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Mirrors > Home > MPE Home > Th. List > Mathboxes > op01dm | Structured version Visualization version GIF version |
Description: Conditions necessary for zero and unit elements to exist. (Contributed by NM, 14-Sep-2018.) |
Ref | Expression |
---|---|
op01dm.b | ⊢ 𝐵 = (Base‘𝐾) |
op01dm.u | ⊢ 𝑈 = (lub‘𝐾) |
op01dm.g | ⊢ 𝐺 = (glb‘𝐾) |
Ref | Expression |
---|---|
op01dm | ⊢ (𝐾 ∈ OP → (𝐵 ∈ dom 𝑈 ∧ 𝐵 ∈ dom 𝐺)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | op01dm.b | . . 3 ⊢ 𝐵 = (Base‘𝐾) | |
2 | op01dm.u | . . 3 ⊢ 𝑈 = (lub‘𝐾) | |
3 | op01dm.g | . . 3 ⊢ 𝐺 = (glb‘𝐾) | |
4 | eqid 2610 | . . 3 ⊢ (le‘𝐾) = (le‘𝐾) | |
5 | eqid 2610 | . . 3 ⊢ (oc‘𝐾) = (oc‘𝐾) | |
6 | eqid 2610 | . . 3 ⊢ (join‘𝐾) = (join‘𝐾) | |
7 | eqid 2610 | . . 3 ⊢ (meet‘𝐾) = (meet‘𝐾) | |
8 | eqid 2610 | . . 3 ⊢ (0.‘𝐾) = (0.‘𝐾) | |
9 | eqid 2610 | . . 3 ⊢ (1.‘𝐾) = (1.‘𝐾) | |
10 | 1, 2, 3, 4, 5, 6, 7, 8, 9 | isopos 33485 | . 2 ⊢ (𝐾 ∈ OP ↔ ((𝐾 ∈ Poset ∧ 𝐵 ∈ dom 𝑈 ∧ 𝐵 ∈ dom 𝐺) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((((oc‘𝐾)‘𝑥) ∈ 𝐵 ∧ ((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 | simpl 472 | . . 3 ⊢ (((𝐵 ∈ dom 𝑈 ∧ 𝐵 ∈ dom 𝐺) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((((oc‘𝐾)‘𝑥) ∈ 𝐵 ∧ ((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))) → (𝐵 ∈ dom 𝑈 ∧ 𝐵 ∈ dom 𝐺)) | |
12 | 11 | 3adantl1 1210 | . 2 ⊢ (((𝐾 ∈ Poset ∧ 𝐵 ∈ dom 𝑈 ∧ 𝐵 ∈ dom 𝐺) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((((oc‘𝐾)‘𝑥) ∈ 𝐵 ∧ ((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))) → (𝐵 ∈ dom 𝑈 ∧ 𝐵 ∈ dom 𝐺)) |
13 | 10, 12 | sylbi 206 | 1 ⊢ (𝐾 ∈ OP → (𝐵 ∈ dom 𝑈 ∧ 𝐵 ∈ dom 𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ∀wral 2896 class class class wbr 4583 dom cdm 5038 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 lecple 15775 occoc 15776 Posetcpo 16763 lubclub 16765 glbcglb 16766 joincjn 16767 meetcmee 16768 0.cp0 16860 1.cp1 16861 OPcops 33477 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-nul 4717 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-dm 5048 df-iota 5768 df-fv 5812 df-ov 6552 df-oposet 33481 |
This theorem is referenced by: op0cl 33489 op1cl 33490 op0le 33491 ople1 33496 lhp2lt 34305 |
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