |
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 2166 ax-ext 2696 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 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 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3961 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-int 4946 df-iun 4994 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7866 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8718 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-card 9957 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-nn 12238 df-n0 12498 df-z 12584 df-uz 12848 df-rp 13002 df-fz 13512 df-fzo 13655 df-hash 14317 df-word 14492 df-lsw 14540 df-concat 14548 df-s1 14573 |
This theorem is referenced by: clwlkclwwlk2 29852 |
Copyright terms: Public domain | W3C validator |