Nuprl Lemma : free-dma-lift-0
∀T:Type. ∀eq:EqDecider(T). ∀dm:DeMorganAlgebra. ∀eq2:EqDecider(Point(dm)). ∀f:T ⟶ Point(dm).
∀x:Point(free-DeMorgan-algebra(T;eq)).
(free-dma-lift(T;eq;dm;eq2;f) x) = 0 ∈ Point(dm) supposing x = 0 ∈ Point(free-DeMorgan-algebra(T;eq))
Proof
Definitions occuring in Statement :
free-dma-lift: free-dma-lift(T;eq;dm;eq2;f),
free-DeMorgan-algebra: free-DeMorgan-algebra(T;eq),
DeMorgan-algebra: DeMorganAlgebra,
lattice-0: 0,
lattice-point: Point(l),
deq: EqDecider(T),
uimplies: b supposing a,
all: ∀x:A. B[x],
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
dma-hom: dma-hom(dma1;dma2),
bounded-lattice-hom: Hom(l1;l2),
lattice-hom: Hom(l1;l2),
subtype_rel: A ⊆r B,
so_apply: x[s],
prop: ℙ,
implies: P ⇒ Q,
and: P ∧ Q,
DeMorgan-algebra: DeMorganAlgebra,
guard: {T}
Lemmas referenced :
free-dma-lift_wf,
set_wf,
dma-hom_wf,
free-DeMorgan-algebra_wf,
all_wf,
equal_wf,
dminc_wf,
free-dma-point-subtype,
lattice-0_wf,
subtype_rel_set,
DeMorgan-algebra-structure_wf,
bounded-lattice-structure_wf,
lattice-axioms_wf,
bounded-lattice-structure-subtype,
DeMorgan-algebra-structure-subtype,
subtype_rel_transitivity,
lattice-structure_wf,
bounded-lattice-axioms_wf,
uall_wf,
lattice-point_wf,
lattice-meet_wf,
lattice-join_wf,
DeMorgan-algebra-axioms_wf,
deq_wf,
DeMorgan-algebra_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
isect_memberFormation,
cut,
hypothesis,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
cumulativity,
hypothesisEquality,
functionExtensionality,
applyEquality,
isectElimination,
because_Cache,
sqequalRule,
lambdaEquality,
setElimination,
rename,
productElimination,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
hyp_replacement,
applyLambdaEquality,
setEquality,
instantiate,
productEquality,
independent_isectElimination,
universeEquality,
functionEquality
Latex:
\mforall{}T:Type. \mforall{}eq:EqDecider(T). \mforall{}dm:DeMorganAlgebra. \mforall{}eq2:EqDecider(Point(dm)). \mforall{}f:T {}\mrightarrow{} Point(dm).
\mforall{}x:Point(free-DeMorgan-algebra(T;eq)).
(free-dma-lift(T;eq;dm;eq2;f) x) = 0 supposing x = 0
Date html generated:
2020_05_20-AM-08_57_02
Last ObjectModification:
2017_07_28-AM-09_17_36
Theory : lattices
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