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Mirrors > Home > MPE Home > Th. List > swrdccatin12d | 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 Mario Carneiro/AV, 21-Oct-2018.) |
Ref | Expression |
---|---|
swrdccatind.l | ⊢ (𝜑 → (#‘𝐴) = 𝐿) |
swrdccatind.w | ⊢ (𝜑 → (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉)) |
swrdccatin12d.1 | ⊢ (𝜑 → 𝑀 ∈ (0...𝐿)) |
swrdccatin12d.2 | ⊢ (𝜑 → 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵)))) |
Ref | Expression |
---|---|
swrdccatin12d | ⊢ (𝜑 → ((𝐴 ++ 𝐵) substr 〈𝑀, 𝑁〉) = ((𝐴 substr 〈𝑀, 𝐿〉) ++ (𝐵 substr 〈0, (𝑁 − 𝐿)〉))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | swrdccatind.l | . 2 ⊢ (𝜑 → (#‘𝐴) = 𝐿) | |
2 | swrdccatind.w | . . . . . 6 ⊢ (𝜑 → (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉)) | |
3 | 2 | adantl 482 | . . . . 5 ⊢ (((#‘𝐴) = 𝐿 ∧ 𝜑) → (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉)) |
4 | swrdccatin12d.1 | . . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ (0...𝐿)) | |
5 | swrdccatin12d.2 | . . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵)))) | |
6 | 4, 5 | jca 554 | . . . . . . 7 ⊢ (𝜑 → (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) |
7 | 6 | adantl 482 | . . . . . 6 ⊢ (((#‘𝐴) = 𝐿 ∧ 𝜑) → (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) |
8 | oveq2 6658 | . . . . . . . . 9 ⊢ ((#‘𝐴) = 𝐿 → (0...(#‘𝐴)) = (0...𝐿)) | |
9 | 8 | eleq2d 2687 | . . . . . . . 8 ⊢ ((#‘𝐴) = 𝐿 → (𝑀 ∈ (0...(#‘𝐴)) ↔ 𝑀 ∈ (0...𝐿))) |
10 | id 22 | . . . . . . . . . 10 ⊢ ((#‘𝐴) = 𝐿 → (#‘𝐴) = 𝐿) | |
11 | oveq1 6657 | . . . . . . . . . 10 ⊢ ((#‘𝐴) = 𝐿 → ((#‘𝐴) + (#‘𝐵)) = (𝐿 + (#‘𝐵))) | |
12 | 10, 11 | oveq12d 6668 | . . . . . . . . 9 ⊢ ((#‘𝐴) = 𝐿 → ((#‘𝐴)...((#‘𝐴) + (#‘𝐵))) = (𝐿...(𝐿 + (#‘𝐵)))) |
13 | 12 | eleq2d 2687 | . . . . . . . 8 ⊢ ((#‘𝐴) = 𝐿 → (𝑁 ∈ ((#‘𝐴)...((#‘𝐴) + (#‘𝐵))) ↔ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) |
14 | 9, 13 | anbi12d 747 | . . . . . . 7 ⊢ ((#‘𝐴) = 𝐿 → ((𝑀 ∈ (0...(#‘𝐴)) ∧ 𝑁 ∈ ((#‘𝐴)...((#‘𝐴) + (#‘𝐵)))) ↔ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵)))))) |
15 | 14 | adantr 481 | . . . . . 6 ⊢ (((#‘𝐴) = 𝐿 ∧ 𝜑) → ((𝑀 ∈ (0...(#‘𝐴)) ∧ 𝑁 ∈ ((#‘𝐴)...((#‘𝐴) + (#‘𝐵)))) ↔ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵)))))) |
