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Mirrors > Home > MPE Home > Th. List > Mathboxes > lswn0 | Structured version Visualization version GIF version |
Description: The last symbol of a not empty word exists. The empty set must be excluded as symbol, because otherwise, it cannot be distinguished between valid cases (∅ is the last symbol) and invalid cases (∅ means that no last symbol exists. This is because of the special definition of a function in set.mm. (Contributed by Alexander van der Vekens, 18-Mar-2018.) |
Ref | Expression |
---|---|
lswn0 | ⊢ ((𝑊 ∈ Word 𝑉 ∧ ∅ ∉ 𝑉 ∧ (#‘𝑊) ≠ 0) → ( lastS ‘𝑊) ≠ ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lsw 13351 | . . 3 ⊢ (𝑊 ∈ Word 𝑉 → ( lastS ‘𝑊) = (𝑊‘((#‘𝑊) − 1))) | |
2 | 1 | 3ad2ant1 1082 | . 2 ⊢ ((𝑊 ∈ Word 𝑉 ∧ ∅ ∉ 𝑉 ∧ (#‘𝑊) ≠ 0) → ( lastS ‘𝑊) = (𝑊‘((#‘𝑊) − 1))) |
3 | wrdf 13310 | . . . . . 6 ⊢ (𝑊 ∈ Word 𝑉 → 𝑊:(0..^(#‘𝑊))⟶𝑉) | |
4 | lencl 13324 | . . . . . 6 ⊢ (𝑊 ∈ Word 𝑉 → (#‘𝑊) ∈ ℕ0) | |
5 | simpll 790 | . . . . . . . 8 ⊢ (((𝑊:(0..^(#‘𝑊))⟶𝑉 ∧ (#‘𝑊) ∈ ℕ0) ∧ (#‘𝑊) ≠ 0) → 𝑊:(0..^(#‘𝑊))⟶𝑉) | |
6 | elnnne0 11306 | . . . . . . . . . . . 12 ⊢ ((#‘𝑊) ∈ ℕ ↔ ((#‘𝑊) ∈ ℕ0 ∧ (#‘𝑊) ≠ 0)) | |
7 | 6 | biimpri 218 | . . . . . . . . . . 11 ⊢ (((#‘𝑊) ∈ ℕ0 ∧ (#‘𝑊) ≠ 0) → (#‘𝑊) ∈ ℕ) |
8 | nnm1nn0 11334 | . . . . . . . . . . 11 ⊢ ((#‘𝑊) ∈ ℕ → ((#‘𝑊) − 1) ∈ ℕ0) | |
9 | 7, 8 | syl 17 | . . . . . . . . . 10 ⊢ (((#‘𝑊) ∈ ℕ0 ∧ (#‘𝑊) ≠ 0) → ((#‘𝑊) − 1) ∈ ℕ0) |
10 | nn0re 11301 | . . . . . . . . . . . 12 ⊢ ((#‘𝑊) ∈ ℕ0 → (#‘𝑊) ∈ ℝ) | |
11 | 10 | ltm1d 10956 | . . . . . . . . . . 11 ⊢ ((#‘𝑊) ∈ ℕ0 → ((#‘𝑊) − 1) < (#‘𝑊)) |
12 | 11 | adantr 481 | . . . . . . . . . 10 ⊢ (((#‘𝑊) ∈ ℕ0 ∧ (#‘𝑊) ≠ 0) → ((#‘𝑊) − 1) < (#‘𝑊)) |
13 | elfzo0 12508 | . . . . . . . . . 10 ⊢ (((#‘𝑊) − 1) ∈ (0..^(#‘𝑊)) ↔ (((#‘𝑊) − 1) ∈ ℕ0 ∧ (#‘𝑊) ∈ ℕ ∧ ((#‘𝑊) − 1) < (#‘𝑊))) | |
14 | 9, 7, 12, 13 | syl3anbrc 1246 | . . . . . . . . 9 ⊢ (((#‘𝑊) ∈ ℕ0 ∧ (#‘𝑊) ≠ 0) → ((#‘𝑊) − 1) ∈ (0..^(#‘𝑊))) |
