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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 13930 | . . . 4 ⊢ 〈“𝐴𝐵”〉 = (〈“𝐴”〉 ++ 〈“𝐵”〉) | |
2 | df-s2 13930 | . . . 4 ⊢ 〈“𝐶𝐷”〉 = (〈“𝐶”〉 ++ 〈“𝐷”〉) | |
3 | 1, 2 | oveq12i 6889 | . . 3 ⊢ (〈“𝐴𝐵”〉 ∘𝑓 𝑅〈“𝐶𝐷”〉) = ((〈“𝐴”〉 ++ 〈“𝐵”〉) ∘𝑓 𝑅(〈“𝐶”〉 ++ 〈“𝐷”〉)) |
4 | simpll 784 | . . . . 5 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑇 ∧ 𝐷 ∈ 𝑇)) → 𝐴 ∈ 𝑆) | |
5 | 4 | s1cld 13620 | . . . 4 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑇 ∧ 𝐷 ∈ 𝑇)) → 〈“𝐴”〉 ∈ Word 𝑆) |
6 | simplr 786 | . . . . 5 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑇 ∧ 𝐷 ∈ 𝑇)) → 𝐵 ∈ 𝑆) | |
7 | 6 | s1cld 13620 | . . . 4 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑇 ∧ 𝐷 ∈ 𝑇)) → 〈“𝐵”〉 ∈ Word 𝑆) |
8 | simprl 788 | . . . . 5 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑇 ∧ 𝐷 ∈ 𝑇)) → 𝐶 ∈ 𝑇) | |
9 | 8 | s1cld 13620 | . . . 4 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑇 ∧ 𝐷 ∈ 𝑇)) → 〈“𝐶”〉 ∈ Word 𝑇) |
10 | simprr 790 | . . . . 5 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑇 ∧ 𝐷 ∈ 𝑇)) → 𝐷 ∈ 𝑇) | |
11 | 10 | s1cld 13620 | . . . 4 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑇 ∧ 𝐷 ∈ 𝑇)) → 〈“𝐷”〉 ∈ Word 𝑇) |
12 | s1len 13623 | . . . . . 6 ⊢ (♯‘〈“𝐴”〉) = 1 | |
13 | s1len 13623 | . . . . . 6 ⊢ (♯‘〈“𝐶”〉) = 1 | |
14 | 12, 13 | eqtr4i 2823 | . . . . 5 ⊢ (♯‘〈“𝐴”〉) = (♯‘〈“𝐶”〉) |
15 | 14 | a1i 11 | . . . 4 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑇 ∧ 𝐷 ∈ 𝑇)) → (♯‘〈“𝐴”〉) = (♯‘〈“𝐶”〉)) |
16 | s1len 13623 | . . . . . 6 ⊢ (♯‘〈“𝐵”〉) = 1 | |
17 | s1len 13623 | . . . . . 6 ⊢ (♯‘〈“𝐷”〉) = 1 | |
18 | 16, 17 | eqtr4i 2823 | . . . . 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 14048 | . . 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 2844 | . 2 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑇 ∧ 𝐷 ∈ 𝑇)) → (〈“𝐴𝐵”〉 ∘𝑓 𝑅〈“𝐶𝐷”〉) = ((〈“𝐴”〉 ∘𝑓 𝑅〈“𝐶”〉) ++ (〈“𝐵”〉 ∘𝑓 𝑅〈“𝐷”〉))) |
22 | ofs1 14049 | . . . . 5 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐶 ∈ 𝑇) → (〈“𝐴”〉 ∘𝑓 𝑅〈“𝐶”〉) = 〈“(𝐴𝑅𝐶)”〉) | |
23 | 4, 8, 22 | syl2anc 580 | . . . 4 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑇 ∧ 𝐷 ∈ 𝑇)) → (〈“𝐴”〉 ∘𝑓 𝑅〈“𝐶”〉) = 〈“(𝐴𝑅𝐶)”〉) |
24 | ofs1 14049 | . . . . 5 ⊢ ((𝐵 ∈ 𝑆 ∧ 𝐷 ∈ 𝑇) → (〈“𝐵”〉 ∘𝑓 𝑅〈“𝐷”〉) = 〈“(𝐵𝑅𝐷)”〉) | |
25 | 6, 10, 24 | syl2anc 580 | . . . 4 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑇 ∧ 𝐷 ∈ 𝑇)) → (〈“𝐵”〉 ∘𝑓 𝑅〈“𝐷”〉) = 〈“(𝐵𝑅𝐷)”〉) |