16 | 7, 15 | mpbird 247 | . . . . 5 ⊢ (((#‘𝐴) = 𝐿 ∧ 𝜑) → (𝑀 ∈ (0...(#‘𝐴)) ∧ 𝑁 ∈ ((#‘𝐴)...((#‘𝐴) + (#‘𝐵))))) |
17 | eqid 2622 | . . . . . 6 ⊢ (#‘𝐴) = (#‘𝐴) | |
18 | 17 | swrdccatin12 13491 | . . . . 5 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → ((𝑀 ∈ (0...(#‘𝐴)) ∧ 𝑁 ∈ ((#‘𝐴)...((#‘𝐴) + (#‘𝐵)))) → ((𝐴 ++ 𝐵) substr 〈𝑀, 𝑁〉) = ((𝐴 substr 〈𝑀, (#‘𝐴)〉) ++ (𝐵 substr 〈0, (𝑁 − (#‘𝐴))〉)))) |
19 | 3, 16, 18 | sylc 65 | . . . 4 ⊢ (((#‘𝐴) = 𝐿 ∧ 𝜑) → ((𝐴 ++ 𝐵) substr 〈𝑀, 𝑁〉) = ((𝐴 substr 〈𝑀, (#‘𝐴)〉) ++ (𝐵 substr 〈0, (𝑁 − (#‘𝐴))〉))) |
20 | 19 | ex 450 | . . 3 ⊢ ((#‘𝐴) = 𝐿 → (𝜑 → ((𝐴 ++ 𝐵) substr 〈𝑀, 𝑁〉) = ((𝐴 substr 〈𝑀, (#‘𝐴)〉) ++ (𝐵 substr 〈0, (𝑁 − (#‘𝐴))〉)))) |
21 | opeq2 4403 | . . . . . 6 ⊢ ((#‘𝐴) = 𝐿 → 〈𝑀, (#‘𝐴)〉 = 〈𝑀, 𝐿〉) | |
22 | 21 | oveq2d 6666 | . . . . 5 ⊢ ((#‘𝐴) = 𝐿 → (𝐴 substr 〈𝑀, (#‘𝐴)〉) = (𝐴 substr 〈𝑀, 𝐿〉)) |
23 | oveq2 6658 | . . . . . . 7 ⊢ ((#‘𝐴) = 𝐿 → (𝑁 − (#‘𝐴)) = (𝑁 − 𝐿)) | |
24 | 23 | opeq2d 4409 | . . . . . 6 ⊢ ((#‘𝐴) = 𝐿 → 〈0, (𝑁 − (#‘𝐴))〉 = 〈0, (𝑁 − 𝐿)〉) |
25 | 24 | oveq2d 6666 | . . . . 5 ⊢ ((#‘𝐴) = 𝐿 → (𝐵 substr 〈0, (𝑁 − (#‘𝐴))〉) = (𝐵 substr 〈0, (𝑁 − 𝐿)〉)) |
26 | 22, 25 | oveq12d 6668 | . . . 4 ⊢ ((#‘𝐴) = 𝐿 → ((𝐴 substr 〈𝑀, (#‘𝐴)〉) ++ (𝐵 substr 〈0, (𝑁 − (#‘𝐴))〉)) = ((𝐴 substr 〈𝑀, 𝐿〉) ++ (𝐵 substr 〈0, (𝑁 − 𝐿)〉))) |
27 | 26 | eqeq2d 2632 | . . 3 ⊢ ((#‘𝐴) = 𝐿 → (((𝐴 ++ 𝐵) substr 〈𝑀, 𝑁〉) = ((𝐴 substr 〈𝑀, (#‘𝐴)〉) ++ (𝐵 substr 〈0, (𝑁 − (#‘𝐴))〉)) ↔ ((𝐴 ++ 𝐵) substr 〈𝑀, 𝑁〉) = ((𝐴 substr 〈𝑀, 𝐿〉) ++ (𝐵 substr 〈0, (𝑁 − 𝐿)〉)))) |
28 | 20, 27 | sylibd 229 | . 2 ⊢ ((#‘𝐴) = 𝐿 → (𝜑 → ((𝐴 ++ 𝐵) substr 〈𝑀, 𝑁〉) = ((𝐴 substr 〈𝑀, 𝐿〉) ++ (𝐵 substr 〈0, (𝑁 − 𝐿)〉)))) |
29 | 1, 28 | mpcom 38 | 1 ⊢ (𝜑 → ((𝐴 ++ 𝐵) substr 〈𝑀, 𝑁〉) = ((𝐴 substr 〈𝑀, 𝐿〉) ++ (𝐵 substr 〈0, (𝑁 − 𝐿)〉))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 〈cop 4183 ‘cfv 5888 (class class class)co 6650 0cc0 9936 + caddc 9939 − cmin 10266 ...cfz 12326 #chash 13117 Word cword 13291 ++ cconcat 13293 substr csubstr 13295 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-n0 11293 df-z 11378 df-uz 11688 df-fz 12327 df-fzo 12466 df-hash 13118 df-word 13299 df-concat 13301 df-substr 13303 |
This theorem is referenced by: (None) |
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