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Theorem op01dm 33488
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 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)
101, 2, 3, 4, 5, 6, 7, 8, 9isopos 33485 . 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 472 . . 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 1210 . 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 206 1 (𝐾 ∈ OP → (𝐵 ∈ dom 𝑈𝐵 ∈ 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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