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Mirrors > Home > MPE Home > Th. List > swrdccatin2d | Structured version Visualization version GIF version |
Description: The subword of a concatenation of two words within the second of the concatenated words. (Contributed by AV, 31-May-2018.) (Revised by Mario Carneiro/AV, 21-Oct-2018.) |
Ref | Expression |
---|---|
swrdccatind.l | ⊢ (𝜑 → (♯‘𝐴) = 𝐿) |
swrdccatind.w | ⊢ (𝜑 → (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉)) |
swrdccatin2d.1 | ⊢ (𝜑 → 𝑀 ∈ (𝐿...𝑁)) |
swrdccatin2d.2 | ⊢ (𝜑 → 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵)))) |
Ref | Expression |
---|---|
swrdccatin2d | ⊢ (𝜑 → ((𝐴 ++ 𝐵) substr 〈𝑀, 𝑁〉) = (𝐵 substr 〈(𝑀 − 𝐿), (𝑁 − 𝐿)〉)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | swrdccatind.l | . 2 ⊢ (𝜑 → (♯‘𝐴) = 𝐿) | |
2 | swrdccatind.w | . . . . . . 7 ⊢ (𝜑 → (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉)) | |
3 | 2 | adantl 484 | . . . . . 6 ⊢ (((♯‘𝐴) = 𝐿 ∧ 𝜑) → (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉)) |
4 | swrdccatin2d.1 | . . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ (𝐿...𝑁)) | |
5 | swrdccatin2d.2 | . . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵)))) | |
6 | 4, 5 | jca 514 | . . . . . . . 8 ⊢ (𝜑 → (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) |
7 | 6 | adantl 484 | . . . . . . 7 ⊢ (((♯‘𝐴) = 𝐿 ∧ 𝜑) → (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) |
8 | oveq1 7162 | . . . . . . . . . 10 ⊢ ((♯‘𝐴) = 𝐿 → ((♯‘𝐴)...𝑁) = (𝐿...𝑁)) | |
9 | 8 | eleq2d 2898 | . . . . . . . . 9 ⊢ ((♯‘𝐴) = 𝐿 → (𝑀 ∈ ((♯‘𝐴)...𝑁) ↔ 𝑀 ∈ (𝐿...𝑁))) |
10 | id 22 | . . . . . . . . . . 11 ⊢ ((♯‘𝐴) = 𝐿 → (♯‘𝐴) = 𝐿) | |
11 | oveq1 7162 | . . . . . . . . . . 11 ⊢ ((♯‘𝐴) = 𝐿 → ((♯‘𝐴) + (♯‘𝐵)) = (𝐿 + (♯‘𝐵))) | |
12 | 10, 11 | oveq12d 7173 | . . . . . . . . . 10 ⊢ ((♯‘𝐴) = 𝐿 → ((♯‘𝐴)...((♯‘𝐴) + (♯‘𝐵))) = (𝐿...(𝐿 + (♯‘𝐵)))) |
13 | 12 | eleq2d 2898 | . . . . . . . . 9 ⊢ ((♯‘𝐴) = 𝐿 → (𝑁 ∈ ((♯‘𝐴)...((♯‘𝐴) + (♯‘𝐵))) ↔ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) |
14 | 9, 13 | anbi12d 632 | . . . . . . . 8 ⊢ ((♯‘𝐴) = 𝐿 → ((𝑀 ∈ ((♯‘𝐴)...𝑁) ∧ 𝑁 ∈ ((♯‘𝐴)...((♯‘𝐴) + (♯‘𝐵)))) ↔ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵)))))) |
15 | 14 | adantr 483 | . . . . . . 7 ⊢ (((♯‘𝐴) = 𝐿 ∧ 𝜑) → ((𝑀 ∈ ((♯‘𝐴)...𝑁) ∧ 𝑁 ∈ ((♯‘𝐴)...((♯‘𝐴) + (♯‘𝐵)))) ↔ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵)))))) |
16 | 7, 15 | mpbird 259 | . . . . . 6 ⊢ (((♯‘𝐴) = 𝐿 ∧ 𝜑) → (𝑀 ∈ ((♯‘𝐴)...𝑁) ∧ 𝑁 ∈ ((♯‘𝐴)...((♯‘𝐴) + (♯‘𝐵))))) |
