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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 2622 | . . 3 ⊢ (le‘𝐾) = (le‘𝐾) | |
5 | eqid 2622 | . . 3 ⊢ (oc‘𝐾) = (oc‘𝐾) | |
6 | eqid 2622 | . . 3 ⊢ (join‘𝐾) = (join‘𝐾) | |
7 | eqid 2622 | . . 3 ⊢ (meet‘𝐾) = (meet‘𝐾) | |
8 | eqid 2622 | . . 3 ⊢ (0.‘𝐾) = (0.‘𝐾) | |
9 | eqid 2622 | . . 3 ⊢ (1.‘𝐾) = (1.‘𝐾) | |
10 | 1, 2, 3, 4, 5, 6, 7, 8, 9 | isopos 34467 | . 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 473 | . . 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 1217 | . 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 207 | 1 ⊢ (𝐾 ∈ OP → (𝐵 ∈ dom 𝑈 ∧ 𝐵 ∈ dom 𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ∀wral 2912 class class class wbr 4653 dom cdm 5114 ‘cfv 5888 (class class class)co 6650 Basecbs 15857 lecple 15948 occoc 15949 Posetcpo 16940 lubclub 16942 glbcglb 16943 joincjn 16944 meetcmee 16945 0.cp0 17037 1.cp1 17038 OPcops 34459 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-nul 4789 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-dm 5124 df-iota 5851 df-fv 5896 df-ov 6653 df-oposet 34463 |
This theorem is referenced by: op0cl 34471 op1cl 34472 op0le 34473 ople1 34478 lhp2lt 35287 |
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