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Mathbox for Norm Megill |
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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 2798 | . . 3 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
2 | eqid 2798 | . . 3 ⊢ (lub‘𝐾) = (lub‘𝐾) | |
3 | eqid 2798 | . . 3 ⊢ (glb‘𝐾) = (glb‘𝐾) | |
4 | eqid 2798 | . . 3 ⊢ (le‘𝐾) = (le‘𝐾) | |
5 | eqid 2798 | . . 3 ⊢ (oc‘𝐾) = (oc‘𝐾) | |
6 | eqid 2798 | . . 3 ⊢ (join‘𝐾) = (join‘𝐾) | |
7 | eqid 2798 | . . 3 ⊢ (meet‘𝐾) = (meet‘𝐾) | |
8 | eqid 2798 | . . 3 ⊢ (0.‘𝐾) = (0.‘𝐾) | |
9 | eqid 2798 | . . 3 ⊢ (1.‘𝐾) = (1.‘𝐾) | |
10 | 1, 2, 3, 4, 5, 6, 7, 8, 9 | isopos 36476 | . 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 1188 | . 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 220 | 1 ⊢ (𝐾 ∈ OP → 𝐾 ∈ Poset) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ∀wral 3106 class class class wbr 5030 dom cdm 5519 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 lecple 16564 occoc 16565 Posetcpo 17542 lubclub 17544 glbcglb 17545 joincjn 17546 meetcmee 17547 0.cp0 17639 1.cp1 17640 OPcops 36468 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-nul 5174 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-dm 5529 df-iota 6283 df-fv 6332 df-ov 7138 df-oposet 36472 |
This theorem is referenced by: ople0 36483 op1le 36488 opltcon3b 36500 olposN 36511 ncvr1 36568 cvrcmp2 36580 leatb 36588 dalemcea 36956 |
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