17 | 3, 16 | jca 514 | . . . . 5 ⊢ (((♯‘𝐴) = 𝐿 ∧ 𝜑) → ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ ((♯‘𝐴)...𝑁) ∧ 𝑁 ∈ ((♯‘𝐴)...((♯‘𝐴) + (♯‘𝐵)))))) |
18 | 17 | ex 415 | . . . 4 ⊢ ((♯‘𝐴) = 𝐿 → (𝜑 → ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ ((♯‘𝐴)...𝑁) ∧ 𝑁 ∈ ((♯‘𝐴)...((♯‘𝐴) + (♯‘𝐵))))))) |
19 | eqid 2821 | . . . . . 6 ⊢ (♯‘𝐴) = (♯‘𝐴) | |
20 | 19 | swrdccatin2 14090 | . . . . 5 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → ((𝑀 ∈ ((♯‘𝐴)...𝑁) ∧ 𝑁 ∈ ((♯‘𝐴)...((♯‘𝐴) + (♯‘𝐵)))) → ((𝐴 ++ 𝐵) substr 〈𝑀, 𝑁〉) = (𝐵 substr 〈(𝑀 − (♯‘𝐴)), (𝑁 − (♯‘𝐴))〉))) |
21 | 20 | imp 409 | . . . 4 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ ((♯‘𝐴)...𝑁) ∧ 𝑁 ∈ ((♯‘𝐴)...((♯‘𝐴) + (♯‘𝐵))))) → ((𝐴 ++ 𝐵) substr 〈𝑀, 𝑁〉) = (𝐵 substr 〈(𝑀 − (♯‘𝐴)), (𝑁 − (♯‘𝐴))〉)) |
22 | 18, 21 | syl6 35 | . . 3 ⊢ ((♯‘𝐴) = 𝐿 → (𝜑 → ((𝐴 ++ 𝐵) substr 〈𝑀, 𝑁〉) = (𝐵 substr 〈(𝑀 − (♯‘𝐴)), (𝑁 − (♯‘𝐴))〉))) |
23 | oveq2 7163 | . . . . . 6 ⊢ ((♯‘𝐴) = 𝐿 → (𝑀 − (♯‘𝐴)) = (𝑀 − 𝐿)) | |
24 | oveq2 7163 | . . . . . 6 ⊢ ((♯‘𝐴) = 𝐿 → (𝑁 − (♯‘𝐴)) = (𝑁 − 𝐿)) | |
25 | 23, 24 | opeq12d 4810 | . . . . 5 ⊢ ((♯‘𝐴) = 𝐿 → 〈(𝑀 − (♯‘𝐴)), (𝑁 − (♯‘𝐴))〉 = 〈(𝑀 − 𝐿), (𝑁 − 𝐿)〉) |
26 | 25 | oveq2d 7171 | . . . 4 ⊢ ((♯‘𝐴) = 𝐿 → (𝐵 substr 〈(𝑀 − (♯‘𝐴)), (𝑁 − (♯‘𝐴))〉) = (𝐵 substr 〈(𝑀 − 𝐿), (𝑁 − 𝐿)〉)) |
27 | 26 | eqeq2d 2832 | . . 3 ⊢ ((♯‘𝐴) = 𝐿 → (((𝐴 ++ 𝐵) substr 〈𝑀, 𝑁〉) = (𝐵 substr 〈(𝑀 − (♯‘𝐴)), (𝑁 − (♯‘𝐴))〉) ↔ ((𝐴 ++ 𝐵) substr 〈𝑀, 𝑁〉) = (𝐵 substr 〈(𝑀 − 𝐿), (𝑁 − 𝐿)〉))) |
28 | 22, 27 | sylibd 241 | . 2 ⊢ ((♯‘𝐴) = 𝐿 → (𝜑 → ((𝐴 ++ 𝐵) substr 〈𝑀, 𝑁〉) = (𝐵 substr 〈(𝑀 − 𝐿), (𝑁 − 𝐿)〉))) |
29 | 1, 28 | mpcom 38 | 1 ⊢ (𝜑 → ((𝐴 ++ 𝐵) substr 〈𝑀, 𝑁〉) = (𝐵 substr 〈(𝑀 − 𝐿), (𝑁 − 𝐿)〉)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 〈cop 4572 ‘cfv 6354 (class class class)co 7155 + caddc 10539 − cmin 10869 ...cfz 12891 ♯chash 13689 Word cword 13860 ++ cconcat 13921 substr csubstr 14001 |
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 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 |
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 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-int 4876 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-1st 7688 df-2nd 7689 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-1o 8101 df-oadd 8105 df-er 8288 df-en 8509 df-dom 8510 df-sdom 8511 df-fin 8512 df-card 9367 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-nn 11638 df-n0 11897 df-z 11981 df-uz 12243 df-fz 12892 df-fzo 13033 df-hash 13690 df-word 13861 df-concat 13922 df-substr 14002 |
This theorem is referenced by: (None) |
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