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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 12213 | . . . . . . . 8 ⊢ (𝐿 ∈ (0...(#‘𝑆)) → 𝐿 ∈ ℤ) | |
2 | 1 | adantl 481 | . . . . . . 7 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝐿 ∈ (0...(#‘𝑆))) → 𝐿 ∈ ℤ) |
3 | 2 | zcnd 11359 | . . . . . 6 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝐿 ∈ (0...(#‘𝑆))) → 𝐿 ∈ ℂ) |
4 | 3 | subid1d 10260 | . . . . 5 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝐿 ∈ (0...(#‘𝑆))) → (𝐿 − 0) = 𝐿) |
5 | 4 | oveq2d 6565 | . . . 4 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝐿 ∈ (0...(#‘𝑆))) → (0..^(𝐿 − 0)) = (0..^𝐿)) |
6 | 5 | mpteq1d 4666 | . . 3 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝐿 ∈ (0...(#‘𝑆))) → (𝑥 ∈ (0..^(𝐿 − 0)) ↦ (𝑆‘(𝑥 + 0))) = (𝑥 ∈ (0..^𝐿) ↦ (𝑆‘(𝑥 + 0)))) |
7 | elfzoelz 12339 | . . . . . . . 8 ⊢ (𝑥 ∈ (0..^𝐿) → 𝑥 ∈ ℤ) | |
8 | 7 | zcnd 11359 | . . . . . . 7 ⊢ (𝑥 ∈ (0..^𝐿) → 𝑥 ∈ ℂ) |
9 | 8 | addid1d 10115 | . . . . . 6 ⊢ (𝑥 ∈ (0..^𝐿) → (𝑥 + 0) = 𝑥) |
10 | 9 | fveq2d 6107 | . . . . 5 ⊢ (𝑥 ∈ (0..^𝐿) → (𝑆‘(𝑥 + 0)) = (𝑆‘𝑥)) |
11 | 10 | adantl 481 | . . . 4 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝐿 ∈ (0...(#‘𝑆))) ∧ 𝑥 ∈ (0..^𝐿)) → (𝑆‘(𝑥 + 0)) = (𝑆‘𝑥)) |
12 | 11 | mpteq2dva 4672 | . . 3 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝐿 ∈ (0...(#‘𝑆))) → (𝑥 ∈ (0..^𝐿) ↦ (𝑆‘(𝑥 + 0))) = (𝑥 ∈ (0..^𝐿) ↦ (𝑆‘𝑥))) |
13 | 6, 12 | eqtrd 2644 | . 2 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝐿 ∈ (0...(#‘𝑆))) → (𝑥 ∈ (0..^(𝐿 − 0)) ↦ (𝑆‘(𝑥 + 0))) = (𝑥 ∈ (0..^𝐿) ↦ (𝑆‘𝑥))) |
14 | simpl 472 | . . 3 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝐿 ∈ (0...(#‘𝑆))) → 𝑆 ∈ Word 𝐴) | |
15 | elfzuz 12209 | . . . . 5 ⊢ (𝐿 ∈ (0...(#‘𝑆)) → 𝐿 ∈ (ℤ≥‘0)) | |
16 | 15 | adantl 481 | . . . 4 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝐿 ∈ (0...(#‘𝑆))) → 𝐿 ∈ (ℤ≥‘0)) |
17 | eluzfz1 12219 | . . . 4 ⊢ (𝐿 ∈ (ℤ≥‘0) → 0 ∈ (0...𝐿)) | |
18 | 16, 17 | syl 17 | . . 3 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝐿 ∈ (0...(#‘𝑆))) → 0 ∈ (0...𝐿)) |
19 | simpr 476 | . . 3 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝐿 ∈ (0...(#‘𝑆))) → 𝐿 ∈ (0...(#‘𝑆))) | |
20 | swrdval2 13272 | . . 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 1318 | . 2 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝐿 ∈ (0...(#‘𝑆))) → (𝑆 substr 〈0, 𝐿〉) = (𝑥 ∈ (0..^(𝐿 − 0)) ↦ (𝑆‘(𝑥 + 0)))) |
22 | wrdf 13165 | . . . 4 ⊢ (𝑆 ∈ Word 𝐴 → 𝑆:(0..^(#‘𝑆))⟶𝐴) | |
23 | 22 | adantr 480 | . . 3 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝐿 ∈ (0...(#‘𝑆))) → 𝑆:(0..^(#‘𝑆))⟶𝐴) |
24 | elfzuz3 12210 | . . . . 5 ⊢ (𝐿 ∈ (0...(#‘𝑆)) → (#‘𝑆) ∈ (ℤ≥‘𝐿)) | |
25 | 24 | adantl 481 | . . . 4 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝐿 ∈ (0...(#‘𝑆))) → (#‘𝑆) ∈ (ℤ≥‘𝐿)) |
26 | fzoss2 12365 | . . . 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 6160 | . 2 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝐿 ∈ (0...(#‘𝑆))) → (𝑆 ↾ (0..^𝐿)) = (𝑥 ∈ (0..^𝐿) ↦ (𝑆‘𝑥))) |
29 | 13, 21, 28 | 3eqtr4d 2654 | 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 1475 ∈ wcel 1977 ⊆ wss 3540 〈cop 4131 ↦ cmpt 4643 ↾ cres 5040 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 0cc0 9815 + caddc 9818 − cmin 10145 ℤcz 11254 ℤ≥cuz 11563 ...cfz 12197 ..^cfzo 12334 #chash 12979 Word cword 13146 substr csubstr 13150 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-fzo 12335 df-hash 12980 df-word 13154 df-substr 13158 |
This theorem is referenced by: swrd0len 13274 swrdccat1 13309 psgnunilem5 17737 efgsres 17974 efgredlemd 17980 efgredlem 17983 wwlkm1edg 26263 iwrdsplit 29776 wrdsplex 29944 signsvtn0 29973 1wlkreslem0 40877 wwlksm1edg 41078 |
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