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 35257
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 2824 . . 3 (le‘𝐾) = (le‘𝐾)
5 eqid 2824 . . 3 (oc‘𝐾) = (oc‘𝐾)
6 eqid 2824 . . 3 (join‘𝐾) = (join‘𝐾)
7 eqid 2824 . . 3 (meet‘𝐾) = (meet‘𝐾)
8 eqid 2824 . . 3 (0.‘𝐾) = (0.‘𝐾)
9 eqid 2824 . . 3 (1.‘𝐾) = (1.‘𝐾)
101, 2, 3, 4, 5, 6, 7, 8, 9isopos 35254 . 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 476 . . 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 1213 . 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 209 1 (𝐾 ∈ OP → (𝐵 ∈ dom 𝑈𝐵 ∈ dom 𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386  w3a 1113   = wceq 1658  wcel 2166  wral 3116   class class class wbr 4872  dom cdm 5341  cfv 6122  (class class class)co 6904  Basecbs 16221  lecple 16311  occoc 16312  Posetcpo 17292  lubclub 17294  glbcglb 17295  joincjn 17296  meetcmee 17297  0.cp0 17389  1.cp1 17390  OPcops 35246
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2390  ax-ext 2802  ax-nul 5012
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2604  df-eu 2639  df-clab 2811  df-cleq 2817  df-clel 2820  df-nfc 2957  df-ral 3121  df-rex 3122  df-rab 3125  df-v 3415  df-sbc 3662  df-dif 3800  df-un 3802  df-in 3804  df-ss 3811  df-nul 4144  df-if 4306  df-sn 4397  df-pr 4399  df-op 4403  df-uni 4658  df-br 4873  df-dm 5351  df-iota 6085  df-fv 6130  df-ov 6907  df-oposet 35250
This theorem is referenced by:  op0cl  35258  op1cl  35259  op0le  35260  ople1  35265  lhp2lt  36075
  Copyright terms: Public domain W3C validator