Nuprl Lemma : generated-subspace-contains
∀K:Rng. ∀vs:VectorSpace(K). ∀[P:Point(vs) ⟶ ℙ]. ∀v:Point(vs). (P[v] ⇒ (Subspace(x.P[x]) v))
Proof
Definitions occuring in Statement :
generated-subspace: Subspace(v.P[v]),
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]),
guard: {T},
or: P ∨ Q,
member: t ∈ T,
prop: ℙ,
rng: Rng,
so_apply: x[s],
exists: ∃x:A. B[x],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
so_lambda: λ2x.t[x],
top: Top,
vs-tree-val: vs-tree-val(vs;t),
l_tree_node: l_tree_node(val;left_subtree;right_subtree),
l_tree_ind: l_tree_ind,
pi1: fst(t),
l_tree_leaf: l_tree_leaf(val),
true: True,
squash: ↓T,
and: P ∧ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
equal_wf,
vs-point_wf,
vs-0_wf,
vector-space_wf,
rng_wf,
l_tree_node_wf,
subtype_rel_self,
rng_car_wf,
rng_one_wf,
l_tree_leaf_wf,
rng_zero_wf,
vs-tree-val_wf,
l_tree_covariant,
top_wf,
subtype_rel_product,
vs-add_wf,
vs-mul-zero,
vs-mul_wf,
vs-mul-one,
squash_wf,
true_wf,
rng_sig_wf,
vs-mon_ident,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
isect_memberFormation,
sqequalRule,
cut,
hypothesis,
inrFormation,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
because_Cache,
hypothesisEquality,
applyEquality,
functionEquality,
cumulativity,
universeEquality,
dependent_functionElimination,
dependent_pairFormation,
productEquality,
instantiate,
dependent_pairEquality,
independent_isectElimination,
lambdaEquality,
isect_memberEquality,
voidElimination,
voidEquality,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
hyp_replacement,
applyLambdaEquality,
equalityUniverse,
levelHypothesis,
natural_numberEquality,
imageElimination,
productElimination,
imageMemberEquality,
baseClosed
Latex:
\mforall{}K:Rng. \mforall{}vs:VectorSpace(K). \mforall{}[P:Point(vs) {}\mrightarrow{} \mBbbP{}]. \mforall{}v:Point(vs). (P[v] {}\mRightarrow{} (Subspace(x.P[x]) v))
Date html generated:
2018_05_22-PM-09_42_20
Last ObjectModification:
2018_05_20-PM-10_42_13
Theory : linear!algebra
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