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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 2823 | . . 3 ⊢ (le‘𝐾) = (le‘𝐾) | |
5 | eqid 2823 | . . 3 ⊢ (oc‘𝐾) = (oc‘𝐾) | |
6 | eqid 2823 | . . 3 ⊢ (join‘𝐾) = (join‘𝐾) | |
7 | eqid 2823 | . . 3 ⊢ (meet‘𝐾) = (meet‘𝐾) | |
8 | eqid 2823 | . . 3 ⊢ (0.‘𝐾) = (0.‘𝐾) | |
9 | eqid 2823 | . . 3 ⊢ (1.‘𝐾) = (1.‘𝐾) | |
10 | 1, 2, 3, 4, 5, 6, 7, 8, 9 | isopos 36318 | . 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 485 | . . 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 1162 | . 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 219 | 1 ⊢ (𝐾 ∈ OP → (𝐵 ∈ dom 𝑈 ∧ 𝐵 ∈ dom 𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ∀wral 3140 class class class wbr 5068 dom cdm 5557 ‘cfv 6357 (class class class)co 7158 Basecbs 16485 lecple 16574 occoc 16575 Posetcpo 17552 lubclub 17554 glbcglb 17555 joincjn 17556 meetcmee 17557 0.cp0 17649 1.cp1 17650 OPcops 36310 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-nul 5212 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-dm 5567 df-iota 6316 df-fv 6365 df-ov 7161 df-oposet 36314 |
This theorem is referenced by: op0cl 36322 op1cl 36323 op0le 36324 ople1 36329 lhp2lt 37139 |
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