Nuprl Lemma : free-dma-lift-unique2
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[dm:DeMorganAlgebra]. ∀[eq2:EqDecider(Point(dm))]. ∀[f:T ⟶ Point(dm)].
∀[g:dma-hom(free-DeMorgan-algebra(T;eq);dm)].
∀x:Point(free-DeMorgan-algebra(T;eq)). ((free-dma-lift(T;eq;dm;eq2;f) x) = (g x) ∈ Point(dm))
supposing ∀i:T. ((g <i>) = (f i) ∈ Point(dm))
Proof
Definitions occuring in Statement :
free-dma-lift: free-dma-lift(T;eq;dm;eq2;f),
free-DeMorgan-algebra: free-DeMorgan-algebra(T;eq),
dma-hom: dma-hom(dma1;dma2),
DeMorgan-algebra: DeMorganAlgebra,
dminc: <i>,
lattice-point: Point(l),
deq: EqDecider(T),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x],
squash: ↓T,
prop: ℙ,
subtype_rel: A ⊆r B,
DeMorgan-algebra: DeMorganAlgebra,
so_lambda: λ2x.t[x],
and: P ∧ Q,
so_apply: x[s],
guard: {T},
dma-hom: dma-hom(dma1;dma2),
bounded-lattice-hom: Hom(l1;l2),
lattice-hom: Hom(l1;l2),
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
free-dma-lift-unique,
equal_wf,
squash_wf,
true_wf,
lattice-point_wf,
subtype_rel_set,
DeMorgan-algebra-structure_wf,
lattice-structure_wf,
lattice-axioms_wf,
bounded-lattice-axioms_wf,
DeMorgan-algebra-structure-subtype,
uall_wf,
lattice-meet_wf,
lattice-join_wf,
DeMorgan-algebra-axioms_wf,
bounded-lattice-structure-subtype,
subtype_rel_transitivity,
bounded-lattice-structure_wf,
dma-hom_wf,
free-DeMorgan-algebra_wf,
iff_weakening_equal,
all_wf,
dminc_wf,
free-dma-point-subtype,
deq_wf,
DeMorgan-algebra_wf
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
independent_isectElimination,
lambdaFormation,
applyEquality,
lambdaEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
sqequalRule,
instantiate,
productEquality,
because_Cache,
setElimination,
rename,
cumulativity,
dependent_functionElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
productElimination,
independent_functionElimination,
axiomEquality,
functionExtensionality,
functionEquality
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[dm:DeMorganAlgebra]. \mforall{}[eq2:EqDecider(Point(dm))].
\mforall{}[f:T {}\mrightarrow{} Point(dm)]. \mforall{}[g:dma-hom(free-DeMorgan-algebra(T;eq);dm)].
\mforall{}x:Point(free-DeMorgan-algebra(T;eq)). ((free-dma-lift(T;eq;dm;eq2;f) x) = (g x))
supposing \mforall{}i:T. ((g <i>) = (f i))
Date html generated:
2020_05_20-AM-08_57_17
Last ObjectModification:
2017_07_28-AM-09_17_47
Theory : lattices
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