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Mirrors > Home > MPE Home > Th. List > splid | Structured version Visualization version GIF version |
Description: Splicing a subword for the same subword makes no difference. (Contributed by Stefan O'Rear, 20-Aug-2015.) (Proof shortened by AV, 14-Oct-2022.) |
Ref | Expression |
---|---|
splid | ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...(♯‘𝑆)))) → (𝑆 splice ⟨𝑋, 𝑌, (𝑆 substr ⟨𝑋, 𝑌⟩)⟩) = 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ovex 7395 | . . 3 ⊢ (𝑆 substr ⟨𝑋, 𝑌⟩) ∈ V | |
2 | splval 14646 | . . 3 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...(♯‘𝑆)) ∧ (𝑆 substr ⟨𝑋, 𝑌⟩) ∈ V)) → (𝑆 splice ⟨𝑋, 𝑌, (𝑆 substr ⟨𝑋, 𝑌⟩)⟩) = (((𝑆 prefix 𝑋) ++ (𝑆 substr ⟨𝑋, 𝑌⟩)) ++ (𝑆 substr ⟨𝑌, (♯‘𝑆)⟩))) | |
3 | 1, 2 | mp3anr3 1461 | . 2 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...(♯‘𝑆)))) → (𝑆 splice ⟨𝑋, 𝑌, (𝑆 substr ⟨𝑋, 𝑌⟩)⟩) = (((𝑆 prefix 𝑋) ++ (𝑆 substr ⟨𝑋, 𝑌⟩)) ++ (𝑆 substr ⟨𝑌, (♯‘𝑆)⟩))) |
4 | ccatpfx 14596 | . . . . 5 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...(♯‘𝑆))) → ((𝑆 prefix 𝑋) ++ (𝑆 substr ⟨𝑋, 𝑌⟩)) = (𝑆 prefix 𝑌)) | |
5 | 4 | 3expb 1121 | . . . 4 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...(♯‘𝑆)))) → ((𝑆 prefix 𝑋) ++ (𝑆 substr ⟨𝑋, 𝑌⟩)) = (𝑆 prefix 𝑌)) |
6 | 5 | oveq1d 7377 | . . 3 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...(♯‘𝑆)))) → (((𝑆 prefix 𝑋) ++ (𝑆 substr ⟨𝑋, 𝑌⟩)) ++ (𝑆 substr ⟨𝑌, (♯‘𝑆)⟩)) = ((𝑆 prefix 𝑌) ++ (𝑆 substr ⟨𝑌, (♯‘𝑆)⟩))) |
7 | simpl 484 | . . . . 5 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...(♯‘𝑆)))) → 𝑆 ∈ Word 𝐴) | |
8 | simprr 772 | . . . . 5 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...(♯‘𝑆)))) → 𝑌 ∈ (0...(♯‘𝑆))) | |
9 | elfzuz2 13453 | . . . . . . 7 ⊢ (𝑌 ∈ (0...(♯‘𝑆)) → (♯‘𝑆) ∈ (ℤ≥‘0)) | |
10 | 9 | ad2antll 728 | . . . . . 6 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...(♯‘𝑆)))) → (♯‘𝑆) ∈ (ℤ≥‘0)) |
11 | eluzfz2 13456 | . . . . . 6 ⊢ ((♯‘𝑆) ∈ (ℤ≥‘0) → (♯‘𝑆) ∈ (0...(♯‘𝑆))) | |
12 | 10, 11 | syl 17 | . . . . 5 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...(♯‘𝑆)))) → (♯‘𝑆) ∈ (0...(♯‘𝑆))) |
13 | ccatpfx 14596 | . . . . 5 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑌 ∈ (0...(♯‘𝑆)) ∧ (♯‘𝑆) ∈ (0...(♯‘𝑆))) → ((𝑆 prefix 𝑌) ++ (𝑆 substr ⟨𝑌, (♯‘𝑆)⟩)) = (𝑆 prefix (♯‘𝑆))) | |
14 | 7, 8, 12, 13 | syl3anc 1372 | . . . 4 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...(♯‘𝑆)))) → ((𝑆 prefix 𝑌) ++ (𝑆 substr ⟨𝑌, (♯‘𝑆)⟩)) = (𝑆 prefix (♯‘𝑆))) |
15 | pfxid 14579 | . . . . 5 ⊢ (𝑆 ∈ Word 𝐴 → (𝑆 prefix (♯‘𝑆)) = 𝑆) | |
16 | 15 | adantr 482 | . . . 4 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...(♯‘𝑆)))) → (𝑆 prefix (♯‘𝑆)) = 𝑆) |
17 | 14, 16 | eqtrd 2777 | . . 3 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...(♯‘𝑆)))) → ((𝑆 prefix 𝑌) ++ (𝑆 substr ⟨𝑌, (♯‘𝑆)⟩)) = 𝑆) |
18 | 6, 17 | eqtrd 2777 | . 2 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...(♯‘𝑆)))) → (((𝑆 prefix 𝑋) ++ (𝑆 substr ⟨𝑋, 𝑌⟩)) ++ (𝑆 substr ⟨𝑌, (♯‘𝑆)⟩)) = 𝑆) |
19 | 3, 18 | eqtrd 2777 | 1 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...(♯‘𝑆)))) → (𝑆 splice ⟨𝑋, 𝑌, (𝑆 substr ⟨𝑋, 𝑌⟩)⟩) = 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 Vcvv 3448 ⟨cop 4597 ⟨cotp 4599 ‘cfv 6501 (class class class)co 7362 0cc0 11058 ℤ≥cuz 12770 ...cfz 13431 ♯chash 14237 Word cword 14409 ++ cconcat 14465 substr csubstr 14535 prefix cpfx 14565 splice csplice 14644 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-cnex 11114 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 ax-pre-mulgt0 11135 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-ot 4600 df-uni 4871 df-int 4913 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-1st 7926 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-1o 8417 df-er 8655 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-card 9882 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-nn 12161 df-n0 12421 df-z 12507 df-uz 12771 df-fz 13432 df-fzo 13575 df-hash 14238 df-word 14410 df-concat 14466 df-substr 14536 df-pfx 14566 df-splice 14645 |
This theorem is referenced by: psgnunilem2 19284 |
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