Nuprl Lemma : lattice-le-meet
∀l:Lattice. ∀a,b,c:Point(l). (c ≤ a ∧ b ⇐⇒ c ≤ a ∧ c ≤ b)
Proof
Definitions occuring in Statement :
lattice-le: a ≤ b,
lattice: Lattice,
lattice-meet: 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,
greatest-lower-bound: greatest-lower-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-meet_wf,
lattice-le_wf,
lattice-meet-is-glb
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). (c \mleq{} a \mwedge{} b \mLeftarrow{}{}\mRightarrow{} c \mleq{} a \mwedge{} c \mleq{} b)
Date html generated:
2020_05_20-AM-08_25_37
Last ObjectModification:
2016_01_19-PM-02_36_58
Theory : lattices
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