Nuprl Lemma : lattice-join-le
∀l:Lattice. ∀a,b,c:Point(l). (a ∨ b ≤ c ⇐⇒ a ≤ c ∧ b ≤ c)
Proof
Definitions occuring in Statement :
lattice-le: a ≤ b,
lattice: Lattice,
lattice-join: a ∨ b,
lattice-point: Point(l),
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
least-upper-bound: least-upper-bound(T;x,y.R[x; y];a;b;c),
and: P ∧ Q,
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
lattice: Lattice,
rev_implies: P ⇐ Q,
prop: ℙ,
guard: {T},
uimplies: b supposing a
Lemmas referenced :
lattice-le_transitivity,
lattice_wf,
lattice-point_wf,
and_wf,
lattice-join_wf,
lattice-le_wf,
lattice-join-is-lub
Rules used in proof :
cut,
lemma_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
productElimination,
independent_pairFormation,
isectElimination,
setElimination,
rename,
independent_functionElimination,
independent_isectElimination
Latex:
\mforall{}l:Lattice. \mforall{}a,b,c:Point(l). (a \mvee{} b \mleq{} c \mLeftarrow{}{}\mRightarrow{} a \mleq{} c \mwedge{} b \mleq{} c)
Date html generated:
2020_05_20-AM-08_25_41
Last ObjectModification:
2016_01_19-PM-02_35_04
Theory : lattices
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