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Mirrors > Home > MPE Home > Th. List > Mathboxes > lhp1cvr | Structured version Visualization version GIF version |
Description: The lattice unit covers a co-atom (lattice hyperplane). (Contributed by NM, 18-May-2012.) |
Ref | Expression |
---|---|
lhp1cvr.u | ⊢ 1 = (1.‘𝐾) |
lhp1cvr.c | ⊢ 𝐶 = ( ⋖ ‘𝐾) |
lhp1cvr.h | ⊢ 𝐻 = (LHyp‘𝐾) |
Ref | Expression |
---|---|
lhp1cvr | ⊢ ((𝐾 ∈ 𝐴 ∧ 𝑊 ∈ 𝐻) → 𝑊𝐶 1 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2737 | . . 3 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
2 | lhp1cvr.u | . . 3 ⊢ 1 = (1.‘𝐾) | |
3 | lhp1cvr.c | . . 3 ⊢ 𝐶 = ( ⋖ ‘𝐾) | |
4 | lhp1cvr.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
5 | 1, 2, 3, 4 | islhp 37747 | . 2 ⊢ (𝐾 ∈ 𝐴 → (𝑊 ∈ 𝐻 ↔ (𝑊 ∈ (Base‘𝐾) ∧ 𝑊𝐶 1 ))) |
6 | 5 | simplbda 503 | 1 ⊢ ((𝐾 ∈ 𝐴 ∧ 𝑊 ∈ 𝐻) → 𝑊𝐶 1 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2110 class class class wbr 5053 ‘cfv 6380 Basecbs 16760 1.cp1 17930 ⋖ ccvr 37013 LHypclh 37735 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pr 5322 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3410 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-br 5054 df-opab 5116 df-mpt 5136 df-id 5455 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-iota 6338 df-fun 6382 df-fv 6388 df-lhyp 37739 |
This theorem is referenced by: lhplt 37751 lhp2lt 37752 lhpexlt 37753 lhpexnle 37757 lhpjat1 37771 lhpmcvr 37774 cdlemb2 37792 lhpat 37794 dih1 39037 |
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