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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 2622 | . . 3 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
2 | eqid 2622 | . . 3 ⊢ (le‘𝐾) = (le‘𝐾) | |
3 | eqid 2622 | . . 3 ⊢ (join‘𝐾) = (join‘𝐾) | |
4 | eqid 2622 | . . 3 ⊢ (meet‘𝐾) = (meet‘𝐾) | |
5 | eqid 2622 | . . 3 ⊢ (oc‘𝐾) = (oc‘𝐾) | |
6 | 1, 2, 3, 4, 5 | isoml 34525 | . 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 476 | 1 ⊢ (𝐾 ∈ OML → 𝐾 ∈ OL) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 ∀wral 2912 class class class wbr 4653 ‘cfv 5888 (class class class)co 6650 Basecbs 15857 lecple 15948 occoc 15949 joincjn 16944 meetcmee 16945 OLcol 34461 OMLcoml 34462 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-iota 5851 df-fv 5896 df-ov 6653 df-oml 34466 |
This theorem is referenced by: omlop 34528 omllat 34529 omllaw3 34532 omllaw4 34533 cmtcomlemN 34535 cmtbr2N 34540 cmtbr3N 34541 omlfh1N 34545 omlfh3N 34546 omlspjN 34548 hlol 34648 |
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