15 | 14 | adantll 750 | . . . . . . . 8 ⊢ (((𝑊:(0..^(#‘𝑊))⟶𝑉 ∧ (#‘𝑊) ∈ ℕ0) ∧ (#‘𝑊) ≠ 0) → ((#‘𝑊) − 1) ∈ (0..^(#‘𝑊))) |
16 | 5, 15 | ffvelrnd 6360 | . . . . . . 7 ⊢ (((𝑊:(0..^(#‘𝑊))⟶𝑉 ∧ (#‘𝑊) ∈ ℕ0) ∧ (#‘𝑊) ≠ 0) → (𝑊‘((#‘𝑊) − 1)) ∈ 𝑉) |
17 | 16 | ex 450 | . . . . . 6 ⊢ ((𝑊:(0..^(#‘𝑊))⟶𝑉 ∧ (#‘𝑊) ∈ ℕ0) → ((#‘𝑊) ≠ 0 → (𝑊‘((#‘𝑊) − 1)) ∈ 𝑉)) |
18 | 3, 4, 17 | syl2anc 693 | . . . . 5 ⊢ (𝑊 ∈ Word 𝑉 → ((#‘𝑊) ≠ 0 → (𝑊‘((#‘𝑊) − 1)) ∈ 𝑉)) |
19 | eleq1a 2696 | . . . . . . . . . 10 ⊢ ((𝑊‘((#‘𝑊) − 1)) ∈ 𝑉 → (∅ = (𝑊‘((#‘𝑊) − 1)) → ∅ ∈ 𝑉)) | |
20 | 19 | com12 32 | . . . . . . . . 9 ⊢ (∅ = (𝑊‘((#‘𝑊) − 1)) → ((𝑊‘((#‘𝑊) − 1)) ∈ 𝑉 → ∅ ∈ 𝑉)) |
21 | 20 | eqcoms 2630 | . . . . . . . 8 ⊢ ((𝑊‘((#‘𝑊) − 1)) = ∅ → ((𝑊‘((#‘𝑊) − 1)) ∈ 𝑉 → ∅ ∈ 𝑉)) |
22 | 21 | com12 32 | . . . . . . 7 ⊢ ((𝑊‘((#‘𝑊) − 1)) ∈ 𝑉 → ((𝑊‘((#‘𝑊) − 1)) = ∅ → ∅ ∈ 𝑉)) |
23 | nnel 2906 | . . . . . . 7 ⊢ (¬ ∅ ∉ 𝑉 ↔ ∅ ∈ 𝑉) | |
24 | 22, 23 | syl6ibr 242 | . . . . . 6 ⊢ ((𝑊‘((#‘𝑊) − 1)) ∈ 𝑉 → ((𝑊‘((#‘𝑊) − 1)) = ∅ → ¬ ∅ ∉ 𝑉)) |
25 | 24 | necon2ad 2809 | . . . . 5 ⊢ ((𝑊‘((#‘𝑊) − 1)) ∈ 𝑉 → (∅ ∉ 𝑉 → (𝑊‘((#‘𝑊) − 1)) ≠ ∅)) |
26 | 18, 25 | syl6 35 | . . . 4 ⊢ (𝑊 ∈ Word 𝑉 → ((#‘𝑊) ≠ 0 → (∅ ∉ 𝑉 → (𝑊‘((#‘𝑊) − 1)) ≠ ∅))) |
27 | 26 | com23 86 | . . 3 ⊢ (𝑊 ∈ Word 𝑉 → (∅ ∉ 𝑉 → ((#‘𝑊) ≠ 0 → (𝑊‘((#‘𝑊) − 1)) ≠ ∅))) |
28 | 27 | 3imp 1256 | . 2 ⊢ ((𝑊 ∈ Word 𝑉 ∧ ∅ ∉ 𝑉 ∧ (#‘𝑊) ≠ 0) → (𝑊‘((#‘𝑊) − 1)) ≠ ∅) |
29 | 2, 28 | eqnetrd 2861 | 1 ⊢ ((𝑊 ∈ Word 𝑉 ∧ ∅ ∉ 𝑉 ∧ (#‘𝑊) ≠ 0) → ( lastS ‘𝑊) ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∉ wnel 2897 ∅c0 3915 class class class wbr 4653 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 0cc0 9936 1c1 9937 < clt 10074 − cmin 10266 ℕcn 11020 ℕ0cn0 11292 ..^cfzo 12465 #chash 13117 Word cword 13291 lastS clsw 13292 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-n0 11293 df-z 11378 df-uz 11688 df-fz 12327 df-fzo 12466 df-hash 13118 df-word 13299 df-lsw 13300 |
This theorem is referenced by: (None) |
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