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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > omlol | Structured version Visualization version GIF version |
Description: An orthomodular lattice is an ortholattice. (Contributed by NM, 18-Sep-2011.) |
Ref | Expression |
---|---|
omlol | ⊢ (𝐾 ∈ OML → 𝐾 ∈ OL) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2735 | . . 3 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
2 | eqid 2735 | . . 3 ⊢ (le‘𝐾) = (le‘𝐾) | |
3 | eqid 2735 | . . 3 ⊢ (join‘𝐾) = (join‘𝐾) | |
4 | eqid 2735 | . . 3 ⊢ (meet‘𝐾) = (meet‘𝐾) | |
5 | eqid 2735 | . . 3 ⊢ (oc‘𝐾) = (oc‘𝐾) | |
6 | 1, 2, 3, 4, 5 | isoml 39220 | . 2 ⊢ (𝐾 ∈ OML ↔ (𝐾 ∈ OL ∧ ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)(𝑥(le‘𝐾)𝑦 → 𝑦 = (𝑥(join‘𝐾)(𝑦(meet‘𝐾)((oc‘𝐾)‘𝑥)))))) |
7 | 6 | simplbi 497 | 1 ⊢ (𝐾 ∈ OML → 𝐾 ∈ OL) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 ∀wral 3059 class class class wbr 5148 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 lecple 17305 occoc 17306 joincjn 18369 meetcmee 18370 OLcol 39156 OMLcoml 39157 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-iota 6516 df-fv 6571 df-ov 7434 df-oml 39161 |
This theorem is referenced by: omlop 39223 omllat 39224 omllaw3 39227 omllaw4 39228 cmtcomlemN 39230 cmtbr2N 39235 cmtbr3N 39236 omlfh1N 39240 omlfh3N 39241 omlspjN 39243 hlol 39343 |
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