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Mirrors > Home > MPE Home > Th. List > ofs2 | Structured version Visualization version GIF version |
Description: Letterwise operations on a double letter word. (Contributed by Thierry Arnoux, 7-Oct-2018.) |
Ref | Expression |
---|---|
ofs2 | ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑇 ∧ 𝐷 ∈ 𝑇)) → (〈“𝐴𝐵”〉 ∘f 𝑅〈“𝐶𝐷”〉) = 〈“(𝐴𝑅𝐶)(𝐵𝑅𝐷)”〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-s2 14210 | . . . 4 ⊢ 〈“𝐴𝐵”〉 = (〈“𝐴”〉 ++ 〈“𝐵”〉) | |
2 | df-s2 14210 | . . . 4 ⊢ 〈“𝐶𝐷”〉 = (〈“𝐶”〉 ++ 〈“𝐷”〉) | |
3 | 1, 2 | oveq12i 7168 | . . 3 ⊢ (〈“𝐴𝐵”〉 ∘f 𝑅〈“𝐶𝐷”〉) = ((〈“𝐴”〉 ++ 〈“𝐵”〉) ∘f 𝑅(〈“𝐶”〉 ++ 〈“𝐷”〉)) |
4 | simpll 765 | . . . . 5 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑇 ∧ 𝐷 ∈ 𝑇)) → 𝐴 ∈ 𝑆) | |
5 | 4 | s1cld 13957 | . . . 4 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑇 ∧ 𝐷 ∈ 𝑇)) → 〈“𝐴”〉 ∈ Word 𝑆) |
6 | simplr 767 | . . . . 5 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑇 ∧ 𝐷 ∈ 𝑇)) → 𝐵 ∈ 𝑆) | |
7 | 6 | s1cld 13957 | . . . 4 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑇 ∧ 𝐷 ∈ 𝑇)) → 〈“𝐵”〉 ∈ Word 𝑆) |
8 | simprl 769 | . . . . 5 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑇 ∧ 𝐷 ∈ 𝑇)) → 𝐶 ∈ 𝑇) | |
9 | 8 | s1cld 13957 | . . . 4 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑇 ∧ 𝐷 ∈ 𝑇)) → 〈“𝐶”〉 ∈ Word 𝑇) |
10 | simprr 771 | . . . . 5 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑇 ∧ 𝐷 ∈ 𝑇)) → 𝐷 ∈ 𝑇) | |
11 | 10 | s1cld 13957 | . . . 4 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑇 ∧ 𝐷 ∈ 𝑇)) → 〈“𝐷”〉 ∈ Word 𝑇) |
12 | s1len 13960 | . . . . . 6 ⊢ (♯‘〈“𝐴”〉) = 1 | |
13 | s1len 13960 | . . . . . 6 ⊢ (♯‘〈“𝐶”〉) = 1 | |
14 | 12, 13 | eqtr4i 2847 | . . . . 5 ⊢ (♯‘〈“𝐴”〉) = (♯‘〈“𝐶”〉) |
15 | 14 | a1i 11 | . . . 4 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑇 ∧ 𝐷 ∈ 𝑇)) → (♯‘〈“𝐴”〉) = (♯‘〈“𝐶”〉)) |
16 | s1len 13960 | . . . . . 6 ⊢ (♯‘〈“𝐵”〉) = 1 | |
17 | s1len 13960 | . . . . . 6 ⊢ (♯‘〈“𝐷”〉) = 1 | |
18 | 16, 17 | eqtr4i 2847 | . . . . 5 ⊢ (♯‘〈“𝐵”〉) = (♯‘〈“𝐷”〉) |
19 | 18 | a1i 11 | . . . 4 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑇 ∧ 𝐷 ∈ 𝑇)) → (♯‘〈“𝐵”〉) = (♯‘〈“𝐷”〉)) |
20 | 5, 7, 9, 11, 15, 19 | ofccat 14329 | . . 3 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑇 ∧ 𝐷 ∈ 𝑇)) → ((〈“𝐴”〉 ++ 〈“𝐵”〉) ∘f 𝑅(〈“𝐶”〉 ++ 〈“𝐷”〉)) = ((〈“𝐴”〉 ∘f 𝑅〈“𝐶”〉) ++ (〈“𝐵”〉 ∘f 𝑅〈“𝐷”〉))) |
21 | 3, 20 | syl5eq 2868 | . 2 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑇 ∧ 𝐷 ∈ 𝑇)) → (〈“𝐴𝐵”〉 ∘f 𝑅〈“𝐶𝐷”〉) = ((〈“𝐴”〉 ∘f 𝑅〈“𝐶”〉) ++ (〈“𝐵”〉 ∘f 𝑅〈“𝐷”〉))) |
22 | ofs1 14330 | . . . . 5 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐶 ∈ 𝑇) → (〈“𝐴”〉 ∘f 𝑅〈“𝐶”〉) = 〈“(𝐴𝑅𝐶)”〉) | |
23 | 4, 8, 22 | syl2anc 586 | . . . 4 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑇 ∧ 𝐷 ∈ 𝑇)) → (〈“𝐴”〉 ∘f 𝑅〈“𝐶”〉) = 〈“(𝐴𝑅𝐶)”〉) |
24 | ofs1 14330 | . . . . 5 ⊢ ((𝐵 ∈ 𝑆 ∧ 𝐷 ∈ 𝑇) → (〈“𝐵”〉 ∘f 𝑅〈“𝐷”〉) = 〈“(𝐵𝑅𝐷)”〉) | |
25 | 6, 10, 24 | syl2anc 586 | . . . 4 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑇 ∧ 𝐷 ∈ 𝑇)) → (〈“𝐵”〉 ∘f 𝑅〈“𝐷”〉) = 〈“(𝐵𝑅𝐷)”〉) |
26 | 23, 25 | oveq12d 7174 | . . 3 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑇 ∧ 𝐷 ∈ 𝑇)) → ((〈“𝐴”〉 ∘f 𝑅〈“𝐶”〉) ++ (〈“𝐵”〉 ∘f 𝑅〈“𝐷”〉)) = (〈“(𝐴𝑅𝐶)”〉 ++ 〈“(𝐵𝑅𝐷)”〉)) |
27 | df-s2 14210 | . . 3 ⊢ 〈“(𝐴𝑅𝐶)(𝐵𝑅𝐷)”〉 = (〈“(𝐴𝑅𝐶)”〉 ++ 〈“(𝐵𝑅𝐷)”〉) | |
28 | 26, 27 | syl6eqr 2874 | . 2 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑇 ∧ 𝐷 ∈ 𝑇)) → ((〈“𝐴”〉 ∘f 𝑅〈“𝐶”〉) ++ (〈“𝐵”〉 ∘f 𝑅〈“𝐷”〉)) = 〈“(𝐴𝑅𝐶)(𝐵𝑅𝐷)”〉) |
29 | 21, 28 | eqtrd 2856 | 1 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑇 ∧ 𝐷 ∈ 𝑇)) → (〈“𝐴𝐵”〉 ∘f 𝑅〈“𝐶𝐷”〉) = 〈“(𝐴𝑅𝐶)(𝐵𝑅𝐷)”〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ‘cfv 6355 (class class class)co 7156 ∘f cof 7407 1c1 10538 ♯chash 13691 ++ cconcat 13922 〈“cs1 13949 〈“cs2 14203 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-of 7409 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-n0 11899 df-z 11983 df-uz 12245 df-fz 12894 df-fzo 13035 df-hash 13692 df-word 13863 df-concat 13923 df-s1 13950 df-s2 14210 |
This theorem is referenced by: amgmw2d 44925 |
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