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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 7438 | . . . . . . . . . 10 ⊢ ((♯‘𝐴) = 𝐿 → ((♯‘𝐴)...𝑁) = (𝐿...𝑁)) | |
9 | 8 | eleq2d 2825 | . . . . . . . . 9 ⊢ ((♯‘𝐴) = 𝐿 → (𝑀 ∈ ((♯‘𝐴)...𝑁) ↔ 𝑀 ∈ (𝐿...𝑁))) |
10 | id 22 | . . . . . . . . . . 11 ⊢ ((♯‘𝐴) = 𝐿 → (♯‘𝐴) = 𝐿) | |
11 | oveq1 7438 | . . . . . . . . . . 11 ⊢ ((♯‘𝐴) = 𝐿 → ((♯‘𝐴) + (♯‘𝐵)) = (𝐿 + (♯‘𝐵))) | |
12 | 10, 11 | oveq12d 7449 | . . . . . . . . . 10 ⊢ ((♯‘𝐴) = 𝐿 → ((♯‘𝐴)...((♯‘𝐴) + (♯‘𝐵))) = (𝐿...(𝐿 + (♯‘𝐵)))) |
13 | 12 | eleq2d 2825 | . . . . . . . . 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 2735 | . . . . . 6 ⊢ (♯‘𝐴) = (♯‘𝐴) | |
20 | 19 | swrdccatin2 14764 | . . . . 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 7439 | . . . . . 6 ⊢ ((♯‘𝐴) = 𝐿 → (𝑀 − (♯‘𝐴)) = (𝑀 − 𝐿)) | |
24 | oveq2 7439 | . . . . . 6 ⊢ ((♯‘𝐴) = 𝐿 → (𝑁 − (♯‘𝐴)) = (𝑁 − 𝐿)) | |
25 | 23, 24 | opeq12d 4886 | . . . . 5 ⊢ ((♯‘𝐴) = 𝐿 → 〈(𝑀 − (♯‘𝐴)), (𝑁 − (♯‘𝐴))〉 = 〈(𝑀 − 𝐿), (𝑁 − 𝐿)〉) |
26 | 25 | oveq2d 7447 | . . . 4 ⊢ ((♯‘𝐴) = 𝐿 → (𝐵 substr 〈(𝑀 − (♯‘𝐴)), (𝑁 − (♯‘𝐴))〉) = (𝐵 substr 〈(𝑀 − 𝐿), (𝑁 − 𝐿)〉)) |
27 | 26 | eqeq2d 2746 | . . 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 1537 ∈ wcel 2106 〈cop 4637 ‘cfv 6563 (class class class)co 7431 + caddc 11156 − cmin 11490 ...cfz 13544 ♯chash 14366 Word cword 14549 ++ cconcat 14605 substr csubstr 14675 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-n0 12525 df-z 12612 df-uz 12877 df-fz 13545 df-fzo 13692 df-hash 14367 df-word 14550 df-concat 14606 df-substr 14676 |
This theorem is referenced by: (None) |
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