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Mirrors > Home > MPE Home > Th. List > s2eq2s1eq | Structured version Visualization version GIF version |
Description: Two length 2 words are equal iff the corresponding singleton words consisting of their symbols are equal. (Contributed by Alexander van der Vekens, 24-Sep-2018.) |
Ref | Expression |
---|---|
s2eq2s1eq | ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → (〈“𝐴𝐵”〉 = 〈“𝐶𝐷”〉 ↔ (〈“𝐴”〉 = 〈“𝐶”〉 ∧ 〈“𝐵”〉 = 〈“𝐷”〉))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-s2 13802 | . . . 4 ⊢ 〈“𝐴𝐵”〉 = (〈“𝐴”〉 ++ 〈“𝐵”〉) | |
2 | 1 | a1i 11 | . . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → 〈“𝐴𝐵”〉 = (〈“𝐴”〉 ++ 〈“𝐵”〉)) |
3 | df-s2 13802 | . . . 4 ⊢ 〈“𝐶𝐷”〉 = (〈“𝐶”〉 ++ 〈“𝐷”〉) | |
4 | 3 | a1i 11 | . . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → 〈“𝐶𝐷”〉 = (〈“𝐶”〉 ++ 〈“𝐷”〉)) |
5 | 2, 4 | eqeq12d 2786 | . 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → (〈“𝐴𝐵”〉 = 〈“𝐶𝐷”〉 ↔ (〈“𝐴”〉 ++ 〈“𝐵”〉) = (〈“𝐶”〉 ++ 〈“𝐷”〉))) |
6 | s1cl 13582 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → 〈“𝐴”〉 ∈ Word 𝑉) | |
7 | s1cl 13582 | . . . . 5 ⊢ (𝐵 ∈ 𝑉 → 〈“𝐵”〉 ∈ Word 𝑉) | |
8 | 6, 7 | anim12i 600 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (〈“𝐴”〉 ∈ Word 𝑉 ∧ 〈“𝐵”〉 ∈ Word 𝑉)) |
9 | 8 | adantr 466 | . . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → (〈“𝐴”〉 ∈ Word 𝑉 ∧ 〈“𝐵”〉 ∈ Word 𝑉)) |
10 | s1cl 13582 | . . . . 5 ⊢ (𝐶 ∈ 𝑉 → 〈“𝐶”〉 ∈ Word 𝑉) | |
11 | s1cl 13582 | . . . . 5 ⊢ (𝐷 ∈ 𝑉 → 〈“𝐷”〉 ∈ Word 𝑉) | |
12 | 10, 11 | anim12i 600 | . . . 4 ⊢ ((𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉) → (〈“𝐶”〉 ∈ Word 𝑉 ∧ 〈“𝐷”〉 ∈ Word 𝑉)) |
13 | 12 | adantl 467 | . . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → (〈“𝐶”〉 ∈ Word 𝑉 ∧ 〈“𝐷”〉 ∈ Word 𝑉)) |
14 | s1len 13586 | . . . . 5 ⊢ (♯‘〈“𝐴”〉) = 1 | |
15 | s1len 13586 | . . . . 5 ⊢ (♯‘〈“𝐶”〉) = 1 | |
16 | 14, 15 | eqtr4i 2796 | . . . 4 ⊢ (♯‘〈“𝐴”〉) = (♯‘〈“𝐶”〉) |
17 | 16 | a1i 11 | . . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → (♯‘〈“𝐴”〉) = (♯‘〈“𝐶”〉)) |
18 | ccatopth 13679 | . . 3 ⊢ (((〈“𝐴”〉 ∈ Word 𝑉 ∧ 〈“𝐵”〉 ∈ Word 𝑉) ∧ (〈“𝐶”〉 ∈ Word 𝑉 ∧ 〈“𝐷”〉 ∈ Word 𝑉) ∧ (♯‘〈“𝐴”〉) = (♯‘〈“𝐶”〉)) → ((〈“𝐴”〉 ++ 〈“𝐵”〉) = (〈“𝐶”〉 ++ 〈“𝐷”〉) ↔ (〈“𝐴”〉 = 〈“𝐶”〉 ∧ 〈“𝐵”〉 = 〈“𝐷”〉))) | |
19 | 9, 13, 17, 18 | syl3anc 1476 | . 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → ((〈“𝐴”〉 ++ 〈“𝐵”〉) = (〈“𝐶”〉 ++ 〈“𝐷”〉) ↔ (〈“𝐴”〉 = 〈“𝐶”〉 ∧ 〈“𝐵”〉 = 〈“𝐷”〉))) |
20 | 5, 19 | bitrd 268 | 1 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → (〈“𝐴𝐵”〉 = 〈“𝐶𝐷”〉 ↔ (〈“𝐴”〉 = 〈“𝐶”〉 ∧ 〈“𝐵”〉 = 〈“𝐷”〉))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 382 = wceq 1631 ∈ wcel 2145 ‘cfv 6030 (class class class)co 6796 1c1 10143 ♯chash 13321 Word cword 13487 ++ cconcat 13489 〈“cs1 13490 〈“cs2 13795 |
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 4905 ax-sep 4916 ax-nul 4924 ax-pow 4975 ax-pr 5035 ax-un 7100 ax-cnex 10198 ax-resscn 10199 ax-1cn 10200 ax-icn 10201 ax-addcl 10202 ax-addrcl 10203 ax-mulcl 10204 ax-mulrcl 10205 ax-mulcom 10206 ax-addass 10207 ax-mulass 10208 ax-distr 10209 ax-i2m1 10210 ax-1ne0 10211 ax-1rid 10212 ax-rnegex 10213 ax-rrecex 10214 ax-cnre 10215 ax-pre-lttri 10216 ax-pre-lttrn 10217 ax-pre-ltadd 10218 ax-pre-mulgt0 10219 |
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-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 4227 df-pw 4300 df-sn 4318 df-pr 4320 df-tp 4322 df-op 4324 df-uni 4576 df-int 4613 df-iun 4657 df-br 4788 df-opab 4848 df-mpt 4865 df-tr 4888 df-id 5158 df-eprel 5163 df-po 5171 df-so 5172 df-fr 5209 df-we 5211 df-xp 5256 df-rel 5257 df-cnv 5258 df-co 5259 df-dm 5260 df-rn 5261 df-res 5262 df-ima 5263 df-pred 5822 df-ord 5868 df-on 5869 df-lim 5870 df-suc 5871 df-iota 5993 df-fun 6032 df-fn 6033 df-f 6034 df-f1 6035 df-fo 6036 df-f1o 6037 df-fv 6038 df-riota 6757 df-ov 6799 df-oprab 6800 df-mpt2 6801 df-om 7217 df-1st 7319 df-2nd 7320 df-wrecs 7563 df-recs 7625 df-rdg 7663 df-1o 7717 df-oadd 7721 df-er 7900 df-en 8114 df-dom 8115 df-sdom 8116 df-fin 8117 df-card 8969 df-pnf 10282 df-mnf 10283 df-xr 10284 df-ltxr 10285 df-le 10286 df-sub 10474 df-neg 10475 df-nn 11227 df-n0 11500 df-z 11585 df-uz 11894 df-fz 12534 df-fzo 12674 df-hash 13322 df-word 13495 df-concat 13497 df-s1 13498 df-substr 13499 df-s2 13802 |
This theorem is referenced by: s2eq2seq 13891 2swrd2eqwrdeq 13906 |
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