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Mirrors > Home > MPE Home > Th. List > swrd0val | Structured version Visualization version GIF version |
Description: Value of the subword extractor for left-anchored subwords. (Contributed by Stefan O'Rear, 24-Aug-2015.) |
Ref | Expression |
---|---|
swrd0val | ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝐿 ∈ (0...(♯‘𝑆))) → (𝑆 substr 〈0, 𝐿〉) = (𝑆 ↾ (0..^𝐿))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfzelz 12527 | . . . . . . . 8 ⊢ (𝐿 ∈ (0...(♯‘𝑆)) → 𝐿 ∈ ℤ) | |
2 | 1 | adantl 473 | . . . . . . 7 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝐿 ∈ (0...(♯‘𝑆))) → 𝐿 ∈ ℤ) |
3 | 2 | zcnd 11667 | . . . . . 6 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝐿 ∈ (0...(♯‘𝑆))) → 𝐿 ∈ ℂ) |
4 | 3 | subid1d 10565 | . . . . 5 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝐿 ∈ (0...(♯‘𝑆))) → (𝐿 − 0) = 𝐿) |
5 | 4 | oveq2d 6821 | . . . 4 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝐿 ∈ (0...(♯‘𝑆))) → (0..^(𝐿 − 0)) = (0..^𝐿)) |
6 | 5 | mpteq1d 4882 | . . 3 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝐿 ∈ (0...(♯‘𝑆))) → (𝑥 ∈ (0..^(𝐿 − 0)) ↦ (𝑆‘(𝑥 + 0))) = (𝑥 ∈ (0..^𝐿) ↦ (𝑆‘(𝑥 + 0)))) |
7 | elfzoelz 12656 | . . . . . . . 8 ⊢ (𝑥 ∈ (0..^𝐿) → 𝑥 ∈ ℤ) | |
8 | 7 | zcnd 11667 | . . . . . . 7 ⊢ (𝑥 ∈ (0..^𝐿) → 𝑥 ∈ ℂ) |
9 | 8 | addid1d 10420 | . . . . . 6 ⊢ (𝑥 ∈ (0..^𝐿) → (𝑥 + 0) = 𝑥) |
10 | 9 | fveq2d 6348 | . . . . 5 ⊢ (𝑥 ∈ (0..^𝐿) → (𝑆‘(𝑥 + 0)) = (𝑆‘𝑥)) |
11 | 10 | adantl 473 | . . . 4 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝐿 ∈ (0...(♯‘𝑆))) ∧ 𝑥 ∈ (0..^𝐿)) → (𝑆‘(𝑥 + 0)) = (𝑆‘𝑥)) |
12 | 11 | mpteq2dva 4888 | . . 3 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝐿 ∈ (0...(♯‘𝑆))) → (𝑥 ∈ (0..^𝐿) ↦ (𝑆‘(𝑥 + 0))) = (𝑥 ∈ (0..^𝐿) ↦ (𝑆‘𝑥))) |
13 | 6, 12 | eqtrd 2786 | . 2 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝐿 ∈ (0...(♯‘𝑆))) → (𝑥 ∈ (0..^(𝐿 − 0)) ↦ (𝑆‘(𝑥 + 0))) = (𝑥 ∈ (0..^𝐿) ↦ (𝑆‘𝑥))) |
14 | simpl 474 | . . 3 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝐿 ∈ (0...(♯‘𝑆))) → 𝑆 ∈ Word 𝐴) | |
15 | elfzuz 12523 | . . . . 5 ⊢ (𝐿 ∈ (0...(♯‘𝑆)) → 𝐿 ∈ (ℤ≥‘0)) | |
16 | 15 | adantl 473 | . . . 4 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝐿 ∈ (0...(♯‘𝑆))) → 𝐿 ∈ (ℤ≥‘0)) |
17 | eluzfz1 12533 | . . . 4 ⊢ (𝐿 ∈ (ℤ≥‘0) → 0 ∈ (0...𝐿)) | |
18 | 16, 17 | syl 17 | . . 3 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝐿 ∈ (0...(♯‘𝑆))) → 0 ∈ (0...𝐿)) |
19 | simpr 479 | . . 3 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝐿 ∈ (0...(♯‘𝑆))) → 𝐿 ∈ (0...(♯‘𝑆))) | |
20 | swrdval2 13611 | . . 3 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 0 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(♯‘𝑆))) → (𝑆 substr 〈0, 𝐿〉) = (𝑥 ∈ (0..^(𝐿 − 0)) ↦ (𝑆‘(𝑥 + 0)))) | |
21 | 14, 18, 19, 20 | syl3anc 1473 | . 2 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝐿 ∈ (0...(♯‘𝑆))) → (𝑆 substr 〈0, 𝐿〉) = (𝑥 ∈ (0..^(𝐿 − 0)) ↦ (𝑆‘(𝑥 + 0)))) |
22 | wrdf 13488 | . . . 4 ⊢ (𝑆 ∈ Word 𝐴 → 𝑆:(0..^(♯‘𝑆))⟶𝐴) | |
23 | 22 | adantr 472 | . . 3 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝐿 ∈ (0...(♯‘𝑆))) → 𝑆:(0..^(♯‘𝑆))⟶𝐴) |
