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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 | ⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 0) → (lastS‘𝑊) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lsw 13904 | . . 3 ⊢ (𝑊 ∈ Word 𝑉 → (lastS‘𝑊) = (𝑊‘((♯‘𝑊) − 1))) | |
2 | 1 | adantr 481 | . 2 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 0) → (lastS‘𝑊) = (𝑊‘((♯‘𝑊) − 1))) |
3 | fvoveq1 7168 | . . 3 ⊢ ((♯‘𝑊) = 0 → (𝑊‘((♯‘𝑊) − 1)) = (𝑊‘(0 − 1))) | |
4 | wrddm 13856 | . . . 4 ⊢ (𝑊 ∈ Word 𝑉 → dom 𝑊 = (0..^(♯‘𝑊))) | |
5 | 1nn 11637 | . . . . . . . 8 ⊢ 1 ∈ ℕ | |
6 | nnnle0 11658 | . . . . . . . 8 ⊢ (1 ∈ ℕ → ¬ 1 ≤ 0) | |
7 | 5, 6 | ax-mp 5 | . . . . . . 7 ⊢ ¬ 1 ≤ 0 |
8 | 0re 10631 | . . . . . . . 8 ⊢ 0 ∈ ℝ | |
9 | 1re 10629 | . . . . . . . 8 ⊢ 1 ∈ ℝ | |
10 | 8, 9 | subge0i 11181 | . . . . . . 7 ⊢ (0 ≤ (0 − 1) ↔ 1 ≤ 0) |
11 | 7, 10 | mtbir 324 | . . . . . 6 ⊢ ¬ 0 ≤ (0 − 1) |
12 | elfzole1 13034 | . . . . . 6 ⊢ ((0 − 1) ∈ (0..^(♯‘𝑊)) → 0 ≤ (0 − 1)) | |
13 | 11, 12 | mto 198 | . . . . 5 ⊢ ¬ (0 − 1) ∈ (0..^(♯‘𝑊)) |
14 | eleq2 2898 | . . . . 5 ⊢ (dom 𝑊 = (0..^(♯‘𝑊)) → ((0 − 1) ∈ dom 𝑊 ↔ (0 − 1) ∈ (0..^(♯‘𝑊)))) | |
15 | 13, 14 | mtbiri 328 | . . . 4 ⊢ (dom 𝑊 = (0..^(♯‘𝑊)) → ¬ (0 − 1) ∈ dom 𝑊) |
16 | ndmfv 6693 | . . . 4 ⊢ (¬ (0 − 1) ∈ dom 𝑊 → (𝑊‘(0 − 1)) = ∅) | |
17 | 4, 15, 16 | 3syl 18 | . . 3 ⊢ (𝑊 ∈ Word 𝑉 → (𝑊‘(0 − 1)) = ∅) |
18 | 3, 17 | sylan9eqr 2875 | . 2 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 0) → (𝑊‘((♯‘𝑊) − 1)) = ∅) |
19 | 2, 18 | eqtrd 2853 | 1 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 0) → (lastS‘𝑊) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ∅c0 4288 class class class wbr 5057 dom cdm 5548 ‘cfv 6348 (class class class)co 7145 0cc0 10525 1c1 10526 ≤ cle 10664 − cmin 10858 ℕcn 11626 ..^cfzo 13021 ♯chash 13678 Word cword 13849 lastSclsw 13902 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-card 9356 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12881 df-fzo 13022 df-hash 13679 df-word 13850 df-lsw 13903 |
This theorem is referenced by: lsw0g 13906 |
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