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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 2818 | . . 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 37012 | . 2 ⊢ (𝐾 ∈ 𝐴 → (𝑊 ∈ 𝐻 ↔ (𝑊 ∈ (Base‘𝐾) ∧ 𝑊𝐶 1 ))) |
6 | 5 | simplbda 500 | 1 ⊢ ((𝐾 ∈ 𝐴 ∧ 𝑊 ∈ 𝐻) → 𝑊𝐶 1 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 class class class wbr 5057 ‘cfv 6348 Basecbs 16471 1.cp1 17636 ⋖ ccvr 36278 LHypclh 37000 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-iota 6307 df-fun 6350 df-fv 6356 df-lhyp 37004 |
This theorem is referenced by: lhplt 37016 lhp2lt 37017 lhpexlt 37018 lhpexnle 37022 lhpjat1 37036 lhpmcvr 37039 cdlemb2 37057 lhpat 37059 dih1 38302 |
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