26 | 23, 25 | oveq12d 6895 | . . 3 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑇 ∧ 𝐷 ∈ 𝑇)) → ((〈“𝐴”〉 ∘𝑓 𝑅〈“𝐶”〉) ++ (〈“𝐵”〉 ∘𝑓 𝑅〈“𝐷”〉)) = (〈“(𝐴𝑅𝐶)”〉 ++ 〈“(𝐵𝑅𝐷)”〉)) |
27 | df-s2 13930 | . . 3 ⊢ 〈“(𝐴𝑅𝐶)(𝐵𝑅𝐷)”〉 = (〈“(𝐴𝑅𝐶)”〉 ++ 〈“(𝐵𝑅𝐷)”〉) | |
28 | 26, 27 | syl6eqr 2850 | . 2 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑇 ∧ 𝐷 ∈ 𝑇)) → ((〈“𝐴”〉 ∘𝑓 𝑅〈“𝐶”〉) ++ (〈“𝐵”〉 ∘𝑓 𝑅〈“𝐷”〉)) = 〈“(𝐴𝑅𝐶)(𝐵𝑅𝐷)”〉) |
29 | 21, 28 | eqtrd 2832 | 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 385 = wceq 1653 ∈ wcel 2157 ‘cfv 6100 (class class class)co 6877 ∘𝑓 cof 7128 1c1 10224 ♯chash 13367 ++ cconcat 13587 〈“cs1 13612 〈“cs2 13923 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2776 ax-rep 4963 ax-sep 4974 ax-nul 4982 ax-pow 5034 ax-pr 5096 ax-un 7182 ax-cnex 10279 ax-resscn 10280 ax-1cn 10281 ax-icn 10282 ax-addcl 10283 ax-addrcl 10284 ax-mulcl 10285 ax-mulrcl 10286 ax-mulcom 10287 ax-addass 10288 ax-mulass 10289 ax-distr 10290 ax-i2m1 10291 ax-1ne0 10292 ax-1rid 10293 ax-rnegex 10294 ax-rrecex 10295 ax-cnre 10296 ax-pre-lttri 10297 ax-pre-lttrn 10298 ax-pre-ltadd 10299 ax-pre-mulgt0 10300 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2785 df-cleq 2791 df-clel 2794 df-nfc 2929 df-ne 2971 df-nel 3074 df-ral 3093 df-rex 3094 df-reu 3095 df-rab 3097 df-v 3386 df-sbc 3633 df-csb 3728 df-dif 3771 df-un 3773 df-in 3775 df-ss 3782 df-pss 3784 df-nul 4115 df-if 4277 df-pw 4350 df-sn 4368 df-pr 4370 df-tp 4372 df-op 4374 df-uni 4628 df-int 4667 df-iun 4711 df-br 4843 df-opab 4905 df-mpt 4922 df-tr 4945 df-id 5219 df-eprel 5224 df-po 5232 df-so 5233 df-fr 5270 df-we 5272 df-xp 5317 df-rel 5318 df-cnv 5319 df-co 5320 df-dm 5321 df-rn 5322 df-res 5323 df-ima 5324 df-pred 5897 df-ord 5943 df-on 5944 df-lim 5945 df-suc 5946 df-iota 6063 df-fun 6102 df-fn 6103 df-f 6104 df-f1 6105 df-fo 6106 df-f1o 6107 df-fv 6108 df-riota 6838 df-ov 6880 df-oprab 6881 df-mpt2 6882 df-of 7130 df-om 7299 df-1st 7400 df-2nd 7401 df-wrecs 7644 df-recs 7706 df-rdg 7744 df-1o 7798 df-oadd 7802 df-er 7981 df-en 8195 df-dom 8196 df-sdom 8197 df-fin 8198 df-card 9050 df-pnf 10364 df-mnf 10365 df-xr 10366 df-ltxr 10367 df-le 10368 df-sub 10557 df-neg 10558 df-nn 11312 df-n0 11578 df-z 11664 df-uz 11928 df-fz 12578 df-fzo 12718 df-hash 13368 df-word 13532 df-concat 13588 df-s1 13613 df-s2 13930 |
This theorem is referenced by: amgmw2d 43341 |
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