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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > islhp | Structured version Visualization version GIF version |
Description: The predicate "is a co-atom (lattice hyperplane)". (Contributed by NM, 11-May-2012.) |
Ref | Expression |
---|---|
lhpset.b | ⊢ 𝐵 = (Base‘𝐾) |
lhpset.u | ⊢ 1 = (1.‘𝐾) |
lhpset.c | ⊢ 𝐶 = ( ⋖ ‘𝐾) |
lhpset.h | ⊢ 𝐻 = (LHyp‘𝐾) |
Ref | Expression |
---|---|
islhp | ⊢ (𝐾 ∈ 𝐴 → (𝑊 ∈ 𝐻 ↔ (𝑊 ∈ 𝐵 ∧ 𝑊𝐶 1 ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lhpset.b | . . . 4 ⊢ 𝐵 = (Base‘𝐾) | |
2 | lhpset.u | . . . 4 ⊢ 1 = (1.‘𝐾) | |
3 | lhpset.c | . . . 4 ⊢ 𝐶 = ( ⋖ ‘𝐾) | |
4 | lhpset.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
5 | 1, 2, 3, 4 | lhpset 36071 | . . 3 ⊢ (𝐾 ∈ 𝐴 → 𝐻 = {𝑤 ∈ 𝐵 ∣ 𝑤𝐶 1 }) |
6 | 5 | eleq2d 2893 | . 2 ⊢ (𝐾 ∈ 𝐴 → (𝑊 ∈ 𝐻 ↔ 𝑊 ∈ {𝑤 ∈ 𝐵 ∣ 𝑤𝐶 1 })) |
7 | breq1 4877 | . . 3 ⊢ (𝑤 = 𝑊 → (𝑤𝐶 1 ↔ 𝑊𝐶 1 )) | |
8 | 7 | elrab 3586 | . 2 ⊢ (𝑊 ∈ {𝑤 ∈ 𝐵 ∣ 𝑤𝐶 1 } ↔ (𝑊 ∈ 𝐵 ∧ 𝑊𝐶 1 )) |
9 | 6, 8 | syl6bb 279 | 1 ⊢ (𝐾 ∈ 𝐴 → (𝑊 ∈ 𝐻 ↔ (𝑊 ∈ 𝐵 ∧ 𝑊𝐶 1 ))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1658 ∈ wcel 2166 {crab 3122 class class class wbr 4874 ‘cfv 6124 Basecbs 16223 1.cp1 17392 ⋖ ccvr 35338 LHypclh 36060 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-sep 5006 ax-nul 5014 ax-pr 5128 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ral 3123 df-rex 3124 df-rab 3127 df-v 3417 df-sbc 3664 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-nul 4146 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4660 df-br 4875 df-opab 4937 df-mpt 4954 df-id 5251 df-xp 5349 df-rel 5350 df-cnv 5351 df-co 5352 df-dm 5353 df-iota 6087 df-fun 6126 df-fv 6132 df-lhyp 36064 |
This theorem is referenced by: islhp2 36073 lhpbase 36074 lhp1cvr 36075 |
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