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Mirrors > Home > MPE Home > Th. List > s3eqs2s1eq | Structured version Visualization version GIF version |
Description: Two length 3 words are equal iff the corresponding length 2 words and singleton words consisting of their symbols are equal. (Contributed by AV, 4-Jan-2022.) |
Ref | Expression |
---|---|
s3eqs2s1eq | ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ (𝐷 ∈ 𝑉 ∧ 𝐸 ∈ 𝑉 ∧ 𝐹 ∈ 𝑉)) → (〈“𝐴𝐵𝐶”〉 = 〈“𝐷𝐸𝐹”〉 ↔ (〈“𝐴𝐵”〉 = 〈“𝐷𝐸”〉 ∧ 〈“𝐶”〉 = 〈“𝐹”〉))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-s3 14191 | . . . 4 ⊢ 〈“𝐴𝐵𝐶”〉 = (〈“𝐴𝐵”〉 ++ 〈“𝐶”〉) | |
2 | 1 | a1i 11 | . . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ (𝐷 ∈ 𝑉 ∧ 𝐸 ∈ 𝑉 ∧ 𝐹 ∈ 𝑉)) → 〈“𝐴𝐵𝐶”〉 = (〈“𝐴𝐵”〉 ++ 〈“𝐶”〉)) |
3 | df-s3 14191 | . . . 4 ⊢ 〈“𝐷𝐸𝐹”〉 = (〈“𝐷𝐸”〉 ++ 〈“𝐹”〉) | |
4 | 3 | a1i 11 | . . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ (𝐷 ∈ 𝑉 ∧ 𝐸 ∈ 𝑉 ∧ 𝐹 ∈ 𝑉)) → 〈“𝐷𝐸𝐹”〉 = (〈“𝐷𝐸”〉 ++ 〈“𝐹”〉)) |
5 | 2, 4 | eqeq12d 2836 | . 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ (𝐷 ∈ 𝑉 ∧ 𝐸 ∈ 𝑉 ∧ 𝐹 ∈ 𝑉)) → (〈“𝐴𝐵𝐶”〉 = 〈“𝐷𝐸𝐹”〉 ↔ (〈“𝐴𝐵”〉 ++ 〈“𝐶”〉) = (〈“𝐷𝐸”〉 ++ 〈“𝐹”〉))) |
6 | s2cl 14220 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 〈“𝐴𝐵”〉 ∈ Word 𝑉) | |
7 | s1cl 13936 | . . . . . 6 ⊢ (𝐶 ∈ 𝑉 → 〈“𝐶”〉 ∈ Word 𝑉) | |
8 | 6, 7 | anim12i 614 | . . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝐶 ∈ 𝑉) → (〈“𝐴𝐵”〉 ∈ Word 𝑉 ∧ 〈“𝐶”〉 ∈ Word 𝑉)) |
9 | 8 | 3impa 1106 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (〈“𝐴𝐵”〉 ∈ Word 𝑉 ∧ 〈“𝐶”〉 ∈ Word 𝑉)) |
10 | 9 | adantr 483 | . . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ (𝐷 ∈ 𝑉 ∧ 𝐸 ∈ 𝑉 ∧ 𝐹 ∈ 𝑉)) → (〈“𝐴𝐵”〉 ∈ Word 𝑉 ∧ 〈“𝐶”〉 ∈ Word 𝑉)) |
11 | s2cl 14220 | . . . . . 6 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐸 ∈ 𝑉) → 〈“𝐷𝐸”〉 ∈ Word 𝑉) | |
12 | s1cl 13936 | . . . . . 6 ⊢ (𝐹 ∈ 𝑉 → 〈“𝐹”〉 ∈ Word 𝑉) | |
13 | 11, 12 | anim12i 614 | . . . . 5 ⊢ (((𝐷 ∈ 𝑉 ∧ 𝐸 ∈ 𝑉) ∧ 𝐹 ∈ 𝑉) → (〈“𝐷𝐸”〉 ∈ Word 𝑉 ∧ 〈“𝐹”〉 ∈ Word 𝑉)) |
14 | 13 | 3impa 1106 | . . . 4 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐸 ∈ 𝑉 ∧ 𝐹 ∈ 𝑉) → (〈“𝐷𝐸”〉 ∈ Word 𝑉 ∧ 〈“𝐹”〉 ∈ Word 𝑉)) |
15 | 14 | adantl 484 | . . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ (𝐷 ∈ 𝑉 ∧ 𝐸 ∈ 𝑉 ∧ 𝐹 ∈ 𝑉)) → (〈“𝐷𝐸”〉 ∈ Word 𝑉 ∧ 〈“𝐹”〉 ∈ Word 𝑉)) |
16 | s2len 14231 | . . . . 5 ⊢ (♯‘〈“𝐴𝐵”〉) = 2 | |
17 | s2len 14231 | . . . . 5 ⊢ (♯‘〈“𝐷𝐸”〉) = 2 | |
18 | 16, 17 | eqtr4i 2846 | . . . 4 ⊢ (♯‘〈“𝐴𝐵”〉) = (♯‘〈“𝐷𝐸”〉) |
19 | 18 | a1i 11 | . . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ (𝐷 ∈ 𝑉 ∧ 𝐸 ∈ 𝑉 ∧ 𝐹 ∈ 𝑉)) → (♯‘〈“𝐴𝐵”〉) = (♯‘〈“𝐷𝐸”〉)) |
