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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 1061 | . . 3 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(#‘𝑆))) → 𝑆 ∈ Word 𝐴) | |
2 | elfzelz 12342 | . . . 4 ⊢ (𝐹 ∈ (0...𝐿) → 𝐹 ∈ ℤ) | |
3 | 2 | 3ad2ant2 1083 | . . 3 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(#‘𝑆))) → 𝐹 ∈ ℤ) |
4 | elfzelz 12342 | . . . 4 ⊢ (𝐿 ∈ (0...(#‘𝑆)) → 𝐿 ∈ ℤ) | |
5 | 4 | 3ad2ant3 1084 | . . 3 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(#‘𝑆))) → 𝐿 ∈ ℤ) |
6 | swrdval 13417 | . . 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 1326 | . 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 12338 | . . . . . . 7 ⊢ (𝐹 ∈ (0...𝐿) → 𝐹 ∈ (ℤ≥‘0)) | |
9 | 8 | 3ad2ant2 1083 | . . . . . 6 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(#‘𝑆))) → 𝐹 ∈ (ℤ≥‘0)) |
10 | fzoss1 12495 | . . . . . 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 12339 | . . . . . . 7 ⊢ (𝐿 ∈ (0...(#‘𝑆)) → (#‘𝑆) ∈ (ℤ≥‘𝐿)) | |
13 | 12 | 3ad2ant3 1084 | . . . . . 6 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(#‘𝑆))) → (#‘𝑆) ∈ (ℤ≥‘𝐿)) |
14 | fzoss2 12496 | . . . . . 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 3613 | . . . 4 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(#‘𝑆))) → (𝐹..^𝐿) ⊆ (0..^(#‘𝑆))) |
17 | wrddm 13312 | . . . . 5 ⊢ (𝑆 ∈ Word 𝐴 → dom 𝑆 = (0..^(#‘𝑆))) | |
18 | 17 | 3ad2ant1 1082 | . . . 4 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(#‘𝑆))) → dom 𝑆 = (0..^(#‘𝑆))) |
19 | 16, 18 | sseqtr4d 3642 | . . 3 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(#‘𝑆))) → (𝐹..^𝐿) ⊆ dom 𝑆) |
20 | 19 | iftrued 4094 | . 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 2656 | 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 1037 = wceq 1483 ∈ wcel 1990 ⊆ wss 3574 ∅c0 3915 ifcif 4086 〈cop 4183 ↦ cmpt 4729 dom cdm 5114 ‘cfv 5888 (class class class)co 6650 0cc0 9936 + caddc 9939 − cmin 10266 ℤcz 11377 ℤ≥cuz 11687 ...cfz 12326 ..^cfzo 12465 #chash 13117 Word cword 13291 substr csubstr 13295 |
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-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-substr 13303 |
This theorem is referenced by: swrd0val 13421 swrdlen 13423 swrdfv 13424 swrdswrd 13460 pfxmpt 41387 |
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