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Mirrors > Home > MPE Home > Th. List > swrdval2 | Structured version Visualization version GIF version |
Description: Value of the subword extractor in its intended domain. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Proof shortened by AV, 2-May-2020.) |
Ref | Expression |
---|---|
swrdval2 | ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(♯‘𝑆))) → (𝑆 substr 〈𝐹, 𝐿〉) = (𝑥 ∈ (0..^(𝐿 − 𝐹)) ↦ (𝑆‘(𝑥 + 𝐹)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 1132 | . . 3 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(♯‘𝑆))) → 𝑆 ∈ Word 𝐴) | |
2 | elfzelz 12907 | . . . 4 ⊢ (𝐹 ∈ (0...𝐿) → 𝐹 ∈ ℤ) | |
3 | 2 | 3ad2ant2 1130 | . . 3 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(♯‘𝑆))) → 𝐹 ∈ ℤ) |
4 | elfzelz 12907 | . . . 4 ⊢ (𝐿 ∈ (0...(♯‘𝑆)) → 𝐿 ∈ ℤ) | |
5 | 4 | 3ad2ant3 1131 | . . 3 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(♯‘𝑆))) → 𝐿 ∈ ℤ) |
6 | swrdval 14004 | . . 3 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → (𝑆 substr 〈𝐹, 𝐿〉) = if((𝐹..^𝐿) ⊆ dom 𝑆, (𝑥 ∈ (0..^(𝐿 − 𝐹)) ↦ (𝑆‘(𝑥 + 𝐹))), ∅)) | |
7 | 1, 3, 5, 6 | syl3anc 1367 | . 2 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(♯‘𝑆))) → (𝑆 substr 〈𝐹, 𝐿〉) = if((𝐹..^𝐿) ⊆ dom 𝑆, (𝑥 ∈ (0..^(𝐿 − 𝐹)) ↦ (𝑆‘(𝑥 + 𝐹))), ∅)) |
8 | elfzuz 12903 | . . . . . . 7 ⊢ (𝐹 ∈ (0...𝐿) → 𝐹 ∈ (ℤ≥‘0)) | |
9 | 8 | 3ad2ant2 1130 | . . . . . 6 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(♯‘𝑆))) → 𝐹 ∈ (ℤ≥‘0)) |
10 | fzoss1 13063 | . . . . . 6 ⊢ (𝐹 ∈ (ℤ≥‘0) → (𝐹..^𝐿) ⊆ (0..^𝐿)) | |
11 | 9, 10 | syl 17 | . . . . 5 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(♯‘𝑆))) → (𝐹..^𝐿) ⊆ (0..^𝐿)) |
12 | elfzuz3 12904 | . . . . . . 7 ⊢ (𝐿 ∈ (0...(♯‘𝑆)) → (♯‘𝑆) ∈ (ℤ≥‘𝐿)) | |
13 | 12 | 3ad2ant3 1131 | . . . . . 6 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(♯‘𝑆))) → (♯‘𝑆) ∈ (ℤ≥‘𝐿)) |
14 | fzoss2 13064 | . . . . . 6 ⊢ ((♯‘𝑆) ∈ (ℤ≥‘𝐿) → (0..^𝐿) ⊆ (0..^(♯‘𝑆))) | |
15 | 13, 14 | syl 17 | . . . . 5 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(♯‘𝑆))) → (0..^𝐿) ⊆ (0..^(♯‘𝑆))) |
16 | 11, 15 | sstrd 3976 | . . . 4 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(♯‘𝑆))) → (𝐹..^𝐿) ⊆ (0..^(♯‘𝑆))) |
17 | wrddm 13867 | . . . . 5 ⊢ (𝑆 ∈ Word 𝐴 → dom 𝑆 = (0..^(♯‘𝑆))) | |
18 | 17 | 3ad2ant1 1129 | . . . 4 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(♯‘𝑆))) → dom 𝑆 = (0..^(♯‘𝑆))) |
19 | 16, 18 | sseqtrrd 4007 | . . 3 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(♯‘𝑆))) → (𝐹..^𝐿) ⊆ dom 𝑆) |
20 | 19 | iftrued 4474 | . 2 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(♯‘𝑆))) → if((𝐹..^𝐿) ⊆ dom 𝑆, (𝑥 ∈ (0..^(𝐿 − 𝐹)) ↦ (𝑆‘(𝑥 + 𝐹))), ∅) = (𝑥 ∈ (0..^(𝐿 − 𝐹)) ↦ (𝑆‘(𝑥 + 𝐹)))) |
21 | 7, 20 | eqtrd 2856 | 1 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(♯‘𝑆))) → (𝑆 substr 〈𝐹, 𝐿〉) = (𝑥 ∈ (0..^(𝐿 − 𝐹)) ↦ (𝑆‘(𝑥 + 𝐹)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ⊆ wss 3935 ∅c0 4290 ifcif 4466 〈cop 4572 ↦ cmpt 5145 dom cdm 5554 ‘cfv 6354 (class class class)co 7155 0cc0 10536 + caddc 10539 − cmin 10869 ℤcz 11980 ℤ≥cuz 12242 ...cfz 12891 ..^cfzo 13032 ♯chash 13689 Word cword 13860 substr csubstr 14001 |
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 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 |
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 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-int 4876 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-1st 7688 df-2nd 7689 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-1o 8101 df-er 8288 df-en 8509 df-dom 8510 df-sdom 8511 df-fin 8512 df-card 9367 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-nn 11638 df-n0 11897 df-z 11981 df-uz 12243 df-fz 12892 df-fzo 13033 df-hash 13690 df-word 13861 df-substr 14002 |
This theorem is referenced by: swrdlen 14008 swrdfv 14009 swrdwrdsymb 14023 pfxmpt 14039 swrdswrd 14066 swrdrn2 30628 swrdrn3 30629 cshw1s2 30634 |
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