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Mirrors > Home > MPE Home > Th. List > Mathboxes > op01dm | Structured version Visualization version GIF version |
Description: Conditions necessary for zero and unity 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 2734 | . . 3 ⊢ (le‘𝐾) = (le‘𝐾) | |
5 | eqid 2734 | . . 3 ⊢ (oc‘𝐾) = (oc‘𝐾) | |
6 | eqid 2734 | . . 3 ⊢ (join‘𝐾) = (join‘𝐾) | |
7 | eqid 2734 | . . 3 ⊢ (meet‘𝐾) = (meet‘𝐾) | |
8 | eqid 2734 | . . 3 ⊢ (0.‘𝐾) = (0.‘𝐾) | |
9 | eqid 2734 | . . 3 ⊢ (1.‘𝐾) = (1.‘𝐾) | |
10 | 1, 2, 3, 4, 5, 6, 7, 8, 9 | isopos 39161 | . 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 482 | . . 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 1165 | . 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 217 | 1 ⊢ (𝐾 ∈ OP → (𝐵 ∈ dom 𝑈 ∧ 𝐵 ∈ dom 𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1536 ∈ wcel 2105 ∀wral 3058 class class class wbr 5147 dom cdm 5688 ‘cfv 6562 (class class class)co 7430 Basecbs 17244 lecple 17304 occoc 17305 Posetcpo 18364 lubclub 18366 glbcglb 18367 joincjn 18368 meetcmee 18369 0.cp0 18480 1.cp1 18481 OPcops 39153 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 ax-nul 5311 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-ne 2938 df-ral 3059 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-dm 5698 df-iota 6515 df-fv 6570 df-ov 7433 df-oposet 39157 |
This theorem is referenced by: op0cl 39165 op1cl 39166 op0le 39167 ople1 39172 lhp2lt 39983 |
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