24 | elfzuz3 12524 | . . . . 5 ⊢ (𝐿 ∈ (0...(♯‘𝑆)) → (♯‘𝑆) ∈ (ℤ≥‘𝐿)) | |
25 | 24 | adantl 473 | . . . 4 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝐿 ∈ (0...(♯‘𝑆))) → (♯‘𝑆) ∈ (ℤ≥‘𝐿)) |
26 | fzoss2 12682 | . . . 4 ⊢ ((♯‘𝑆) ∈ (ℤ≥‘𝐿) → (0..^𝐿) ⊆ (0..^(♯‘𝑆))) | |
27 | 25, 26 | syl 17 | . . 3 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝐿 ∈ (0...(♯‘𝑆))) → (0..^𝐿) ⊆ (0..^(♯‘𝑆))) |
28 | 23, 27 | feqresmpt 6404 | . 2 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝐿 ∈ (0...(♯‘𝑆))) → (𝑆 ↾ (0..^𝐿)) = (𝑥 ∈ (0..^𝐿) ↦ (𝑆‘𝑥))) |
29 | 13, 21, 28 | 3eqtr4d 2796 | 1 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝐿 ∈ (0...(♯‘𝑆))) → (𝑆 substr 〈0, 𝐿〉) = (𝑆 ↾ (0..^𝐿))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1624 ∈ wcel 2131 ⊆ wss 3707 〈cop 4319 ↦ cmpt 4873 ↾ cres 5260 ⟶wf 6037 ‘cfv 6041 (class class class)co 6805 0cc0 10120 + caddc 10123 − cmin 10450 ℤcz 11561 ℤ≥cuz 11871 ...cfz 12511 ..^cfzo 12651 ♯chash 13303 Word cword 13469 substr csubstr 13473 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1863 ax-4 1878 ax-5 1980 ax-6 2046 ax-7 2082 ax-8 2133 ax-9 2140 ax-10 2160 ax-11 2175 ax-12 2188 ax-13 2383 ax-ext 2732 ax-rep 4915 ax-sep 4925 ax-nul 4933 ax-pow 4984 ax-pr 5047 ax-un 7106 ax-cnex 10176 ax-resscn 10177 ax-1cn 10178 ax-icn 10179 ax-addcl 10180 ax-addrcl 10181 ax-mulcl 10182 ax-mulrcl 10183 ax-mulcom 10184 ax-addass 10185 ax-mulass 10186 ax-distr 10187 ax-i2m1 10188 ax-1ne0 10189 ax-1rid 10190 ax-rnegex 10191 ax-rrecex 10192 ax-cnre 10193 ax-pre-lttri 10194 ax-pre-lttrn 10195 ax-pre-ltadd 10196 ax-pre-mulgt0 10197 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1627 df-ex 1846 df-nf 1851 df-sb 2039 df-eu 2603 df-mo 2604 df-clab 2739 df-cleq 2745 df-clel 2748 df-nfc 2883 df-ne 2925 df-nel 3028 df-ral 3047 df-rex 3048 df-reu 3049 df-rab 3051 df-v 3334 df-sbc 3569 df-csb 3667 df-dif 3710 df-un 3712 df-in 3714 df-ss 3721 df-pss 3723 df-nul 4051 df-if 4223 df-pw 4296 df-sn 4314 df-pr 4316 df-tp 4318 df-op 4320 df-uni 4581 df-int 4620 df-iun 4666 df-br 4797 df-opab 4857 df-mpt 4874 df-tr 4897 df-id 5166 df-eprel 5171 df-po 5179 df-so 5180 df-fr 5217 df-we 5219 df-xp 5264 df-rel 5265 df-cnv 5266 df-co 5267 df-dm 5268 df-rn 5269 df-res 5270 df-ima 5271 df-pred 5833 df-ord 5879 df-on 5880 df-lim 5881 df-suc 5882 df-iota 6004 df-fun 6043 df-fn 6044 df-f 6045 df-f1 6046 df-fo 6047 df-f1o 6048 df-fv 6049 df-riota 6766 df-ov 6808 df-oprab 6809 df-mpt2 6810 df-om 7223 df-1st 7325 df-2nd 7326 df-wrecs 7568 df-recs 7629 df-rdg 7667 df-1o 7721 df-er 7903 df-en 8114 df-dom 8115 df-sdom 8116 df-fin 8117 df-card 8947 df-pnf 10260 df-mnf 10261 df-xr 10262 df-ltxr 10263 df-le 10264 df-sub 10452 df-neg 10453 df-nn 11205 df-n0 11477 df-z 11562 df-uz 11872 df-fz 12512 df-fzo 12652 df-hash 13304 df-word 13477 df-substr 13481 |
This theorem is referenced by: swrd0len 13613 swrdccat1 13649 psgnunilem5 18106 efgsres 18343 efgredlemd 18349 efgredlem 18352 wlkreslem0 26767 wwlksm1edg 26982 iwrdsplit 30750 wrdsplex 30919 signsvtn0 30948 |
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