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 | rgen2 3203 | . 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 36368 | . 2 ⊢ (𝐾 ∈ OML ↔ (𝐾 ∈ OL ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 → 𝑦 = (𝑥 ∨ (𝑦 ∧ ( ⊥ ‘𝑥)))))) |
10 | 1, 3, 9 | mpbir2an 709 | 1 ⊢ 𝐾 ∈ OML |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∀wral 3138 class class class wbr 5058 ‘cfv 6349 (class class class)co 7150 Basecbs 16477 lecple 16566 occoc 16567 joincjn 17548 meetcmee 17549 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-oml 36309 |
This theorem is referenced by: (None) |
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