Nuprl Lemma : lattice-0-equal-lattice-1-implies
∀L:BoundedLattice. ((1 = 0 ∈ Point(L)) ⇒ (∀x:Point(L). (0 = x ∈ Point(L))))
Proof
Definitions occuring in Statement :
bdd-lattice: BoundedLattice,
lattice-0: 0,
lattice-1: 1,
lattice-point: Point(l),
all: ∀x:A. B[x],
implies: P ⇒ Q,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
subtype_rel: A ⊆r B,
order: Order(T;x,y.R[x; y]),
refl: Refl(T;x,y.E[x; y]),
trans: Trans(T;x,y.E[x; y]),
anti_sym: AntiSym(T;x,y.R[x; y]),
and: P ∧ Q,
uall: ∀[x:A]. B[x],
squash: ↓T,
prop: ℙ,
true: True,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
bdd-lattice: BoundedLattice,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
bdd-lattice_wf,
lattice-0_wf,
lattice-1_wf,
equal_wf,
bounded-lattice-axioms_wf,
bounded-lattice-structure-subtype,
lattice-axioms_wf,
and_wf,
bounded-lattice-structure_wf,
subtype_rel_set,
le-lattice-1,
iff_weakening_equal,
lattice-structure_wf,
lattice-point_wf,
true_wf,
squash_wf,
lattice-le_wf,
lattice-0-le,
bdd-lattice-subtype-lattice,
lattice-le-order
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
applyEquality,
hypothesis,
sqequalRule,
productElimination,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
isectElimination,
lambdaEquality,
imageElimination,
because_Cache,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_isectElimination,
instantiate,
cumulativity,
setElimination,
rename
Latex:
\mforall{}L:BoundedLattice. ((1 = 0) {}\mRightarrow{} (\mforall{}x:Point(L). (0 = x)))
Date html generated:
2020_05_20-AM-08_26_16
Last ObjectModification:
2016_01_17-PM-00_42_38
Theory : lattices
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