Nuprl Lemma : vs-map-0
∀[K:Rng]. ∀[A,B:VectorSpace(K)]. ∀[f:A ⟶ B]. ((f 0) = 0 ∈ Point(B))
Proof
Definitions occuring in Statement :
vs-map: A ⟶ B,
vs-0: 0,
vector-space: VectorSpace(K),
vs-point: Point(vs),
uall: ∀[x:A]. B[x],
apply: f a,
equal: s = t ∈ T,
rng: Rng
Definitions unfolded in proof :
rng: Rng,
all: ∀x:A. B[x],
and: P ∧ Q,
vs-map: A ⟶ B,
member: t ