Nuprl Lemma : vs-tree-val_wf_subspace
∀[K:RngSig]. ∀[vs:VectorSpace(K)]. ∀[P:Point(vs) ⟶ ℙ].
∀[t:l_tree(v:Point(vs) × P[v];|K|)]. (vs-tree-val(vs;t) ∈ {v:Point(vs)| P[v]} ) supposing vs-subspace(K;vs;x.P[x])
Proof
Definitions occuring in Statement :
vs-tree-val: vs-tree-val(vs;t),
vs-subspace: vs-subspace(K;vs;x.P[x]),
vector-space: VectorSpace(K),
vs-point: Point(vs),
l_tree: l_tree(L;T),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
member: t ∈ T,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
product: x:A × B[x],
rng_car: |r|,
rng_sig: RngSig
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
vs-tree-val: vs-tree-val(vs;t),
so_apply: x[s],
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
pi1: fst(t),
so_lambda: so_lambda(x,y,z,w,v.t[x; y; z; w; v]),
all: ∀x:A. B[x],
so_apply: x[s1;s2;s3;s4;s5],
prop: ℙ,
guard: {T},
vs-subspace: vs-subspace(K;vs;x.P[x]),
and: P ∧ Q,
implies: P ⇒ Q
Lemmas referenced :
l_tree_ind_wf_simple,
vs-point_wf,
rng_car_wf,
l_tree_wf,
subtype_rel_self,
vs-subspace_wf,
vector-space_wf,
rng_sig_wf,
vs-mul_wf,
vs-add_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
productEquality,
hypothesisEquality,
hypothesis,
applyEquality,
functionExtensionality,
because_Cache,
setEquality,
lambdaEquality,
productElimination,
dependent_set_memberEquality,
lambdaFormation,
setElimination,
rename,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
instantiate,
universeEquality,
isect_memberEquality,
functionEquality,
cumulativity,
dependent_functionElimination,
independent_functionElimination
Latex:
\mforall{}[K:RngSig]. \mforall{}[vs:VectorSpace(K)]. \mforall{}[P:Point(vs) {}\mrightarrow{} \mBbbP{}].
\mforall{}[t:l\_tree(v:Point(vs) \mtimes{} P[v];|K|)]. (vs-tree-val(vs;t) \mmember{} \{v:Point(vs)| P[v]\} ) supposing vs-subsp\000Cace(K;vs;x.P[x])
Date html generated:
2018_05_22-PM-09_42_08
Last ObjectModification:
2018_05_20-PM-10_42_00
Theory : linear!algebra
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