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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 13384 | . . 3 ⊢ (𝑊 ∈ Word 𝑉 → ( lastS ‘𝑊) = (𝑊‘((#‘𝑊) − 1))) | |
2 | 1 | adantr 480 | . 2 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = 0) → ( lastS ‘𝑊) = (𝑊‘((#‘𝑊) − 1))) |
3 | oveq1 6697 | . . . 4 ⊢ ((#‘𝑊) = 0 → ((#‘𝑊) − 1) = (0 − 1)) | |
4 | 3 | fveq2d 6233 | . . 3 ⊢ ((#‘𝑊) = 0 → (𝑊‘((#‘𝑊) − 1)) = (𝑊‘(0 − 1))) |
5 | wrddm 13344 | . . . 4 ⊢ (𝑊 ∈ Word 𝑉 → dom 𝑊 = (0..^(#‘𝑊))) | |
6 | 1nn 11069 | . . . . . . . 8 ⊢ 1 ∈ ℕ | |
7 | nnnle0 11089 | . . . . . . . 8 ⊢ (1 ∈ ℕ → ¬ 1 ≤ 0) | |
8 | 6, 7 | ax-mp 5 | . . . . . . 7 ⊢ ¬ 1 ≤ 0 |
9 | 0re 10078 | . . . . . . . 8 ⊢ 0 ∈ ℝ | |
10 | 1re 10077 | . . . . . . . 8 ⊢ 1 ∈ ℝ | |
11 | 9, 10 | subge0i 10619 | . . . . . . 7 ⊢ (0 ≤ (0 − 1) ↔ 1 ≤ 0) |
12 | 8, 11 | mtbir 312 | . . . . . 6 ⊢ ¬ 0 ≤ (0 − 1) |
13 | elfzole1 12517 | . . . . . 6 ⊢ ((0 − 1) ∈ (0..^(#‘𝑊)) → 0 ≤ (0 − 1)) | |
14 | 12, 13 | mto 188 | . . . . 5 ⊢ ¬ (0 − 1) ∈ (0..^(#‘𝑊)) |
15 | eleq2 2719 | . . . . 5 ⊢ (dom 𝑊 = (0..^(#‘𝑊)) → ((0 − 1) ∈ dom 𝑊 ↔ (0 − 1) ∈ (0..^(#‘𝑊)))) | |
16 | 14, 15 | mtbiri 316 | . . . 4 ⊢ (dom 𝑊 = (0..^(#‘𝑊)) → ¬ (0 − 1) ∈ dom 𝑊) |
17 | ndmfv 6256 | . . . 4 ⊢ (¬ (0 − 1) ∈ dom 𝑊 → (𝑊‘(0 − 1)) = ∅) | |
18 | 5, 16, 17 | 3syl 18 | . . 3 ⊢ (𝑊 ∈ Word 𝑉 → (𝑊‘(0 − 1)) = ∅) |
19 | 4, 18 | sylan9eqr 2707 | . 2 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = 0) → (𝑊‘((#‘𝑊) − 1)) = ∅) |
20 | 2, 19 | eqtrd 2685 | 1 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = 0) → ( lastS ‘𝑊) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 383 = wceq 1523 ∈ wcel 2030 ∅c0 3948 class class class wbr 4685 dom cdm 5143 ‘cfv 5926 (class class class)co 6690 0cc0 9974 1c1 9975 ≤ cle 10113 − cmin 10304 ℕcn 11058 ..^cfzo 12504 #chash 13157 Word cword 13323 lastS clsw 13324 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-int 4508 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-om 7108 df-1st 7210 df-2nd 7211 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-1o 7605 df-er 7787 df-en 7998 df-dom 7999 df-sdom 8000 df-fin 8001 df-card 8803 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-nn 11059 df-n0 11331 df-z 11416 df-uz 11726 df-fz 12365 df-fzo 12505 df-hash 13158 df-word 13331 df-lsw 13332 |
This theorem is referenced by: lsw0g 13386 |
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