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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 14496 | . . 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 7416 | . . 3 ⊢ ((♯‘𝑊) = 0 → (𝑊‘((♯‘𝑊) − 1)) = (𝑊‘(0 − 1))) | |
4 | wrddm 14453 | . . . 4 ⊢ (𝑊 ∈ Word 𝑉 → dom 𝑊 = (0..^(♯‘𝑊))) | |
5 | 1nn 12205 | . . . . . . . 8 ⊢ 1 ∈ ℕ | |
6 | nnnle0 12227 | . . . . . . . 8 ⊢ (1 ∈ ℕ → ¬ 1 ≤ 0) | |
7 | 5, 6 | ax-mp 5 | . . . . . . 7 ⊢ ¬ 1 ≤ 0 |
8 | 0re 11198 | . . . . . . . 8 ⊢ 0 ∈ ℝ | |
9 | 1re 11196 | . . . . . . . 8 ⊢ 1 ∈ ℝ | |
10 | 8, 9 | subge0i 11749 | . . . . . . 7 ⊢ (0 ≤ (0 − 1) ↔ 1 ≤ 0) |
11 | 7, 10 | mtbir 322 | . . . . . 6 ⊢ ¬ 0 ≤ (0 − 1) |
12 | elfzole1 13622 | . . . . . 6 ⊢ ((0 − 1) ∈ (0..^(♯‘𝑊)) → 0 ≤ (0 − 1)) | |
13 | 11, 12 | mto 196 | . . . . 5 ⊢ ¬ (0 − 1) ∈ (0..^(♯‘𝑊)) |
14 | eleq2 2821 | . . . . 5 ⊢ (dom 𝑊 = (0..^(♯‘𝑊)) → ((0 − 1) ∈ dom 𝑊 ↔ (0 − 1) ∈ (0..^(♯‘𝑊)))) | |
15 | 13, 14 | mtbiri 326 | . . . 4 ⊢ (dom 𝑊 = (0..^(♯‘𝑊)) → ¬ (0 − 1) ∈ dom 𝑊) |
16 | ndmfv 6913 | . . . 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 2793 | . 2 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 0) → (𝑊‘((♯‘𝑊) − 1)) = ∅) |
19 | 2, 18 | eqtrd 2771 | 1 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 0) → (lastS‘𝑊) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∅c0 4318 class class class wbr 5141 dom cdm 5669 ‘cfv 6532 (class class class)co 7393 0cc0 11092 1c1 11093 ≤ cle 11231 − cmin 11426 ℕcn 12194 ..^cfzo 13609 ♯chash 14272 Word cword 14446 lastSclsw 14494 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7708 ax-cnex 11148 ax-resscn 11149 ax-1cn 11150 ax-icn 11151 ax-addcl 11152 ax-addrcl 11153 ax-mulcl 11154 ax-mulrcl 11155 ax-mulcom 11156 ax-addass 11157 ax-mulass 11158 ax-distr 11159 ax-i2m1 11160 ax-1ne0 11161 ax-1rid 11162 ax-rnegex 11163 ax-rrecex 11164 ax-cnre 11165 ax-pre-lttri 11166 ax-pre-lttrn 11167 ax-pre-ltadd 11168 ax-pre-mulgt0 11169 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 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 6289 df-ord 6356 df-on 6357 df-lim 6358 df-suc 6359 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-f1 6537 df-fo 6538 df-f1o 6539 df-fv 6540 df-riota 7349 df-ov 7396 df-oprab 7397 df-mpo 7398 df-om 7839 df-1st 7957 df-2nd 7958 df-frecs 8248 df-wrecs 8279 df-recs 8353 df-rdg 8392 df-1o 8448 df-er 8686 df-en 8923 df-dom 8924 df-sdom 8925 df-fin 8926 df-card 9916 df-pnf 11232 df-mnf 11233 df-xr 11234 df-ltxr 11235 df-le 11236 df-sub 11428 df-neg 11429 df-nn 12195 df-n0 12455 df-z 12541 df-uz 12805 df-fz 13467 df-fzo 13610 df-hash 14273 df-word 14447 df-lsw 14495 |
This theorem is referenced by: lsw0g 14498 |
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