Nuprl Lemma : dM-lift-join
∀[I,J:fset(ℕ)]. ∀[f:I ⟶ J]. ∀[x,y:Point(dM(J))].
((dM-lift(I;J;f) x ∨ y) = dM-lift(I;J;f) x ∨ dM-lift(I;J;f) y ∈ Point(dM(I)))
Proof
Definitions occuring in Statement :
dM-lift: dM-lift(I;J;f),
names-hom: I ⟶ J,
dM: dM(I),
lattice-join: a ∨ b,
lattice-point: Point(l),
fset: fset(T),
nat: ℕ,
uall: ∀[x:A]. B[x],
apply: f a,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
dma-hom: dma-hom(dma1;dma2),
bounded-lattice-hom: Hom(l1;l2),
lattice-hom: Hom(l1;l2),
names-hom: I ⟶ J,
so_apply: x[s],
prop: ℙ,
implies: P ⇒ Q,
and: P ∧ Q,
DeMorgan-algebra: DeMorganAlgebra,
guard: {T},
uimplies: b supposing a
Lemmas referenced :
dM-lift_wf,
set_wf,
dma-hom_wf,
dM_wf,
all_wf,
names_wf,
equal_wf,
lattice-point_wf,
dM_inc_wf,
subtype_rel_set,
DeMorgan-algebra-structure_wf,
lattice-structure_wf,
lattice-axioms_wf,
bounded-lattice-structure-subtype,
DeMorgan-algebra-structure-subtype,
subtype_rel_transitivity,
bounded-lattice-structure_wf,
bounded-lattice-axioms_wf,
uall_wf,
lattice-meet_wf,
lattice-join_wf,
DeMorgan-algebra-axioms_wf,
names-hom_wf,
fset_wf,
nat_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
dependent_functionElimination,
sqequalRule,
lambdaEquality,
applyEquality,
because_Cache,
setElimination,
rename,
lambdaFormation,
equalitySymmetry,
productElimination,
equalityTransitivity,
independent_functionElimination,
instantiate,
productEquality,
independent_isectElimination,
cumulativity,
universeEquality,
isect_memberEquality,
axiomEquality
Latex:
\mforall{}[I,J:fset(\mBbbN{})]. \mforall{}[f:I {}\mrightarrow{} J]. \mforall{}[x,y:Point(dM(J))].
((dM-lift(I;J;f) x \mvee{} y) = dM-lift(I;J;f) x \mvee{} dM-lift(I;J;f) y)
Date html generated:
2016_05_18-AM-11_58_14
Last ObjectModification:
2015_12_28-PM-03_07_51
Theory : cubical!type!theory
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