Nuprl Lemma : vs-subspace_functionality
∀K:RngSig. ∀vs:VectorSpace(K).
∀[P,Q:Point(vs) ⟶ ℙ]. ((∀x:Point(vs). (P[x] ⇐⇒ Q[x])) ⇒ {vs-subspace(K;vs;x.P[x]) ⇒ vs-subspace(K;vs;x.Q[x])})
Proof
Definitions occuring in Statement :
vs-subspace: vs-subspace(K;vs;x.P[x]),
vector-space: VectorSpace(K),
vs-point: Point(vs),
uall: ∀[x:A]. B[x],
prop: ℙ,
guard: {T},
so_apply: x[s],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
function: x:A ⟶ B[x],
rng_sig: RngSig
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
guard: {T},
vs-subspace: vs-subspace(K;vs;x.P[x]),
and: P ∧ Q,
member: t ∈ T,
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ,
iff: P ⇐⇒ Q,
subtype_rel: A ⊆r B,
rev_implies: P ⇐ Q
Lemmas referenced :
vs-subspace_wf,
vs-point_wf,
subtype_rel_self,
vector-space_wf,
rng_sig_wf,
vs-0_wf,
vs-mul_wf,
rng_car_wf,
vs-add_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
isect_memberFormation_alt,
sqequalHypSubstitution,
independent_pairFormation,
productElimination,
thin,
promote_hyp,
universeIsType,
cut,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
sqequalRule,
lambdaEquality_alt,
applyEquality,
hypothesis,
functionIsType,
productIsType,
instantiate,
universeEquality,
because_Cache,
inhabitedIsType,
dependent_functionElimination,
independent_functionElimination
Latex:
\mforall{}K:RngSig. \mforall{}vs:VectorSpace(K).
\mforall{}[P,Q:Point(vs) {}\mrightarrow{} \mBbbP{}].
((\mforall{}x:Point(vs). (P[x] \mLeftarrow{}{}\mRightarrow{} Q[x])) {}\mRightarrow{} \{vs-subspace(K;vs;x.P[x]) {}\mRightarrow{} vs-subspace(K;vs;x.Q[x])\})
Date html generated:
2019_10_31-AM-06_26_47
Last ObjectModification:
2019_08_12-PM-03_12_42
Theory : linear!algebra
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