Nuprl Lemma : generated-subspace-is-least
∀K:Rng. ∀vs:VectorSpace(K).
∀[P:Point(vs) ⟶ ℙ]
∀S:Point(vs) ⟶ ℙ
(vs-subspace(K;vs;w.S[w]) ⇒ (∀v:Point(vs). (P[v] ⇒ S[v])) ⇒ (∀v:Point(vs). ((Subspace(x.P[x]) v) ⇒ S[v])))
Proof
Definitions occuring in Statement :
generated-subspace: Subspace(v.P[v]),
vs-subspace: vs-subspace(K;vs;x.P[x]),
vector-space: VectorSpace(K),
vs-point: Point(vs),
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
apply: f a,
function: x:A ⟶ B[x],
rng: Rng
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
generated-subspace: Subspace(v.P[v]),
or: P ∨ Q,
so_apply: x[s],
member: t ∈ T,
subtype_rel: A ⊆r B,
prop: ℙ,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
exists: ∃x:A. B[x],
rng: Rng,
so_lambda: λ2x.t[x],
vs-subspace: vs-subspace(K;vs;x.P[x]),
top: Top,
vs-tree-val: vs-tree-val(vs;t),
l_tree_leaf: l_tree_leaf(val),
l_tree_ind: l_tree_ind,
l_tree_node: l_tree_node(val;left_subtree;right_subtree),
pi1: fst(t)
Lemmas referenced :
subtype_rel_self,
iff_weakening_equal,
generated-subspace_wf,
vs-point_wf,
all_wf,
vs-subspace_wf,
vector-space_wf,
rng_wf,
l_tree-induction,
rng_car_wf,
vs-tree-val_wf,
l_tree_covariant,
top_wf,
subtype_rel_product,
l_tree_wf,
vs-add_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
isect_memberFormation,
sqequalHypSubstitution,
unionElimination,
thin,
cut,
applyEquality,
hypothesisEquality,
hypothesis,
sqequalRule,
instantiate,
introduction,
extract_by_obid,
isectElimination,
universeEquality,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
productElimination,
independent_functionElimination,
because_Cache,
setElimination,
rename,
lambdaEquality,
functionEquality,
cumulativity,
dependent_functionElimination,
productEquality,
functionExtensionality,
isect_memberEquality,
voidElimination,
voidEquality
Latex:
\mforall{}K:Rng. \mforall{}vs:VectorSpace(K).
\mforall{}[P:Point(vs) {}\mrightarrow{} \mBbbP{}]
\mforall{}S:Point(vs) {}\mrightarrow{} \mBbbP{}
(vs-subspace(K;vs;w.S[w])
{}\mRightarrow{} (\mforall{}v:Point(vs). (P[v] {}\mRightarrow{} S[v]))
{}\mRightarrow{} (\mforall{}v:Point(vs). ((Subspace(x.P[x]) v) {}\mRightarrow{} S[v])))
Date html generated:
2018_05_22-PM-09_42_24
Last ObjectModification:
2018_05_20-PM-10_42_43
Theory : linear!algebra
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