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Mirrors > Home > MPE Home > Th. List > pfxccatin12d | Structured version Visualization version GIF version |
Description: The subword of a concatenation of two words within both of the concatenated words. (Contributed by AV, 31-May-2018.) (Revised by AV, 10-May-2020.) |
Ref | Expression |
---|---|
swrdccatind.l | ⊢ (𝜑 → (♯‘𝐴) = 𝐿) |
swrdccatind.w | ⊢ (𝜑 → (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉)) |
pfxccatin12d.m | ⊢ (𝜑 → 𝑀 ∈ (0...𝐿)) |
pfxccatin12d.n | ⊢ (𝜑 → 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵)))) |
Ref | Expression |
---|---|
pfxccatin12d | ⊢ (𝜑 → ((𝐴 ++ 𝐵) substr 〈𝑀, 𝑁〉) = ((𝐴 substr 〈𝑀, 𝐿〉) ++ (𝐵 prefix (𝑁 − 𝐿)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | swrdccatind.w | . . 3 ⊢ (𝜑 → (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉)) | |
2 | pfxccatin12d.m | . . . 4 ⊢ (𝜑 → 𝑀 ∈ (0...𝐿)) | |
3 | pfxccatin12d.n | . . . 4 ⊢ (𝜑 → 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵)))) | |
4 | swrdccatind.l | . . . . . . 7 ⊢ (𝜑 → (♯‘𝐴) = 𝐿) | |
5 | 4 | oveq2d 7367 | . . . . . 6 ⊢ (𝜑 → (0...(♯‘𝐴)) = (0...𝐿)) |
6 | 5 | eleq2d 2823 | . . . . 5 ⊢ (𝜑 → (𝑀 ∈ (0...(♯‘𝐴)) ↔ 𝑀 ∈ (0...𝐿))) |
7 | 4 | oveq1d 7366 | . . . . . . 7 ⊢ (𝜑 → ((♯‘𝐴) + (♯‘𝐵)) = (𝐿 + (♯‘𝐵))) |
8 | 4, 7 | oveq12d 7369 | . . . . . 6 ⊢ (𝜑 → ((♯‘𝐴)...((♯‘𝐴) + (♯‘𝐵))) = (𝐿...(𝐿 + (♯‘𝐵)))) |
9 | 8 | eleq2d 2823 | . . . . 5 ⊢ (𝜑 → (𝑁 ∈ ((♯‘𝐴)...((♯‘𝐴) + (♯‘𝐵))) ↔ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) |
10 | 6, 9 | anbi12d 631 | . . . 4 ⊢ (𝜑 → ((𝑀 ∈ (0...(♯‘𝐴)) ∧ 𝑁 ∈ ((♯‘𝐴)...((♯‘𝐴) + (♯‘𝐵)))) ↔ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵)))))) |
11 | 2, 3, 10 | mpbir2and 711 | . . 3 ⊢ (𝜑 → (𝑀 ∈ (0...(♯‘𝐴)) ∧ 𝑁 ∈ ((♯‘𝐴)...((♯‘𝐴) + (♯‘𝐵))))) |
12 | eqid 2737 | . . . 4 ⊢ (♯‘𝐴) = (♯‘𝐴) | |
13 | 12 | pfxccatin12 14579 | . . 3 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → ((𝑀 ∈ (0...(♯‘𝐴)) ∧ 𝑁 ∈ ((♯‘𝐴)...((♯‘𝐴) + (♯‘𝐵)))) → ((𝐴 ++ 𝐵) substr 〈𝑀, 𝑁〉) = ((𝐴 substr 〈𝑀, (♯‘𝐴)〉) ++ (𝐵 prefix (𝑁 − (♯‘𝐴)))))) |
14 | 1, 11, 13 | sylc 65 | . 2 ⊢ (𝜑 → ((𝐴 ++ 𝐵) substr 〈𝑀, 𝑁〉) = ((𝐴 substr 〈𝑀, (♯‘𝐴)〉) ++ (𝐵 prefix (𝑁 − (♯‘𝐴))))) |
15 | 4 | opeq2d 4835 | . . . 4 ⊢ (𝜑 → 〈𝑀, (♯‘𝐴)〉 = 〈𝑀, 𝐿〉) |
16 | 15 | oveq2d 7367 | . . 3 ⊢ (𝜑 → (𝐴 substr 〈𝑀, (♯‘𝐴)〉) = (𝐴 substr 〈𝑀, 𝐿〉)) |
17 | 4 | oveq2d 7367 | . . . 4 ⊢ (𝜑 → (𝑁 − (♯‘𝐴)) = (𝑁 − 𝐿)) |
18 | 17 | oveq2d 7367 | . . 3 ⊢ (𝜑 → (𝐵 prefix (𝑁 − (♯‘𝐴))) = (𝐵 prefix (𝑁 − 𝐿))) |
19 | 16, 18 | oveq12d 7369 | . 2 ⊢ (𝜑 → ((𝐴 substr 〈𝑀, (♯‘𝐴)〉) ++ (𝐵 prefix (𝑁 − (♯‘𝐴)))) = ((𝐴 substr 〈𝑀, 𝐿〉) ++ (𝐵 prefix (𝑁 − 𝐿)))) |
20 | 14, 19 | eqtrd 2777 | 1 ⊢ (𝜑 → ((𝐴 ++ 𝐵) substr 〈𝑀, 𝑁〉) = ((𝐴 substr 〈𝑀, 𝐿〉) ++ (𝐵 prefix (𝑁 − 𝐿)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 〈cop 4590 ‘cfv 6493 (class class class)co 7351 0cc0 11009 + caddc 11012 − cmin 11343 ...cfz 13378 ♯chash 14184 Word cword 14356 ++ cconcat 14412 substr csubstr 14486 prefix cpfx 14516 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-int 4906 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7307 df-ov 7354 df-oprab 7355 df-mpo 7356 df-om 7795 df-1st 7913 df-2nd 7914 df-frecs 8204 df-wrecs 8235 df-recs 8309 df-rdg 8348 df-1o 8404 df-er 8606 df-en 8842 df-dom 8843 df-sdom 8844 df-fin 8845 df-card 9833 df-pnf 11149 df-mnf 11150 df-xr 11151 df-ltxr 11152 df-le 11153 df-sub 11345 df-neg 11346 df-nn 12112 df-n0 12372 df-z 12458 df-uz 12722 df-fz 13379 df-fzo 13522 df-hash 14185 df-word 14357 df-concat 14413 df-substr 14487 df-pfx 14517 |
This theorem is referenced by: (None) |
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