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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 13910 | . . 3 ⊢ (𝑊 ∈ Word 𝑉 → (lastS‘𝑊) = (𝑊‘((♯‘𝑊) − 1))) | |
2 | 1 | adantr 483 | . 2 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 0) → (lastS‘𝑊) = (𝑊‘((♯‘𝑊) − 1))) |
3 | fvoveq1 7173 | . . 3 ⊢ ((♯‘𝑊) = 0 → (𝑊‘((♯‘𝑊) − 1)) = (𝑊‘(0 − 1))) | |
4 | wrddm 13862 | . . . 4 ⊢ (𝑊 ∈ Word 𝑉 → dom 𝑊 = (0..^(♯‘𝑊))) | |
5 | 1nn 11643 | . . . . . . . 8 ⊢ 1 ∈ ℕ | |
6 | nnnle0 11664 | . . . . . . . 8 ⊢ (1 ∈ ℕ → ¬ 1 ≤ 0) | |
7 | 5, 6 | ax-mp 5 | . . . . . . 7 ⊢ ¬ 1 ≤ 0 |
8 | 0re 10637 | . . . . . . . 8 ⊢ 0 ∈ ℝ | |
9 | 1re 10635 | . . . . . . . 8 ⊢ 1 ∈ ℝ | |
10 | 8, 9 | subge0i 11187 | . . . . . . 7 ⊢ (0 ≤ (0 − 1) ↔ 1 ≤ 0) |
11 | 7, 10 | mtbir 325 | . . . . . 6 ⊢ ¬ 0 ≤ (0 − 1) |
12 | elfzole1 13040 | . . . . . 6 ⊢ ((0 − 1) ∈ (0..^(♯‘𝑊)) → 0 ≤ (0 − 1)) | |
13 | 11, 12 | mto 199 | . . . . 5 ⊢ ¬ (0 − 1) ∈ (0..^(♯‘𝑊)) |
14 | eleq2 2901 | . . . . 5 ⊢ (dom 𝑊 = (0..^(♯‘𝑊)) → ((0 − 1) ∈ dom 𝑊 ↔ (0 − 1) ∈ (0..^(♯‘𝑊)))) | |
15 | 13, 14 | mtbiri 329 | . . . 4 ⊢ (dom 𝑊 = (0..^(♯‘𝑊)) → ¬ (0 − 1) ∈ dom 𝑊) |
16 | ndmfv 6695 | . . . 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 2878 | . 2 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 0) → (𝑊‘((♯‘𝑊) − 1)) = ∅) |
19 | 2, 18 | eqtrd 2856 | 1 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 0) → (lastS‘𝑊) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∅c0 4291 class class class wbr 5059 dom cdm 5550 ‘cfv 6350 (class class class)co 7150 0cc0 10531 1c1 10532 ≤ cle 10670 − cmin 10864 ℕcn 11632 ..^cfzo 13027 ♯chash 13684 Word cword 13855 lastSclsw 13908 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-n0 11892 df-z 11976 df-uz 12238 df-fz 12887 df-fzo 13028 df-hash 13685 df-word 13856 df-lsw 13909 |
This theorem is referenced by: lsw0g 13912 |
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