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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 481 | . . . . . 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 511 | . . . . . . . 8 ⊢ (𝜑 → (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) | 
| 7 | 6 | adantl 481 | . . . . . . 7 ⊢ (((♯‘𝐴) = 𝐿 ∧ 𝜑) → (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) | 
| 8 | oveq1 7439 | . . . . . . . . . 10 ⊢ ((♯‘𝐴) = 𝐿 → ((♯‘𝐴)...𝑁) = (𝐿...𝑁)) | |
| 9 | 8 | eleq2d 2826 | . . . . . . . . 9 ⊢ ((♯‘𝐴) = 𝐿 → (𝑀 ∈ ((♯‘𝐴)...𝑁) ↔ 𝑀 ∈ (𝐿...𝑁))) | 
| 10 | id 22 | . . . . . . . . . . 11 ⊢ ((♯‘𝐴) = 𝐿 → (♯‘𝐴) = 𝐿) | |
| 11 | oveq1 7439 | . . . . . . . . . . 11 ⊢ ((♯‘𝐴) = 𝐿 → ((♯‘𝐴) + (♯‘𝐵)) = (𝐿 + (♯‘𝐵))) | |
| 12 | 10, 11 | oveq12d 7450 | . . . . . . . . . 10 ⊢ ((♯‘𝐴) = 𝐿 → ((♯‘𝐴)...((♯‘𝐴) + (♯‘𝐵))) = (𝐿...(𝐿 + (♯‘𝐵)))) | 
| 13 | 12 | eleq2d 2826 | . . . . . . . . 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 480 | . . . . . . 7 ⊢ (((♯‘𝐴) = 𝐿 ∧ 𝜑) → ((𝑀 ∈ ((♯‘𝐴)...𝑁) ∧ 𝑁 ∈ ((♯‘𝐴)...((♯‘𝐴) + (♯‘𝐵)))) ↔ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵)))))) | 
| 16 | 7, 15 | mpbird 257 | . . . . . 6 ⊢ (((♯‘𝐴) = 𝐿 ∧ 𝜑) → (𝑀 ∈ ((♯‘𝐴)...𝑁) ∧ 𝑁 ∈ ((♯‘𝐴)...((♯‘𝐴) + (♯‘𝐵))))) | 
| 17 | 3, 16 | jca 511 | . . . . 5 ⊢ (((♯‘𝐴) = 𝐿 ∧ 𝜑) → ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ ((♯‘𝐴)...𝑁) ∧ 𝑁 ∈ ((♯‘𝐴)...((♯‘𝐴) + (♯‘𝐵)))))) | 
| 18 | 17 | ex 412 | . . . 4 ⊢ ((♯‘𝐴) = 𝐿 → (𝜑 → ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ ((♯‘𝐴)...𝑁) ∧ 𝑁 ∈ ((♯‘𝐴)...((♯‘𝐴) + (♯‘𝐵))))))) | 
| 19 | eqid 2736 | . . . . . 6 ⊢ (♯‘𝐴) = (♯‘𝐴) | |
| 20 | 19 | swrdccatin2 14768 | . . . . 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 406 | . . . 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 7440 | . . . . . 6 ⊢ ((♯‘𝐴) = 𝐿 → (𝑀 − (♯‘𝐴)) = (𝑀 − 𝐿)) | |
| 24 | oveq2 7440 | . . . . . 6 ⊢ ((♯‘𝐴) = 𝐿 → (𝑁 − (♯‘𝐴)) = (𝑁 − 𝐿)) | |
| 25 | 23, 24 | opeq12d 4880 | . . . . 5 ⊢ ((♯‘𝐴) = 𝐿 → 〈(𝑀 − (♯‘𝐴)), (𝑁 − (♯‘𝐴))〉 = 〈(𝑀 − 𝐿), (𝑁 − 𝐿)〉) | 
| 26 | 25 | oveq2d 7448 | . . . 4 ⊢ ((♯‘𝐴) = 𝐿 → (𝐵 substr 〈(𝑀 − (♯‘𝐴)), (𝑁 − (♯‘𝐴))〉) = (𝐵 substr 〈(𝑀 − 𝐿), (𝑁 − 𝐿)〉)) | 
| 27 | 26 | eqeq2d 2747 | . . 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 239 | . 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 206 ∧ wa 395 = wceq 1539 ∈ wcel 2107 〈cop 4631 ‘cfv 6560 (class class class)co 7432 + caddc 11159 − cmin 11493 ...cfz 13548 ♯chash 14370 Word cword 14553 ++ cconcat 14609 substr csubstr 14679 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-int 4946 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-1st 8015 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-1o 8507 df-er 8746 df-en 8987 df-dom 8988 df-sdom 8989 df-fin 8990 df-card 9980 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-nn 12268 df-n0 12529 df-z 12616 df-uz 12880 df-fz 13549 df-fzo 13696 df-hash 14371 df-word 14554 df-concat 14610 df-substr 14680 | 
| This theorem is referenced by: (None) | 
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