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Mirrors > Home > MPE Home > Th. List > swrdlsw | Structured version Visualization version GIF version |
Description: Extract the last single symbol from a word. (Contributed by Alexander van der Vekens, 23-Sep-2018.) |
Ref | Expression |
---|---|
swrdlsw | ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑊 ≠ ∅) → (𝑊 substr 〈((♯‘𝑊) − 1), (♯‘𝑊)〉) = 〈“(lastS‘𝑊)”〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hashneq0 13719 | . . . . 5 ⊢ (𝑊 ∈ Word 𝑉 → (0 < (♯‘𝑊) ↔ 𝑊 ≠ ∅)) | |
2 | lencl 13877 | . . . . . 6 ⊢ (𝑊 ∈ Word 𝑉 → (♯‘𝑊) ∈ ℕ0) | |
3 | nn0z 11999 | . . . . . 6 ⊢ ((♯‘𝑊) ∈ ℕ0 → (♯‘𝑊) ∈ ℤ) | |
4 | elnnz 11985 | . . . . . . . 8 ⊢ ((♯‘𝑊) ∈ ℕ ↔ ((♯‘𝑊) ∈ ℤ ∧ 0 < (♯‘𝑊))) | |
5 | fzo0end 13123 | . . . . . . . 8 ⊢ ((♯‘𝑊) ∈ ℕ → ((♯‘𝑊) − 1) ∈ (0..^(♯‘𝑊))) | |
6 | 4, 5 | sylbir 237 | . . . . . . 7 ⊢ (((♯‘𝑊) ∈ ℤ ∧ 0 < (♯‘𝑊)) → ((♯‘𝑊) − 1) ∈ (0..^(♯‘𝑊))) |
7 | 6 | ex 415 | . . . . . 6 ⊢ ((♯‘𝑊) ∈ ℤ → (0 < (♯‘𝑊) → ((♯‘𝑊) − 1) ∈ (0..^(♯‘𝑊)))) |
8 | 2, 3, 7 | 3syl 18 | . . . . 5 ⊢ (𝑊 ∈ Word 𝑉 → (0 < (♯‘𝑊) → ((♯‘𝑊) − 1) ∈ (0..^(♯‘𝑊)))) |
9 | 1, 8 | sylbird 262 | . . . 4 ⊢ (𝑊 ∈ Word 𝑉 → (𝑊 ≠ ∅ → ((♯‘𝑊) − 1) ∈ (0..^(♯‘𝑊)))) |
10 | 9 | imp 409 | . . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑊 ≠ ∅) → ((♯‘𝑊) − 1) ∈ (0..^(♯‘𝑊))) |
11 | swrds1 14022 | . . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ ((♯‘𝑊) − 1) ∈ (0..^(♯‘𝑊))) → (𝑊 substr 〈((♯‘𝑊) − 1), (((♯‘𝑊) − 1) + 1)〉) = 〈“(𝑊‘((♯‘𝑊) − 1))”〉) | |
12 | 10, 11 | syldan 593 | . 2 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑊 ≠ ∅) → (𝑊 substr 〈((♯‘𝑊) − 1), (((♯‘𝑊) − 1) + 1)〉) = 〈“(𝑊‘((♯‘𝑊) − 1))”〉) |
13 | nn0cn 11901 | . . . . . . 7 ⊢ ((♯‘𝑊) ∈ ℕ0 → (♯‘𝑊) ∈ ℂ) | |
14 | ax-1cn 10589 | . . . . . . 7 ⊢ 1 ∈ ℂ | |
15 | 13, 14 | jctir 523 | . . . . . 6 ⊢ ((♯‘𝑊) ∈ ℕ0 → ((♯‘𝑊) ∈ ℂ ∧ 1 ∈ ℂ)) |
16 | npcan 10889 | . . . . . . 7 ⊢ (((♯‘𝑊) ∈ ℂ ∧ 1 ∈ ℂ) → (((♯‘𝑊) − 1) + 1) = (♯‘𝑊)) | |
17 | 16 | eqcomd 2827 | . . . . . 6 ⊢ (((♯‘𝑊) ∈ ℂ ∧ 1 ∈ ℂ) → (♯‘𝑊) = (((♯‘𝑊) − 1) + 1)) |
18 | 2, 15, 17 | 3syl 18 | . . . . 5 ⊢ (𝑊 ∈ Word 𝑉 → (♯‘𝑊) = (((♯‘𝑊) − 1) + 1)) |
19 | 18 | adantr 483 | . . . 4 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑊 ≠ ∅) → (♯‘𝑊) = (((♯‘𝑊) − 1) + 1)) |
20 | 19 | opeq2d 4804 | . . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑊 ≠ ∅) → 〈((♯‘𝑊) − 1), (♯‘𝑊)〉 = 〈((♯‘𝑊) − 1), (((♯‘𝑊) − 1) + 1)〉) |
21 | 20 | oveq2d 7166 | . 2 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑊 ≠ ∅) → (𝑊 substr 〈((♯‘𝑊) − 1), (♯‘𝑊)〉) = (𝑊 substr 〈((♯‘𝑊) − 1), (((♯‘𝑊) − 1) + 1)〉)) |
22 | lsw 13910 | . . . 4 ⊢ (𝑊 ∈ Word 𝑉 → (lastS‘𝑊) = (𝑊‘((♯‘𝑊) − 1))) | |
23 | 22 | adantr 483 | . . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑊 ≠ ∅) → (lastS‘𝑊) = (𝑊‘((♯‘𝑊) − 1))) |
24 | 23 | s1eqd 13949 | . 2 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑊 ≠ ∅) → 〈“(lastS‘𝑊)”〉 = 〈“(𝑊‘((♯‘𝑊) − 1))”〉) |
25 | 12, 21, 24 | 3eqtr4d 2866 | 1 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑊 ≠ ∅) → (𝑊 substr 〈((♯‘𝑊) − 1), (♯‘𝑊)〉) = 〈“(lastS‘𝑊)”〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 ∅c0 4291 〈cop 4567 class class class wbr 5059 ‘cfv 6350 (class class class)co 7150 ℂcc 10529 0cc0 10531 1c1 10532 + caddc 10534 < clt 10669 − cmin 10864 ℕcn 11632 ℕ0cn0 11891 ℤcz 11975 ..^cfzo 13027 ♯chash 13684 Word cword 13855 lastSclsw 13908 〈“cs1 13943 substr csubstr 13996 |
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-oadd 8100 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-xnn0 11962 df-z 11976 df-uz 12238 df-fz 12887 df-fzo 13028 df-hash 13685 df-word 13856 df-lsw 13909 df-s1 13944 df-substr 13997 |
This theorem is referenced by: pfxsuff1eqwrdeq 14055 pfxlswccat 14069 |
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