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Theorem op01dm 36318
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 2821 . . 3 (le‘𝐾) = (le‘𝐾)
5 eqid 2821 . . 3 (oc‘𝐾) = (oc‘𝐾)
6 eqid 2821 . . 3 (join‘𝐾) = (join‘𝐾)
7 eqid 2821 . . 3 (meet‘𝐾) = (meet‘𝐾)
8 eqid 2821 . . 3 (0.‘𝐾) = (0.‘𝐾)
9 eqid 2821 . . 3 (1.‘𝐾) = (1.‘𝐾)
101, 2, 3, 4, 5, 6, 7, 8, 9isopos 36315 . 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 485 . . 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 1162 . 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 219 1 (𝐾 ∈ OP → (𝐵 ∈ dom 𝑈𝐵 ∈ dom 𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398  w3a 1083   = wceq 1533  wcel 2110  wral 3138   class class class wbr 5065  dom cdm 5554  cfv 6354  (class class class)co 7155  Basecbs 16482  lecple 16571  occoc 16572  Posetcpo 17549  lubclub 17551  glbcglb 17552  joincjn 17553  meetcmee 17554  0.cp0 17646  1.cp1 17647  OPcops 36307
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-nul 5209
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4838  df-br 5066  df-dm 5564  df-iota 6313  df-fv 6362  df-ov 7158  df-oposet 36311
This theorem is referenced by:  op0cl  36319  op1cl  36320  op0le  36321  ople1  36326  lhp2lt  37136
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