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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 36370 | . 2 ⊢ (𝐾 ∈ OML → 𝐾 ∈ OL) | |
2 | ollat 36343 | . 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 2110 Latclat 17649 OLcol 36304 OMLcoml 36305 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-iota 6308 df-fv 6357 df-ov 7153 df-ol 36308 df-oml 36309 |
This theorem is referenced by: omllaw2N 36374 omllaw4 36376 omllaw5N 36377 cmtcomlemN 36378 cmt2N 36380 cmtbr2N 36383 cmtbr3N 36384 cmtbr4N 36385 lecmtN 36386 cmtidN 36387 omlfh1N 36388 omlfh3N 36389 omlmod1i2N 36390 omlspjN 36391 |
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