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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 553 | . . . . . . . 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 6556 | . . . . . . . . . 10 ⊢ ((#‘𝐴) = 𝐿 → ((#‘𝐴)...𝑁) = (𝐿...𝑁)) | |
9 | 8 | eleq2d 2673 | . . . . . . . . 9 ⊢ ((#‘𝐴) = 𝐿 → (𝑀 ∈ ((#‘𝐴)...𝑁) ↔ 𝑀 ∈ (𝐿...𝑁))) |
10 | id 22 | . . . . . . . . . . 11 ⊢ ((#‘𝐴) = 𝐿 → (#‘𝐴) = 𝐿) | |
11 | oveq1 6556 | . . . . . . . . . . 11 ⊢ ((#‘𝐴) = 𝐿 → ((#‘𝐴) + (#‘𝐵)) = (𝐿 + (#‘𝐵))) | |
12 | 10, 11 | oveq12d 6567 | . . . . . . . . . 10 ⊢ ((#‘𝐴) = 𝐿 → ((#‘𝐴)...((#‘𝐴) + (#‘𝐵))) = (𝐿...(𝐿 + (#‘𝐵)))) |
13 | 12 | eleq2d 2673 | . . . . . . . . 9 ⊢ ((#‘𝐴) = 𝐿 → (𝑁 ∈ ((#‘𝐴)...((#‘𝐴) + (#‘𝐵))) ↔ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) |
14 | 9, 13 | anbi12d 743 | . . . . . . . 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 246 | . . . . . 6 ⊢ (((#‘𝐴) = 𝐿 ∧ 𝜑) → (𝑀 ∈ ((#‘𝐴)...𝑁) ∧ 𝑁 ∈ ((#‘𝐴)...((#‘𝐴) + (#‘𝐵))))) |
17 | 3, 16 | jca 553 | . . . . 5 ⊢ (((#‘𝐴) = 𝐿 ∧ 𝜑) → ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ ((#‘𝐴)...𝑁) ∧ 𝑁 ∈ ((#‘𝐴)...((#‘𝐴) + (#‘𝐵)))))) |
18 | 17 | ex 449 | . . . 4 ⊢ ((#‘𝐴) = 𝐿 → (𝜑 → ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ ((#‘𝐴)...𝑁) ∧ 𝑁 ∈ ((#‘𝐴)...((#‘𝐴) + (#‘𝐵))))))) |
19 | eqid 2610 | . . . . . 6 ⊢ (#‘𝐴) = (#‘𝐴) | |
20 | 19 | swrdccatin2 13338 | . . . . 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 444 | . . . 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 34 | . . 3 ⊢ ((#‘𝐴) = 𝐿 → (𝜑 → ((𝐴 ++ 𝐵) substr 〈𝑀, 𝑁〉) = (𝐵 substr 〈(𝑀 − (#‘𝐴)), (𝑁 − (#‘𝐴))〉))) |
23 | oveq2 6557 | . . . . . 6 ⊢ ((#‘𝐴) = 𝐿 → (𝑀 − (#‘𝐴)) = (𝑀 − 𝐿)) | |
24 | oveq2 6557 | . . . . . 6 ⊢ ((#‘𝐴) = 𝐿 → (𝑁 − (#‘𝐴)) = (𝑁 − 𝐿)) | |
25 | 23, 24 | opeq12d 4348 | . . . . 5 ⊢ ((#‘𝐴) = 𝐿 → 〈(𝑀 − (#‘𝐴)), (𝑁 − (#‘𝐴))〉 = 〈(𝑀 − 𝐿), (𝑁 − 𝐿)〉) |
26 | 25 | oveq2d 6565 | . . . 4 ⊢ ((#‘𝐴) = 𝐿 → (𝐵 substr 〈(𝑀 − (#‘𝐴)), (𝑁 − (#‘𝐴))〉) = (𝐵 substr 〈(𝑀 − 𝐿), (𝑁 − 𝐿)〉)) |
27 | 26 | eqeq2d 2620 | . . 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 228 | . 2 ⊢ ((#‘𝐴) = 𝐿 → (𝜑 → ((𝐴 ++ 𝐵) substr 〈𝑀, 𝑁〉) = (𝐵 substr 〈(𝑀 − 𝐿), (𝑁 − 𝐿)〉))) |
29 | 1, 28 | mpcom 37 | 1 ⊢ (𝜑 → ((𝐴 ++ 𝐵) substr 〈𝑀, 𝑁〉) = (𝐵 substr 〈(𝑀 − 𝐿), (𝑁 − 𝐿)〉)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 〈cop 4131 ‘cfv 5804 (class class class)co 6549 + caddc 9818 − cmin 10145 ...cfz 12197 #chash 12979 Word cword 13146 ++ cconcat 13148 substr csubstr 13150 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-fzo 12335 df-hash 12980 df-word 13154 df-concat 13156 df-substr 13158 |
This theorem is referenced by: (None) |
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