Nuprl Lemma : lattice-join-0
∀[l:BoundedLattice]. ∀[x:Point(l)].  ((0 ∨ x = x ∈ Point(l)) ∧ (x ∨ 0 = x ∈ Point(l)))
Proof
Definitions occuring in Statement : 
bdd-lattice: BoundedLattice
, 
lattice-0: 0
, 
lattice-join: a ∨ b
, 
lattice-point: Point(l)
, 
uall: ∀[x:A]. B[x]
, 
and: P ∧ Q
, 
equal: s = t ∈ T
Definitions unfolded in proof : 
uall: ∀[x:A]. B[x]
, 
member: t ∈ T
, 
and: P ∧ Q
, 
cand: A c∧ B
, 
bdd-lattice: BoundedLattice
, 
subtype_rel: A ⊆r B
, 
so_lambda: λ2x.t[x]
, 
prop: ℙ
, 
so_apply: x[s]
, 
uimplies: b supposing a
, 
guard: {T}
, 
lattice-axioms: lattice-axioms(l)
, 
squash: ↓T
, 
true: True
, 
iff: P 
⇐⇒ Q
, 
rev_implies: P 
⇐ Q
, 
implies: P 
⇒ Q
, 
bounded-lattice-axioms: bounded-lattice-axioms(l)
Lemmas referenced : 
lattice-point_wf, 
subtype_rel_set, 
bounded-lattice-structure_wf, 
lattice-structure_wf, 
lattice-axioms_wf, 
bounded-lattice-structure-subtype, 
bounded-lattice-axioms_wf, 
bdd-lattice_wf, 
lattice-0_wf, 
and_wf, 
equal_wf, 
squash_wf, 
true_wf, 
iff_weakening_equal
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
isect_memberFormation, 
introduction, 
cut, 
sqequalHypSubstitution, 
setElimination, 
thin, 
rename, 
productElimination, 
hypothesis, 
independent_pairFormation, 
sqequalRule, 
independent_pairEquality, 
axiomEquality, 
extract_by_obid, 
isectElimination, 
hypothesisEquality, 
applyEquality, 
instantiate, 
lambdaEquality, 
productEquality, 
cumulativity, 
independent_isectElimination, 
isect_memberEquality, 
because_Cache, 
hyp_replacement, 
equalitySymmetry, 
dependent_set_memberEquality, 
equalityTransitivity, 
applyLambdaEquality, 
imageElimination, 
universeEquality, 
natural_numberEquality, 
imageMemberEquality, 
baseClosed, 
independent_functionElimination
Latex:
\mforall{}[l:BoundedLattice].  \mforall{}[x:Point(l)].    ((0  \mvee{}  x  =  x)  \mwedge{}  (x  \mvee{}  0  =  x))
Date html generated:
2017_10_05-AM-00_31_21
Last ObjectModification:
2017_07_28-AM-09_13_03
Theory : lattices
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