20 | ccatopth 14058 | . . 3 ⊢ (((〈“𝐴𝐵”〉 ∈ Word 𝑉 ∧ 〈“𝐶”〉 ∈ Word 𝑉) ∧ (〈“𝐷𝐸”〉 ∈ Word 𝑉 ∧ 〈“𝐹”〉 ∈ Word 𝑉) ∧ (♯‘〈“𝐴𝐵”〉) = (♯‘〈“𝐷𝐸”〉)) → ((〈“𝐴𝐵”〉 ++ 〈“𝐶”〉) = (〈“𝐷𝐸”〉 ++ 〈“𝐹”〉) ↔ (〈“𝐴𝐵”〉 = 〈“𝐷𝐸”〉 ∧ 〈“𝐶”〉 = 〈“𝐹”〉))) | |
21 | 10, 15, 19, 20 | syl3anc 1367 | . 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ (𝐷 ∈ 𝑉 ∧ 𝐸 ∈ 𝑉 ∧ 𝐹 ∈ 𝑉)) → ((〈“𝐴𝐵”〉 ++ 〈“𝐶”〉) = (〈“𝐷𝐸”〉 ++ 〈“𝐹”〉) ↔ (〈“𝐴𝐵”〉 = 〈“𝐷𝐸”〉 ∧ 〈“𝐶”〉 = 〈“𝐹”〉))) |
22 | 5, 21 | bitrd 281 | 1 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ (𝐷 ∈ 𝑉 ∧ 𝐸 ∈ 𝑉 ∧ 𝐹 ∈ 𝑉)) → (〈“𝐴𝐵𝐶”〉 = 〈“𝐷𝐸𝐹”〉 ↔ (〈“𝐴𝐵”〉 = 〈“𝐷𝐸”〉 ∧ 〈“𝐶”〉 = 〈“𝐹”〉))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ‘cfv 6336 (class class class)co 7137 2c2 11674 ♯chash 13675 Word cword 13846 ++ cconcat 13902 〈“cs1 13929 〈“cs2 14183 〈“cs3 14184 |
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 2792 ax-rep 5171 ax-sep 5184 ax-nul 5191 ax-pow 5247 ax-pr 5311 ax-un 7442 ax-cnex 10574 ax-resscn 10575 ax-1cn 10576 ax-icn 10577 ax-addcl 10578 ax-addrcl 10579 ax-mulcl 10580 ax-mulrcl 10581 ax-mulcom 10582 ax-addass 10583 ax-mulass 10584 ax-distr 10585 ax-i2m1 10586 ax-1ne0 10587 ax-1rid 10588 ax-rnegex 10589 ax-rrecex 10590 ax-cnre 10591 ax-pre-lttri 10592 ax-pre-lttrn 10593 ax-pre-ltadd 10594 ax-pre-mulgt0 10595 |
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 2653 df-clab 2799 df-cleq 2813 df-clel 2891 df-nfc 2959 df-ne 3012 df-nel 3119 df-ral 3138 df-rex 3139 df-reu 3140 df-rab 3142 df-v 3483 df-sbc 3759 df-csb 3867 df-dif 3922 df-un 3924 df-in 3926 df-ss 3935 df-pss 3937 df-nul 4275 df-if 4449 df-pw 4522 df-sn 4549 df-pr 4551 df-tp 4553 df-op 4555 df-uni 4820 df-int 4858 df-iun 4902 df-br 5048 df-opab 5110 df-mpt 5128 df-tr 5154 df-id 5441 df-eprel 5446 df-po 5455 df-so 5456 df-fr 5495 df-we 5497 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6295 df-fun 6338 df-fn 6339 df-f 6340 df-f1 6341 df-fo 6342 df-f1o 6343 df-fv 6344 df-riota 7095 df-ov 7140 df-oprab 7141 df-mpo 7142 df-om 7562 df-1st 7670 df-2nd 7671 df-wrecs 7928 df-recs 7989 df-rdg 8027 df-1o 8083 df-oadd 8087 df-er 8270 df-en 8491 df-dom 8492 df-sdom 8493 df-fin 8494 df-card 9349 df-pnf 10658 df-mnf 10659 df-xr 10660 df-ltxr 10661 df-le 10662 df-sub 10853 df-neg 10854 df-nn 11620 df-2 11682 df-n0 11880 df-z 11964 df-uz 12226 df-fz 12878 df-fzo 13019 df-hash 13676 df-word 13847 df-concat 13903 df-s1 13930 df-substr 13983 df-pfx 14013 df-s2 14190 df-s3 14191 |
This theorem is referenced by: s3eq3seq 14281 |
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