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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 2824 | . . 3 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
2 | eqid 2824 | . . 3 ⊢ (lub‘𝐾) = (lub‘𝐾) | |
3 | eqid 2824 | . . 3 ⊢ (glb‘𝐾) = (glb‘𝐾) | |
4 | eqid 2824 | . . 3 ⊢ (le‘𝐾) = (le‘𝐾) | |
5 | eqid 2824 | . . 3 ⊢ (oc‘𝐾) = (oc‘𝐾) | |
6 | eqid 2824 | . . 3 ⊢ (join‘𝐾) = (join‘𝐾) | |
7 | eqid 2824 | . . 3 ⊢ (meet‘𝐾) = (meet‘𝐾) | |
8 | eqid 2824 | . . 3 ⊢ (0.‘𝐾) = (0.‘𝐾) | |
9 | eqid 2824 | . . 3 ⊢ (1.‘𝐾) = (1.‘𝐾) | |
10 | 1, 2, 3, 4, 5, 6, 7, 8, 9 | isopos 36320 | . 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 1187 | . 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 219 | 1 ⊢ (𝐾 ∈ OP → 𝐾 ∈ Poset) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1536 ∈ wcel 2113 ∀wral 3141 class class class wbr 5069 dom cdm 5558 ‘cfv 6358 (class class class)co 7159 Basecbs 16486 lecple 16575 occoc 16576 Posetcpo 17553 lubclub 17555 glbcglb 17556 joincjn 17557 meetcmee 17558 0.cp0 17650 1.cp1 17651 OPcops 36312 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-nul 5213 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-sbc 3776 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-dm 5568 df-iota 6317 df-fv 6366 df-ov 7162 df-oposet 36316 |
This theorem is referenced by: ople0 36327 op1le 36332 opltcon3b 36344 olposN 36355 ncvr1 36412 cvrcmp2 36424 leatb 36432 dalemcea 36800 |
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