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| Mirrors > Home > MPE Home > Th. List > Mathboxes > opposet | Structured version Visualization version GIF version | ||
| Description: Every orthoposet is a poset. (Contributed by NM, 12-Oct-2011.) | 
| Ref | Expression | 
|---|---|
| opposet | ⊢ (𝐾 ∈ OP → 𝐾 ∈ Poset) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | eqid 2736 | . . 3 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
| 2 | eqid 2736 | . . 3 ⊢ (lub‘𝐾) = (lub‘𝐾) | |
| 3 | eqid 2736 | . . 3 ⊢ (glb‘𝐾) = (glb‘𝐾) | |
| 4 | eqid 2736 | . . 3 ⊢ (le‘𝐾) = (le‘𝐾) | |
| 5 | eqid 2736 | . . 3 ⊢ (oc‘𝐾) = (oc‘𝐾) | |
| 6 | eqid 2736 | . . 3 ⊢ (join‘𝐾) = (join‘𝐾) | |
| 7 | eqid 2736 | . . 3 ⊢ (meet‘𝐾) = (meet‘𝐾) | |
| 8 | eqid 2736 | . . 3 ⊢ (0.‘𝐾) = (0.‘𝐾) | |
| 9 | eqid 2736 | . . 3 ⊢ (1.‘𝐾) = (1.‘𝐾) | |
| 10 | 1, 2, 3, 4, 5, 6, 7, 8, 9 | isopos 39182 | . 2 ⊢ (𝐾 ∈ OP ↔ ((𝐾 ∈ Poset ∧ (Base‘𝐾) ∈ dom (lub‘𝐾) ∧ (Base‘𝐾) ∈ dom (glb‘𝐾)) ∧ ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)((((oc‘𝐾)‘𝑥) ∈ (Base‘𝐾) ∧ ((oc‘𝐾)‘((oc‘𝐾)‘𝑥)) = 𝑥 ∧ (𝑥(le‘𝐾)𝑦 → ((oc‘𝐾)‘𝑦)(le‘𝐾)((oc‘𝐾)‘𝑥))) ∧ (𝑥(join‘𝐾)((oc‘𝐾)‘𝑥)) = (1.‘𝐾) ∧ (𝑥(meet‘𝐾)((oc‘𝐾)‘𝑥)) = (0.‘𝐾)))) | 
| 11 | simpl1 1191 | . 2 ⊢ (((𝐾 ∈ Poset ∧ (Base‘𝐾) ∈ dom (lub‘𝐾) ∧ (Base‘𝐾) ∈ dom (glb‘𝐾)) ∧ ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)((((oc‘𝐾)‘𝑥) ∈ (Base‘𝐾) ∧ ((oc‘𝐾)‘((oc‘𝐾)‘𝑥)) = 𝑥 ∧ (𝑥(le‘𝐾)𝑦 → ((oc‘𝐾)‘𝑦)(le‘𝐾)((oc‘𝐾)‘𝑥))) ∧ (𝑥(join‘𝐾)((oc‘𝐾)‘𝑥)) = (1.‘𝐾) ∧ (𝑥(meet‘𝐾)((oc‘𝐾)‘𝑥)) = (0.‘𝐾))) → 𝐾 ∈ Poset) | |
| 12 | 10, 11 | sylbi 217 | 1 ⊢ (𝐾 ∈ OP → 𝐾 ∈ Poset) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1539 ∈ wcel 2107 ∀wral 3060 class class class wbr 5142 dom cdm 5684 ‘cfv 6560 (class class class)co 7432 Basecbs 17248 lecple 17305 occoc 17306 Posetcpo 18354 lubclub 18356 glbcglb 18357 joincjn 18358 meetcmee 18359 0.cp0 18469 1.cp1 18470 OPcops 39174 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 ax-nul 5305 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ne 2940 df-ral 3061 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-dm 5694 df-iota 6513 df-fv 6568 df-ov 7435 df-oposet 39178 | 
| This theorem is referenced by: ople0 39189 op1le 39194 opltcon3b 39206 olposN 39217 ncvr1 39274 cvrcmp2 39286 leatb 39294 dalemcea 39663 | 
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