Nuprl Lemma : face-lattice-1-join-irreducible
∀T:Type. ∀eq:EqDecider(T). ∀x,y:Point(face-lattice(T;eq)).
(x ∨ y = 1 ∈ Point(face-lattice(T;eq)) ⇐⇒ (x = 1 ∈ Point(face-lattice(T;eq))) ∨ (y = 1 ∈ Point(face-lattice(T;eq))))
Proof
Definitions occuring in Statement :
face-lattice: face-lattice(T;eq),
lattice-1: 1,
lattice-join: a ∨ b,
lattice-point: Point(l),
deq: EqDecider(T),
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
or: P ∨ Q,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
face-lattice: face-lattice(T;eq),
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
bdd-distributive-lattice: BoundedDistributiveLattice,
uimplies: b supposing a,
rev_implies: P ⇐ Q,
or: P ∨ Q
Lemmas referenced :
or_wf,
equal_wf,
lattice-point_wf,
free-dist-lattice-with-constraints_wf,
union-deq_wf,
face-lattice-constraints_wf,
subtype_rel_set,
bounded-lattice-structure_wf,
lattice-structure_wf,
lattice-axioms_wf,
bounded-lattice-structure-subtype,
bounded-lattice-axioms_wf,
uall_wf,
lattice-meet_wf,
lattice-join_wf,
lattice-1_wf,
bdd-distributive-lattice_wf,
free-dlwc-1-join-irreducible,
iff_wf,
face-lattice_wf,
deq_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
cut,
independent_pairFormation,
hypothesis,
introduction,
extract_by_obid,
isectElimination,
thin,
unionEquality,
cumulativity,
hypothesisEquality,
because_Cache,
sqequalRule,
lambdaEquality,
applyEquality,
instantiate,
productEquality,
universeEquality,
independent_isectElimination,
setElimination,
rename,
addLevel,
productElimination,
impliesFunctionality,
dependent_functionElimination,
independent_functionElimination
Latex:
\mforall{}T:Type. \mforall{}eq:EqDecider(T). \mforall{}x,y:Point(face-lattice(T;eq)). (x \mvee{} y = 1 \mLeftarrow{}{}\mRightarrow{} (x = 1) \mvee{} (y = 1))
Date html generated:
2020_05_20-AM-08_53_26
Last ObjectModification:
2017_07_28-AM-09_16_28
Theory : lattices
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