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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 14902 | . . 3 ⊢ (𝜑 → 〈“𝐴𝐵”〉 = 〈“𝑁𝑂”〉) |
| 4 | s3eqd.3 | . . . 4 ⊢ (𝜑 → 𝐶 = 𝑃) | |
| 5 | 4 | s1eqd 14639 | . . 3 ⊢ (𝜑 → 〈“𝐶”〉 = 〈“𝑃”〉) |
| 6 | 3, 5 | oveq12d 7449 | . 2 ⊢ (𝜑 → (〈“𝐴𝐵”〉 ++ 〈“𝐶”〉) = (〈“𝑁𝑂”〉 ++ 〈“𝑃”〉)) |
| 7 | df-s3 14888 | . 2 ⊢ 〈“𝐴𝐵𝐶”〉 = (〈“𝐴𝐵”〉 ++ 〈“𝐶”〉) | |
| 8 | df-s3 14888 | . 2 ⊢ 〈“𝑁𝑂𝑃”〉 = (〈“𝑁𝑂”〉 ++ 〈“𝑃”〉) | |
| 9 | 6, 7, 8 | 3eqtr4g 2802 | 1 ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉 = 〈“𝑁𝑂𝑃”〉) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 (class class class)co 7431 ++ cconcat 14608 〈“cs1 14633 〈“cs2 14880 〈“cs3 14881 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-iota 6514 df-fv 6569 df-ov 7434 df-s1 14634 df-s2 14887 df-s3 14888 |
| This theorem is referenced by: s4eqd 14904 s3eq2 14909 s3sndisj 15006 s3iunsndisj 15007 ragcgr 28715 perpneq 28722 isperp2 28723 isperp2d 28724 footexALT 28726 footexlem2 28728 foot 28730 perprag 28734 perpdragALT 28735 colperpexlem1 28738 lmiisolem 28804 hypcgrlem1 28807 hypcgrlem2 28808 trgcopyeu 28814 iscgra 28817 iscgra1 28818 iscgrad 28819 sacgr 28839 isleag 28855 isleagd 28856 iseqlg 28875 |
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