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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 13798 | . . . 4 ⊢ 〈“𝐴𝐵”〉 = (〈“𝐴”〉 ++ 〈“𝐵”〉) | |
2 | 1 | oveq1i 6802 | . . 3 ⊢ (〈“𝐴𝐵”〉∘𝑓/𝑐𝑅𝐶) = ((〈“𝐴”〉 ++ 〈“𝐵”〉)∘𝑓/𝑐𝑅𝐶) |
3 | simp1 1130 | . . . . 5 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑇) → 𝐴 ∈ 𝑆) | |
4 | 3 | s1cld 13579 | . . . 4 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑇) → 〈“𝐴”〉 ∈ Word 𝑆) |
5 | simp2 1131 | . . . . 5 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑇) → 𝐵 ∈ 𝑆) | |
6 | 5 | s1cld 13579 | . . . 4 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑇) → 〈“𝐵”〉 ∈ Word 𝑆) |
7 | simp3 1132 | . . . 4 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑇) → 𝐶 ∈ 𝑇) | |
8 | 4, 6, 7 | ofcccat 30956 | . . 3 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑇) → ((〈“𝐴”〉 ++ 〈“𝐵”〉)∘𝑓/𝑐𝑅𝐶) = ((〈“𝐴”〉∘𝑓/𝑐𝑅𝐶) ++ (〈“𝐵”〉∘𝑓/𝑐𝑅𝐶))) |
9 | 2, 8 | syl5eq 2817 | . 2 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑇) → (〈“𝐴𝐵”〉∘𝑓/𝑐𝑅𝐶) = ((〈“𝐴”〉∘𝑓/𝑐𝑅𝐶) ++ (〈“𝐵”〉∘𝑓/𝑐𝑅𝐶))) |
10 | ofcs1 30957 | . . . . 5 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐶 ∈ 𝑇) → (〈“𝐴”〉∘𝑓/𝑐𝑅𝐶) = 〈“(𝐴𝑅𝐶)”〉) | |
11 | 3, 7, 10 | syl2anc 573 | . . . 4 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑇) → (〈“𝐴”〉∘𝑓/𝑐𝑅𝐶) = 〈“(𝐴𝑅𝐶)”〉) |
12 | ofcs1 30957 | . . . . 5 ⊢ ((𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑇) → (〈“𝐵”〉∘𝑓/𝑐𝑅𝐶) = 〈“(𝐵𝑅𝐶)”〉) | |
13 | 5, 7, 12 | syl2anc 573 | . . . 4 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑇) → (〈“𝐵”〉∘𝑓/𝑐𝑅𝐶) = 〈“(𝐵𝑅𝐶)”〉) |
14 | 11, 13 | oveq12d 6810 | . . 3 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑇) → ((〈“𝐴”〉∘𝑓/𝑐𝑅𝐶) ++ (〈“𝐵”〉∘𝑓/𝑐𝑅𝐶)) = (〈“(𝐴𝑅𝐶)”〉 ++ 〈“(𝐵𝑅𝐶)”〉)) |
15 | df-s2 13798 | . . 3 ⊢ 〈“(𝐴𝑅𝐶)(𝐵𝑅𝐶)”〉 = (〈“(𝐴𝑅𝐶)”〉 ++ 〈“(𝐵𝑅𝐶)”〉) | |
16 | 14, 15 | syl6eqr 2823 | . 2 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑇) → ((〈“𝐴”〉∘𝑓/𝑐𝑅𝐶) ++ (〈“𝐵”〉∘𝑓/𝑐𝑅𝐶)) = 〈“(𝐴𝑅𝐶)(𝐵𝑅𝐶)”〉) |
17 | 9, 16 | eqtrd 2805 | 1 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑇) → (〈“𝐴𝐵”〉∘𝑓/𝑐𝑅𝐶) = 〈“(𝐴𝑅𝐶)(𝐵𝑅𝐶)”〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1071 = wceq 1631 ∈ wcel 2145 (class class class)co 6792 ++ cconcat 13485 〈“cs1 13486 〈“cs2 13791 ∘𝑓/𝑐cofc 30493 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4904 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7096 ax-cnex 10194 ax-resscn 10195 ax-1cn 10196 ax-icn 10197 ax-addcl 10198 ax-addrcl 10199 ax-mulcl 10200 ax-mulrcl 10201 ax-mulcom 10202 ax-addass 10203 ax-mulass 10204 ax-distr 10205 ax-i2m1 10206 ax-1ne0 10207 ax-1rid 10208 ax-rnegex 10209 ax-rrecex 10210 ax-cnre 10211 ax-pre-lttri 10212 ax-pre-lttrn 10213 ax-pre-ltadd 10214 ax-pre-mulgt0 10215 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-tp 4321 df-op 4323 df-uni 4575 df-int 4612 df-iun 4656 df-br 4787 df-opab 4847 df-mpt 4864 df-tr 4887 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5821 df-ord 5867 df-on 5868 df-lim 5869 df-suc 5870 df-iota 5992 df-fun 6031 df-fn 6032 df-f 6033 df-f1 6034 df-fo 6035 df-f1o 6036 df-fv 6037 df-riota 6753 df-ov 6795 df-oprab 6796 df-mpt2 6797 df-of 7044 df-om 7213 df-1st 7315 df-2nd 7316 df-wrecs 7559 df-recs 7621 df-rdg 7659 df-1o 7713 df-oadd 7717 df-er 7896 df-en 8110 df-dom 8111 df-sdom 8112 df-fin 8113 df-card 8965 df-cda 9192 df-pnf 10278 df-mnf 10279 df-xr 10280 df-ltxr 10281 df-le 10282 df-sub 10470 df-neg 10471 df-nn 11223 df-n0 11496 df-z 11581 df-uz 11890 df-fz 12530 df-fzo 12670 df-hash 13318 df-word 13491 df-concat 13493 df-s1 13494 df-s2 13798 df-ofc 30494 |
This theorem is referenced by: (None) |
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