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Theorem op01dm 33947
Description: Conditions necessary for zero and unit elements to exist. (Contributed by NM, 14-Sep-2018.)
Hypotheses
Ref Expression
op01dm.b 𝐵 = (Base‘𝐾)
op01dm.u 𝑈 = (lub‘𝐾)
op01dm.g 𝐺 = (glb‘𝐾)
Assertion
Ref Expression
op01dm (𝐾 ∈ OP → (𝐵 ∈ dom 𝑈𝐵 ∈ dom 𝐺))

Proof of Theorem op01dm
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 op01dm.b . . 3 𝐵 = (Base‘𝐾)
2 op01dm.u . . 3 𝑈 = (lub‘𝐾)
3 op01dm.g . . 3 𝐺 = (glb‘𝐾)
4 eqid 2621 . . 3 (le‘𝐾) = (le‘𝐾)
5 eqid 2621 . . 3 (oc‘𝐾) = (oc‘𝐾)
6 eqid 2621 . . 3 (join‘𝐾) = (join‘𝐾)
7 eqid 2621 . . 3 (meet‘𝐾) = (meet‘𝐾)
8 eqid 2621 . . 3 (0.‘𝐾) = (0.‘𝐾)
9 eqid 2621 . . 3 (1.‘𝐾) = (1.‘𝐾)
101, 2, 3, 4, 5, 6, 7, 8, 9isopos 33944 . 2 (𝐾 ∈ OP ↔ ((𝐾 ∈ Poset ∧ 𝐵 ∈ dom 𝑈𝐵 ∈ dom 𝐺) ∧ ∀𝑥𝐵𝑦𝐵 ((((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 𝑈𝐵 ∈ dom 𝐺) ∧ ∀𝑥𝐵𝑦𝐵 ((((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 𝑈𝐵 ∈ dom 𝐺))
12113adantl1 1215 . 2 (((𝐾 ∈ Poset ∧ 𝐵 ∈ dom 𝑈𝐵 ∈ dom 𝐺) ∧ ∀𝑥𝐵𝑦𝐵 ((((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 𝑈𝐵 ∈ dom 𝐺))
1310, 12sylbi 207 1 (𝐾 ∈ OP → (𝐵 ∈ dom 𝑈𝐵 ∈ dom 𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2907   class class class wbr 4613  dom cdm 5074  cfv 5847  (class class class)co 6604  Basecbs 15781  lecple 15869  occoc 15870  Posetcpo 16861  lubclub 16863  glbcglb 16864  joincjn 16865  meetcmee 16866  0.cp0 16958  1.cp1 16959  OPcops 33936
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-nul 4749
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-dm 5084  df-iota 5810  df-fv 5855  df-ov 6607  df-oposet 33940
This theorem is referenced by:  op0cl  33948  op1cl  33949  op0le  33950  ople1  33955  lhp2lt  34764
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