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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 2736 | . . 3 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
2 | eqid 2736 | . . 3 ⊢ (le‘𝐾) = (le‘𝐾) | |
3 | eqid 2736 | . . 3 ⊢ (join‘𝐾) = (join‘𝐾) | |
4 | eqid 2736 | . . 3 ⊢ (meet‘𝐾) = (meet‘𝐾) | |
5 | eqid 2736 | . . 3 ⊢ (oc‘𝐾) = (oc‘𝐾) | |
6 | 1, 2, 3, 4, 5 | isoml 36938 | . 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 501 | 1 ⊢ (𝐾 ∈ OML → 𝐾 ∈ OL) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2112 ∀wral 3051 class class class wbr 5039 ‘cfv 6358 (class class class)co 7191 Basecbs 16666 lecple 16756 occoc 16757 joincjn 17772 meetcmee 17773 OLcol 36874 OMLcoml 36875 |
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 2018 ax-8 2114 ax-9 2122 ax-ext 2708 |
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-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-ral 3056 df-rab 3060 df-v 3400 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-iota 6316 df-fv 6366 df-ov 7194 df-oml 36879 |
This theorem is referenced by: omlop 36941 omllat 36942 omllaw3 36945 omllaw4 36946 cmtcomlemN 36948 cmtbr2N 36953 cmtbr3N 36954 omlfh1N 36958 omlfh3N 36959 omlspjN 36961 hlol 37061 |
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