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Step * 1 of Lemma lattice-axioms-from-order


1. l : LatticeStructure
2. R : Point(l) ⟶ Point(l) ⟶ ℙ
3. ∀[a,b:Point(l)].  least-upper-bound(Point(l);x,y.R[x;y];a;b;a ∨ b)
4. ∀[a,b:Point(l)].  greatest-lower-bound(Point(l);x,y.R[x;y];a;b;a ∧ b)
5. Order(Point(l);x,y.R[x;y])
6. a : Point(l)
7. b : Point(l)
⊢ a ∧ b = b ∧ a ∈ Point(l)
BY
{ (InstLemma `glb-com` [⌜Point(l)⌝;⌜R⌝;⌜λ2a b.a ∧ b⌝]⋅ THEN Auto) }

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Latex:

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Latex:

1.  l  :  LatticeStructure
2.  R  :  Point(l)  {}\mrightarrow{}  Point(l)  {}\mrightarrow{}  \mBbbP{}
3.  \mforall{}[a,b:Point(l)].    least-upper-bound(Point(l);x,y.R[x;y];a;b;a  \mvee{}  b)
4.  \mforall{}[a,b:Point(l)].    greatest-lower-bound(Point(l);x,y.R[x;y];a;b;a  \mwedge{}  b)
5.  Order(Point(l);x,y.R[x;y])
6.  a  :  Point(l)
7.  b  :  Point(l)
\mvdash{}  a  \mwedge{}  b  =  b  \mwedge{}  a


By

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Latex:
(InstLemma  `glb-com`  [\mkleeneopen{}Point(l)\mkleeneclose{};\mkleeneopen{}R\mkleeneclose{};\mkleeneopen{}\mlambda{}\msubtwo{}a  b.a  \mwedge{}  b\mkleeneclose{}]\mcdot{}  THEN  Auto)

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