Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  op01dm Structured version   Visualization version   GIF version

Theorem op01dm 38557
Description: Conditions necessary for zero and unity elements to exist. (Contributed by NM, 14-Sep-2018.)
Hypotheses
Ref Expression
op01dm.b 𝐡 = (Baseβ€˜πΎ)
op01dm.u π‘ˆ = (lubβ€˜πΎ)
op01dm.g 𝐺 = (glbβ€˜πΎ)
Assertion
Ref Expression
op01dm (𝐾 ∈ OP β†’ (𝐡 ∈ dom π‘ˆ ∧ 𝐡 ∈ dom 𝐺))

Proof of Theorem op01dm
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 op01dm.b . . 3 𝐡 = (Baseβ€˜πΎ)
2 op01dm.u . . 3 π‘ˆ = (lubβ€˜πΎ)
3 op01dm.g . . 3 𝐺 = (glbβ€˜πΎ)
4 eqid 2724 . . 3 (leβ€˜πΎ) = (leβ€˜πΎ)
5 eqid 2724 . . 3 (ocβ€˜πΎ) = (ocβ€˜πΎ)
6 eqid 2724 . . 3 (joinβ€˜πΎ) = (joinβ€˜πΎ)
7 eqid 2724 . . 3 (meetβ€˜πΎ) = (meetβ€˜πΎ)
8 eqid 2724 . . 3 (0.β€˜πΎ) = (0.β€˜πΎ)
9 eqid 2724 . . 3 (1.β€˜πΎ) = (1.β€˜πΎ)
101, 2, 3, 4, 5, 6, 7, 8, 9isopos 38554 . 2 (𝐾 ∈ OP ↔ ((𝐾 ∈ Poset ∧ 𝐡 ∈ dom π‘ˆ ∧ 𝐡 ∈ dom 𝐺) ∧ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 ((((ocβ€˜πΎ)β€˜π‘₯) ∈ 𝐡 ∧ ((ocβ€˜πΎ)β€˜((ocβ€˜πΎ)β€˜π‘₯)) = π‘₯ ∧ (π‘₯(leβ€˜πΎ)𝑦 β†’ ((ocβ€˜πΎ)β€˜π‘¦)(leβ€˜πΎ)((ocβ€˜πΎ)β€˜π‘₯))) ∧ (π‘₯(joinβ€˜πΎ)((ocβ€˜πΎ)β€˜π‘₯)) = (1.β€˜πΎ) ∧ (π‘₯(meetβ€˜πΎ)((ocβ€˜πΎ)β€˜π‘₯)) = (0.β€˜πΎ))))
11 simpl 482 . . 3 (((𝐡 ∈ dom π‘ˆ ∧ 𝐡 ∈ dom 𝐺) ∧ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 ((((ocβ€˜πΎ)β€˜π‘₯) ∈ 𝐡 ∧ ((ocβ€˜πΎ)β€˜((ocβ€˜πΎ)β€˜π‘₯)) = π‘₯ ∧ (π‘₯(leβ€˜πΎ)𝑦 β†’ ((ocβ€˜πΎ)β€˜π‘¦)(leβ€˜πΎ)((ocβ€˜πΎ)β€˜π‘₯))) ∧ (π‘₯(joinβ€˜πΎ)((ocβ€˜πΎ)β€˜π‘₯)) = (1.β€˜πΎ) ∧ (π‘₯(meetβ€˜πΎ)((ocβ€˜πΎ)β€˜π‘₯)) = (0.β€˜πΎ))) β†’ (𝐡 ∈ dom π‘ˆ ∧ 𝐡 ∈ dom 𝐺))
12113adantl1 1163 . 2 (((𝐾 ∈ Poset ∧ 𝐡 ∈ dom π‘ˆ ∧ 𝐡 ∈ dom 𝐺) ∧ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 ((((ocβ€˜πΎ)β€˜π‘₯) ∈ 𝐡 ∧ ((ocβ€˜πΎ)β€˜((ocβ€˜πΎ)β€˜π‘₯)) = π‘₯ ∧ (π‘₯(leβ€˜πΎ)𝑦 β†’ ((ocβ€˜πΎ)β€˜π‘¦)(leβ€˜πΎ)((ocβ€˜πΎ)β€˜π‘₯))) ∧ (π‘₯(joinβ€˜πΎ)((ocβ€˜πΎ)β€˜π‘₯)) = (1.β€˜πΎ) ∧ (π‘₯(meetβ€˜πΎ)((ocβ€˜πΎ)β€˜π‘₯)) = (0.β€˜πΎ))) β†’ (𝐡 ∈ dom π‘ˆ ∧ 𝐡 ∈ dom 𝐺))
1310, 12sylbi 216 1 (𝐾 ∈ OP β†’ (𝐡 ∈ dom π‘ˆ ∧ 𝐡 ∈ dom 𝐺))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  βˆ€wral 3053   class class class wbr 5139  dom cdm 5667  β€˜cfv 6534  (class class class)co 7402  Basecbs 17149  lecple 17209  occoc 17210  Posetcpo 18268  lubclub 18270  glbcglb 18271  joincjn 18272  meetcmee 18273  0.cp0 18384  1.cp1 18385  OPcops 38546
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695  ax-nul 5297
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ne 2933  df-ral 3054  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-dm 5677  df-iota 6486  df-fv 6542  df-ov 7405  df-oposet 38550
This theorem is referenced by:  op0cl  38558  op1cl  38559  op0le  38560  ople1  38565  lhp2lt  39376
  Copyright terms: Public domain W3C validator