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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ofcs2 | Structured version Visualization version GIF version |
Description: Letterwise operations on a double letter word. (Contributed by Thierry Arnoux, 9-Oct-2018.) |
Ref | Expression |
---|---|
ofcs2 | ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑇) → (〈“𝐴𝐵”〉∘𝑓/𝑐𝑅𝐶) = 〈“(𝐴𝑅𝐶)(𝐵𝑅𝐶)”〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-s2 14072 | . . . 4 ⊢ 〈“𝐴𝐵”〉 = (〈“𝐴”〉 ++ 〈“𝐵”〉) | |
2 | 1 | oveq1i 6986 | . . 3 ⊢ (〈“𝐴𝐵”〉∘𝑓/𝑐𝑅𝐶) = ((〈“𝐴”〉 ++ 〈“𝐵”〉)∘𝑓/𝑐𝑅𝐶) |
3 | simp1 1116 | . . . . 5 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑇) → 𝐴 ∈ 𝑆) | |
4 | 3 | s1cld 13766 | . . . 4 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑇) → 〈“𝐴”〉 ∈ Word 𝑆) |
5 | simp2 1117 | . . . . 5 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑇) → 𝐵 ∈ 𝑆) | |
6 | 5 | s1cld 13766 | . . . 4 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑇) → 〈“𝐵”〉 ∈ Word 𝑆) |
7 | simp3 1118 | . . . 4 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑇) → 𝐶 ∈ 𝑇) | |
8 | 4, 6, 7 | ofcccat 31465 | . . 3 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑇) → ((〈“𝐴”〉 ++ 〈“𝐵”〉)∘𝑓/𝑐𝑅𝐶) = ((〈“𝐴”〉∘𝑓/𝑐𝑅𝐶) ++ (〈“𝐵”〉∘𝑓/𝑐𝑅𝐶))) |
9 | 2, 8 | syl5eq 2826 | . 2 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑇) → (〈“𝐴𝐵”〉∘𝑓/𝑐𝑅𝐶) = ((〈“𝐴”〉∘𝑓/𝑐𝑅𝐶) ++ (〈“𝐵”〉∘𝑓/𝑐𝑅𝐶))) |
10 | ofcs1 31466 | . . . . 5 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐶 ∈ 𝑇) → (〈“𝐴”〉∘𝑓/𝑐𝑅𝐶) = 〈“(𝐴𝑅𝐶)”〉) | |
11 | 3, 7, 10 | syl2anc 576 | . . . 4 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑇) → (〈“𝐴”〉∘𝑓/𝑐𝑅𝐶) = 〈“(𝐴𝑅𝐶)”〉) |
12 | ofcs1 31466 | . . . . 5 ⊢ ((𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑇) → (〈“𝐵”〉∘𝑓/𝑐𝑅𝐶) = 〈“(𝐵𝑅𝐶)”〉) | |
13 | 5, 7, 12 | syl2anc 576 | . . . 4 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑇) → (〈“𝐵”〉∘𝑓/𝑐𝑅𝐶) = 〈“(𝐵𝑅𝐶)”〉) |
14 | 11, 13 | oveq12d 6994 | . . 3 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑇) → ((〈“𝐴”〉∘𝑓/𝑐𝑅𝐶) ++ (〈“𝐵”〉∘𝑓/𝑐𝑅𝐶)) = (〈“(𝐴𝑅𝐶)”〉 ++ 〈“(𝐵𝑅𝐶)”〉)) |
15 | df-s2 14072 | . . 3 ⊢ 〈“(𝐴𝑅𝐶)(𝐵𝑅𝐶)”〉 = (〈“(𝐴𝑅𝐶)”〉 ++ 〈“(𝐵𝑅𝐶)”〉) | |
16 | 14, 15 | syl6eqr 2832 | . 2 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑇) → ((〈“𝐴”〉∘𝑓/𝑐𝑅𝐶) ++ (〈“𝐵”〉∘𝑓/𝑐𝑅𝐶)) = 〈“(𝐴𝑅𝐶)(𝐵𝑅𝐶)”〉) |
17 | 9, 16 | eqtrd 2814 | 1 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑇) → (〈“𝐴𝐵”〉∘𝑓/𝑐𝑅𝐶) = 〈“(𝐴𝑅𝐶)(𝐵𝑅𝐶)”〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1068 = wceq 1507 ∈ wcel 2050 (class class class)co 6976 ++ cconcat 13733 〈“cs1 13758 〈“cs2 14065 ∘𝑓/𝑐cofc 31004 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-rep 5049 ax-sep 5060 ax-nul 5067 ax-pow 5119 ax-pr 5186 ax-un 7279 ax-cnex 10391 ax-resscn 10392 ax-1cn 10393 ax-icn 10394 ax-addcl 10395 ax-addrcl 10396 ax-mulcl 10397 ax-mulrcl 10398 ax-mulcom 10399 ax-addass 10400 ax-mulass 10401 ax-distr 10402 ax-i2m1 10403 ax-1ne0 10404 ax-1rid 10405 ax-rnegex 10406 ax-rrecex 10407 ax-cnre 10408 ax-pre-lttri 10409 ax-pre-lttrn 10410 ax-pre-ltadd 10411 ax-pre-mulgt0 10412 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ne 2968 df-nel 3074 df-ral 3093 df-rex 3094 df-reu 3095 df-rmo 3096 df-rab 3097 df-v 3417 df-sbc 3682 df-csb 3787 df-dif 3832 df-un 3834 df-in 3836 df-ss 3843 df-pss 3845 df-nul 4179 df-if 4351 df-pw 4424 df-sn 4442 df-pr 4444 df-tp 4446 df-op 4448 df-uni 4713 df-int 4750 df-iun 4794 df-br 4930 df-opab 4992 df-mpt 5009 df-tr 5031 df-id 5312 df-eprel 5317 df-po 5326 df-so 5327 df-fr 5366 df-we 5368 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-rn 5418 df-res 5419 df-ima 5420 df-pred 5986 df-ord 6032 df-on 6033 df-lim 6034 df-suc 6035 df-iota 6152 df-fun 6190 df-fn 6191 df-f 6192 df-f1 6193 df-fo 6194 df-f1o 6195 df-fv 6196 df-riota 6937 df-ov 6979 df-oprab 6980 df-mpo 6981 df-of 7227 df-om 7397 df-1st 7501 df-2nd 7502 df-wrecs 7750 df-recs 7812 df-rdg 7850 df-1o 7905 df-oadd 7909 df-er 8089 df-en 8307 df-dom 8308 df-sdom 8309 df-fin 8310 df-dju 9124 df-card 9162 df-pnf 10476 df-mnf 10477 df-xr 10478 df-ltxr 10479 df-le 10480 df-sub 10672 df-neg 10673 df-nn 11440 df-n0 11708 df-z 11794 df-uz 12059 df-fz 12709 df-fzo 12850 df-hash 13506 df-word 13673 df-concat 13734 df-s1 13759 df-s2 14072 df-ofc 31005 |
This theorem is referenced by: (None) |
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