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Mirrors > Home > MPE Home > Th. List > lsw0 | Structured version Visualization version GIF version |
Description: The last symbol of an empty word does not exist. (Contributed by Alexander van der Vekens, 19-Mar-2018.) (Proof shortened by AV, 2-May-2020.) |
Ref | Expression |
---|---|
lsw0 | β’ ((π β Word π β§ (β―βπ) = 0) β (lastSβπ) = β ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lsw 14520 | . . 3 β’ (π β Word π β (lastSβπ) = (πβ((β―βπ) β 1))) | |
2 | 1 | adantr 479 | . 2 β’ ((π β Word π β§ (β―βπ) = 0) β (lastSβπ) = (πβ((β―βπ) β 1))) |
3 | fvoveq1 7436 | . . 3 β’ ((β―βπ) = 0 β (πβ((β―βπ) β 1)) = (πβ(0 β 1))) | |
4 | wrddm 14477 | . . . 4 β’ (π β Word π β dom π = (0..^(β―βπ))) | |
5 | 1nn 12229 | . . . . . . . 8 β’ 1 β β | |
6 | nnnle0 12251 | . . . . . . . 8 β’ (1 β β β Β¬ 1 β€ 0) | |
7 | 5, 6 | ax-mp 5 | . . . . . . 7 β’ Β¬ 1 β€ 0 |
8 | 0re 11222 | . . . . . . . 8 β’ 0 β β | |
9 | 1re 11220 | . . . . . . . 8 β’ 1 β β | |
10 | 8, 9 | subge0i 11773 | . . . . . . 7 β’ (0 β€ (0 β 1) β 1 β€ 0) |
11 | 7, 10 | mtbir 322 | . . . . . 6 β’ Β¬ 0 β€ (0 β 1) |
12 | elfzole1 13646 | . . . . . 6 β’ ((0 β 1) β (0..^(β―βπ)) β 0 β€ (0 β 1)) | |
13 | 11, 12 | mto 196 | . . . . 5 β’ Β¬ (0 β 1) β (0..^(β―βπ)) |
14 | eleq2 2820 | . . . . 5 β’ (dom π = (0..^(β―βπ)) β ((0 β 1) β dom π β (0 β 1) β (0..^(β―βπ)))) | |
15 | 13, 14 | mtbiri 326 | . . . 4 β’ (dom π = (0..^(β―βπ)) β Β¬ (0 β 1) β dom π) |
16 | ndmfv 6927 | . . . 4 β’ (Β¬ (0 β 1) β dom π β (πβ(0 β 1)) = β ) | |
17 | 4, 15, 16 | 3syl 18 | . . 3 β’ (π β Word π β (πβ(0 β 1)) = β ) |
18 | 3, 17 | sylan9eqr 2792 | . 2 β’ ((π β Word π β§ (β―βπ) = 0) β (πβ((β―βπ) β 1)) = β ) |
19 | 2, 18 | eqtrd 2770 | 1 β’ ((π β Word π β§ (β―βπ) = 0) β (lastSβπ) = β ) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β§ wa 394 = wceq 1539 β wcel 2104 β c0 4323 class class class wbr 5149 dom cdm 5677 βcfv 6544 (class class class)co 7413 0cc0 11114 1c1 11115 β€ cle 11255 β cmin 11450 βcn 12218 ..^cfzo 13633 β―chash 14296 Word cword 14470 lastSclsw 14518 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7729 ax-cnex 11170 ax-resscn 11171 ax-1cn 11172 ax-icn 11173 ax-addcl 11174 ax-addrcl 11175 ax-mulcl 11176 ax-mulrcl 11177 ax-mulcom 11178 ax-addass 11179 ax-mulass 11180 ax-distr 11181 ax-i2m1 11182 ax-1ne0 11183 ax-1rid 11184 ax-rnegex 11185 ax-rrecex 11186 ax-cnre 11187 ax-pre-lttri 11188 ax-pre-lttrn 11189 ax-pre-ltadd 11190 ax-pre-mulgt0 11191 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8270 df-wrecs 8301 df-recs 8375 df-rdg 8414 df-1o 8470 df-er 8707 df-en 8944 df-dom 8945 df-sdom 8946 df-fin 8947 df-card 9938 df-pnf 11256 df-mnf 11257 df-xr 11258 df-ltxr 11259 df-le 11260 df-sub 11452 df-neg 11453 df-nn 12219 df-n0 12479 df-z 12565 df-uz 12829 df-fz 13491 df-fzo 13634 df-hash 14297 df-word 14471 df-lsw 14519 |
This theorem is referenced by: lsw0g 14522 |
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