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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 7166 | . . . . . 6 ⊢ (𝜑 → (0...(♯‘𝐴)) = (0...𝐿)) |
6 | 5 | eleq2d 2898 | . . . . 5 ⊢ (𝜑 → (𝑀 ∈ (0...(♯‘𝐴)) ↔ 𝑀 ∈ (0...𝐿))) |
7 | 4 | oveq1d 7165 | . . . . . . 7 ⊢ (𝜑 → ((♯‘𝐴) + (♯‘𝐵)) = (𝐿 + (♯‘𝐵))) |
8 | 4, 7 | oveq12d 7168 | . . . . . 6 ⊢ (𝜑 → ((♯‘𝐴)...((♯‘𝐴) + (♯‘𝐵))) = (𝐿...(𝐿 + (♯‘𝐵)))) |
9 | 8 | eleq2d 2898 | . . . . 5 ⊢ (𝜑 → (𝑁 ∈ ((♯‘𝐴)...((♯‘𝐴) + (♯‘𝐵))) ↔ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) |
10 | 6, 9 | anbi12d 632 | . . . 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 2821 | . . . 4 ⊢ (♯‘𝐴) = (♯‘𝐴) | |
13 | 12 | pfxccatin12 14089 | . . 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 4804 | . . . 4 ⊢ (𝜑 → 〈𝑀, (♯‘𝐴)〉 = 〈𝑀, 𝐿〉) |
16 | 15 | oveq2d 7166 | . . 3 ⊢ (𝜑 → (𝐴 substr 〈𝑀, (♯‘𝐴)〉) = (𝐴 substr 〈𝑀, 𝐿〉)) |
17 | 4 | oveq2d 7166 | . . . 4 ⊢ (𝜑 → (𝑁 − (♯‘𝐴)) = (𝑁 − 𝐿)) |
18 | 17 | oveq2d 7166 | . . 3 ⊢ (𝜑 → (𝐵 prefix (𝑁 − (♯‘𝐴))) = (𝐵 prefix (𝑁 − 𝐿))) |
19 | 16, 18 | oveq12d 7168 | . 2 ⊢ (𝜑 → ((𝐴 substr 〈𝑀, (♯‘𝐴)〉) ++ (𝐵 prefix (𝑁 − (♯‘𝐴)))) = ((𝐴 substr 〈𝑀, 𝐿〉) ++ (𝐵 prefix (𝑁 − 𝐿)))) |
20 | 14, 19 | eqtrd 2856 | 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 398 = wceq 1533 ∈ wcel 2110 〈cop 4567 ‘cfv 6350 (class class class)co 7150 0cc0 10531 + caddc 10534 − cmin 10864 ...cfz 12886 ♯chash 13684 Word cword 13855 ++ cconcat 13916 substr csubstr 13996 prefix cpfx 14026 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-n0 11892 df-z 11976 df-uz 12238 df-fz 12887 df-fzo 13028 df-hash 13685 df-word 13856 df-concat 13917 df-substr 13997 df-pfx 14027 |
This theorem is referenced by: (None) |
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