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Mirrors > Home > MPE Home > Th. List > s3eqd | Structured version Visualization version GIF version |
Description: Equality theorem for a length 3 word. (Contributed by Mario Carneiro, 27-Feb-2016.) |
Ref | Expression |
---|---|
s2eqd.1 | ⊢ (𝜑 → 𝐴 = 𝑁) |
s2eqd.2 | ⊢ (𝜑 → 𝐵 = 𝑂) |
s3eqd.3 | ⊢ (𝜑 → 𝐶 = 𝑃) |
Ref | Expression |
---|---|
s3eqd | ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉 = 〈“𝑁𝑂𝑃”〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | s2eqd.1 | . . . 4 ⊢ (𝜑 → 𝐴 = 𝑁) | |
2 | s2eqd.2 | . . . 4 ⊢ (𝜑 → 𝐵 = 𝑂) | |
3 | 1, 2 | s2eqd 14576 | . . 3 ⊢ (𝜑 → 〈“𝐴𝐵”〉 = 〈“𝑁𝑂”〉) |
4 | s3eqd.3 | . . . 4 ⊢ (𝜑 → 𝐶 = 𝑃) | |
5 | 4 | s1eqd 14306 | . . 3 ⊢ (𝜑 → 〈“𝐶”〉 = 〈“𝑃”〉) |
6 | 3, 5 | oveq12d 7293 | . 2 ⊢ (𝜑 → (〈“𝐴𝐵”〉 ++ 〈“𝐶”〉) = (〈“𝑁𝑂”〉 ++ 〈“𝑃”〉)) |
7 | df-s3 14562 | . 2 ⊢ 〈“𝐴𝐵𝐶”〉 = (〈“𝐴𝐵”〉 ++ 〈“𝐶”〉) | |
8 | df-s3 14562 | . 2 ⊢ 〈“𝑁𝑂𝑃”〉 = (〈“𝑁𝑂”〉 ++ 〈“𝑃”〉) | |
9 | 6, 7, 8 | 3eqtr4g 2803 | 1 ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉 = 〈“𝑁𝑂𝑃”〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1539 (class class class)co 7275 ++ cconcat 14273 〈“cs1 14300 〈“cs2 14554 〈“cs3 14555 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2709 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-rab 3073 df-v 3434 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5075 df-iota 6391 df-fv 6441 df-ov 7278 df-s1 14301 df-s2 14561 df-s3 14562 |
This theorem is referenced by: s4eqd 14578 s3eq2 14583 s3sndisj 14678 s3iunsndisj 14679 ragcgr 27068 perpneq 27075 isperp2 27076 isperp2d 27077 footexALT 27079 footexlem2 27081 foot 27083 perprag 27087 perpdragALT 27088 colperpexlem1 27091 lmiisolem 27157 hypcgrlem1 27160 hypcgrlem2 27161 trgcopyeu 27167 iscgra 27170 iscgra1 27171 iscgrad 27172 sacgr 27192 isleag 27208 isleagd 27209 iseqlg 27228 |
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