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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 14202 | . . . 4 ⊢ 〈“𝐴𝐵”〉 = (〈“𝐴”〉 ++ 〈“𝐵”〉) | |
2 | 1 | a1i 11 | . . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → 〈“𝐴𝐵”〉 = (〈“𝐴”〉 ++ 〈“𝐵”〉)) |
3 | df-s2 14202 | . . . 4 ⊢ 〈“𝐶𝐷”〉 = (〈“𝐶”〉 ++ 〈“𝐷”〉) | |
4 | 3 | a1i 11 | . . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → 〈“𝐶𝐷”〉 = (〈“𝐶”〉 ++ 〈“𝐷”〉)) |
5 | 2, 4 | eqeq12d 2835 | . 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → (〈“𝐴𝐵”〉 = 〈“𝐶𝐷”〉 ↔ (〈“𝐴”〉 ++ 〈“𝐵”〉) = (〈“𝐶”〉 ++ 〈“𝐷”〉))) |
6 | s1cl 13948 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → 〈“𝐴”〉 ∈ Word 𝑉) | |
7 | s1cl 13948 | . . . . 5 ⊢ (𝐵 ∈ 𝑉 → 〈“𝐵”〉 ∈ Word 𝑉) | |
8 | 6, 7 | anim12i 614 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (〈“𝐴”〉 ∈ Word 𝑉 ∧ 〈“𝐵”〉 ∈ Word 𝑉)) |
9 | 8 | adantr 483 | . . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → (〈“𝐴”〉 ∈ Word 𝑉 ∧ 〈“𝐵”〉 ∈ Word 𝑉)) |
10 | s1cl 13948 | . . . . 5 ⊢ (𝐶 ∈ 𝑉 → 〈“𝐶”〉 ∈ Word 𝑉) | |
11 | s1cl 13948 | . . . . 5 ⊢ (𝐷 ∈ 𝑉 → 〈“𝐷”〉 ∈ Word 𝑉) | |
12 | 10, 11 | anim12i 614 | . . . 4 ⊢ ((𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉) → (〈“𝐶”〉 ∈ Word 𝑉 ∧ 〈“𝐷”〉 ∈ Word 𝑉)) |
13 | 12 | adantl 484 | . . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → (〈“𝐶”〉 ∈ Word 𝑉 ∧ 〈“𝐷”〉 ∈ Word 𝑉)) |
14 | s1len 13952 | . . . . 5 ⊢ (♯‘〈“𝐴”〉) = 1 | |
15 | s1len 13952 | . . . . 5 ⊢ (♯‘〈“𝐶”〉) = 1 | |
16 | 14, 15 | eqtr4i 2845 | . . . 4 ⊢ (♯‘〈“𝐴”〉) = (♯‘〈“𝐶”〉) |
17 | 16 | a1i 11 | . . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → (♯‘〈“𝐴”〉) = (♯‘〈“𝐶”〉)) |
18 | ccatopth 14070 | . . 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 1366 | . 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → ((〈“𝐴”〉 ++ 〈“𝐵”〉) = (〈“𝐶”〉 ++ 〈“𝐷”〉) ↔ (〈“𝐴”〉 = 〈“𝐶”〉 ∧ 〈“𝐵”〉 = 〈“𝐷”〉))) |
20 | 5, 19 | bitrd 281 | 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 208 ∧ wa 398 = wceq 1531 ∈ wcel 2108 ‘cfv 6348 (class class class)co 7148 1c1 10530 ♯chash 13682 Word cword 13853 ++ cconcat 13914 〈“cs1 13941 〈“cs2 14195 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-cnex 10585 ax-resscn 10586 ax-1cn 10587 ax-icn 10588 ax-addcl 10589 ax-addrcl 10590 ax-mulcl 10591 ax-mulrcl 10592 ax-mulcom 10593 ax-addass 10594 ax-mulass 10595 ax-distr 10596 ax-i2m1 10597 ax-1ne0 10598 ax-1rid 10599 ax-rnegex 10600 ax-rrecex 10601 ax-cnre 10602 ax-pre-lttri 10603 ax-pre-lttrn 10604 ax-pre-ltadd 10605 ax-pre-mulgt0 10606 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-nel 3122 df-ral 3141 df-rex 3142 df-reu 3143 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-pss 3952 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7106 df-ov 7151 df-oprab 7152 df-mpo 7153 df-om 7573 df-1st 7681 df-2nd 7682 df-wrecs 7939 df-recs 8000 df-rdg 8038 df-1o 8094 df-oadd 8098 df-er 8281 df-en 8502 df-dom 8503 df-sdom 8504 df-fin 8505 df-card 9360 df-pnf 10669 df-mnf 10670 df-xr 10671 df-ltxr 10672 df-le 10673 df-sub 10864 df-neg 10865 df-nn 11631 df-n0 11890 df-z 11974 df-uz 12236 df-fz 12885 df-fzo 13026 df-hash 13683 df-word 13854 df-concat 13915 df-s1 13942 df-substr 13995 df-pfx 14025 df-s2 14202 |
This theorem is referenced by: s2eq2seq 14291 2swrd2eqwrdeq 14307 |
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