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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 2737 | . . 3 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
| 2 | eqid 2737 | . . 3 ⊢ (le‘𝐾) = (le‘𝐾) | |
| 3 | eqid 2737 | . . 3 ⊢ (join‘𝐾) = (join‘𝐾) | |
| 4 | eqid 2737 | . . 3 ⊢ (meet‘𝐾) = (meet‘𝐾) | |
| 5 | eqid 2737 | . . 3 ⊢ (oc‘𝐾) = (oc‘𝐾) | |
| 6 | 1, 2, 3, 4, 5 | isoml 39239 | . 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 1540 ∈ wcel 2108 ∀wral 3061 class class class wbr 5143 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 lecple 17304 occoc 17305 joincjn 18357 meetcmee 18358 OLcol 39175 OMLcoml 39176 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-iota 6514 df-fv 6569 df-ov 7434 df-oml 39180 | 
| This theorem is referenced by: omlop 39242 omllat 39243 omllaw3 39246 omllaw4 39247 cmtcomlemN 39249 cmtbr2N 39254 cmtbr3N 39255 omlfh1N 39259 omlfh3N 39260 omlspjN 39262 hlol 39362 | 
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