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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 3186 | . 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 35313 | . 2 ⊢ (𝐾 ∈ OML ↔ (𝐾 ∈ OL ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 → 𝑦 = (𝑥 ∨ (𝑦 ∧ ( ⊥ ‘𝑥)))))) |
10 | 1, 3, 9 | mpbir2an 704 | 1 ⊢ 𝐾 ∈ OML |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1658 ∈ wcel 2166 ∀wral 3117 class class class wbr 4873 ‘cfv 6123 (class class class)co 6905 Basecbs 16222 lecple 16312 occoc 16313 joincjn 17297 meetcmee 17298 OLcol 35249 OMLcoml 35250 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4659 df-br 4874 df-iota 6086 df-fv 6131 df-ov 6908 df-oml 35254 |
This theorem is referenced by: (None) |
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