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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 14225 | . . 3 ⊢ (𝜑 → 〈“𝐴𝐵”〉 = 〈“𝑁𝑂”〉) |
4 | s3eqd.3 | . . . 4 ⊢ (𝜑 → 𝐶 = 𝑃) | |
5 | 4 | s1eqd 13955 | . . 3 ⊢ (𝜑 → 〈“𝐶”〉 = 〈“𝑃”〉) |
6 | 3, 5 | oveq12d 7174 | . 2 ⊢ (𝜑 → (〈“𝐴𝐵”〉 ++ 〈“𝐶”〉) = (〈“𝑁𝑂”〉 ++ 〈“𝑃”〉)) |
7 | df-s3 14211 | . 2 ⊢ 〈“𝐴𝐵𝐶”〉 = (〈“𝐴𝐵”〉 ++ 〈“𝐶”〉) | |
8 | df-s3 14211 | . 2 ⊢ 〈“𝑁𝑂𝑃”〉 = (〈“𝑁𝑂”〉 ++ 〈“𝑃”〉) | |
9 | 6, 7, 8 | 3eqtr4g 2881 | 1 ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉 = 〈“𝑁𝑂𝑃”〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 (class class class)co 7156 ++ cconcat 13922 〈“cs1 13949 〈“cs2 14203 〈“cs3 14204 |
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 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-iota 6314 df-fv 6363 df-ov 7159 df-s1 13950 df-s2 14210 df-s3 14211 |
This theorem is referenced by: s4eqd 14227 s3eq2 14232 s3sndisj 14327 s3iunsndisj 14328 ragcgr 26493 perpneq 26500 isperp2 26501 isperp2d 26502 footexALT 26504 footexlem2 26506 foot 26508 perprag 26512 perpdragALT 26513 colperpexlem1 26516 lmiisolem 26582 hypcgrlem1 26585 hypcgrlem2 26586 trgcopyeu 26592 iscgra 26595 iscgra1 26596 iscgrad 26597 sacgr 26617 isleag 26633 isleagd 26634 iseqlg 26653 |
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