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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > omllat | Structured version Visualization version GIF version |
Description: An orthomodular lattice is a lattice. (Contributed by NM, 6-Nov-2011.) |
Ref | Expression |
---|---|
omllat | ⊢ (𝐾 ∈ OML → 𝐾 ∈ Lat) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | omlol 38566 | . 2 ⊢ (𝐾 ∈ OML → 𝐾 ∈ OL) | |
2 | ollat 38539 | . 2 ⊢ (𝐾 ∈ OL → 𝐾 ∈ Lat) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝐾 ∈ OML → 𝐾 ∈ Lat) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2098 Latclat 18385 OLcol 38500 OMLcoml 38501 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2695 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-ral 3054 df-rab 3425 df-v 3468 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-br 5139 df-iota 6485 df-fv 6541 df-ov 7404 df-ol 38504 df-oml 38505 |
This theorem is referenced by: omllaw2N 38570 omllaw4 38572 omllaw5N 38573 cmtcomlemN 38574 cmt2N 38576 cmtbr2N 38579 cmtbr3N 38580 cmtbr4N 38581 lecmtN 38582 cmtidN 38583 omlfh1N 38584 omlfh3N 38585 omlmod1i2N 38586 omlspjN 38587 |
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