Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > isomliN | Structured version Visualization version GIF version |
Description: Properties that determine an orthomodular lattice. (Contributed by NM, 18-Sep-2011.) (New usage is discouraged.) |
Ref | Expression |
---|---|
isomli.0 | ⊢ 𝐾 ∈ OL |
isomli.b | ⊢ 𝐵 = (Base‘𝐾) |
isomli.l | ⊢ ≤ = (le‘𝐾) |
isomli.j | ⊢ ∨ = (join‘𝐾) |
isomli.m | ⊢ ∧ = (meet‘𝐾) |
isomli.o | ⊢ ⊥ = (oc‘𝐾) |
isomli.7 | ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 ≤ 𝑦 → 𝑦 = (𝑥 ∨ (𝑦 ∧ ( ⊥ ‘𝑥))))) |
Ref | Expression |
---|---|
isomliN | ⊢ 𝐾 ∈ OML |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isomli.0 | . 2 ⊢ 𝐾 ∈ OL | |
2 | isomli.7 | . . 3 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 ≤ 𝑦 → 𝑦 = (𝑥 ∨ (𝑦 ∧ ( ⊥ ‘𝑥))))) | |
3 | 2 | rgen2a 2960 | . 2 ⊢ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 → 𝑦 = (𝑥 ∨ (𝑦 ∧ ( ⊥ ‘𝑥)))) |
4 | isomli.b | . . 3 ⊢ 𝐵 = (Base‘𝐾) | |
5 | isomli.l | . . 3 ⊢ ≤ = (le‘𝐾) | |
6 | isomli.j | . . 3 ⊢ ∨ = (join‘𝐾) | |
7 | isomli.m | . . 3 ⊢ ∧ = (meet‘𝐾) | |
8 | isomli.o | . . 3 ⊢ ⊥ = (oc‘𝐾) | |
9 | 4, 5, 6, 7, 8 | isoml 33543 | . 2 ⊢ (𝐾 ∈ OML ↔ (𝐾 ∈ OL ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 → 𝑦 = (𝑥 ∨ (𝑦 ∧ ( ⊥ ‘𝑥)))))) |
10 | 1, 3, 9 | mpbir2an 957 | 1 ⊢ 𝐾 ∈ OML |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 lecple 15775 occoc 15776 joincjn 16767 meetcmee 16768 OLcol 33479 OMLcoml 33480 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-iota 5768 df-fv 5812 df-ov 6552 df-oml 33484 |
This theorem is referenced by: (None) |
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