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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 14327 | . . 3 ⊢ (𝜑 → 〈“𝐴𝐵”〉 = 〈“𝑁𝑂”〉) |
4 | s3eqd.3 | . . . 4 ⊢ (𝜑 → 𝐶 = 𝑃) | |
5 | 4 | s1eqd 14057 | . . 3 ⊢ (𝜑 → 〈“𝐶”〉 = 〈“𝑃”〉) |
6 | 3, 5 | oveq12d 7201 | . 2 ⊢ (𝜑 → (〈“𝐴𝐵”〉 ++ 〈“𝐶”〉) = (〈“𝑁𝑂”〉 ++ 〈“𝑃”〉)) |
7 | df-s3 14313 | . 2 ⊢ 〈“𝐴𝐵𝐶”〉 = (〈“𝐴𝐵”〉 ++ 〈“𝐶”〉) | |
8 | df-s3 14313 | . 2 ⊢ 〈“𝑁𝑂𝑃”〉 = (〈“𝑁𝑂”〉 ++ 〈“𝑃”〉) | |
9 | 6, 7, 8 | 3eqtr4g 2799 | 1 ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉 = 〈“𝑁𝑂𝑃”〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 (class class class)co 7183 ++ cconcat 14024 〈“cs1 14051 〈“cs2 14305 〈“cs3 14306 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-ext 2711 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-ex 1787 df-sb 2075 df-clab 2718 df-cleq 2731 df-clel 2812 df-v 3402 df-un 3858 df-in 3860 df-ss 3870 df-sn 4527 df-pr 4529 df-op 4533 df-uni 4807 df-br 5041 df-iota 6308 df-fv 6358 df-ov 7186 df-s1 14052 df-s2 14312 df-s3 14313 |
This theorem is referenced by: s4eqd 14329 s3eq2 14334 s3sndisj 14429 s3iunsndisj 14430 ragcgr 26666 perpneq 26673 isperp2 26674 isperp2d 26675 footexALT 26677 footexlem2 26679 foot 26681 perprag 26685 perpdragALT 26686 colperpexlem1 26689 lmiisolem 26755 hypcgrlem1 26758 hypcgrlem2 26759 trgcopyeu 26765 iscgra 26768 iscgra1 26769 iscgrad 26770 sacgr 26790 isleag 26806 isleagd 26807 iseqlg 26826 |
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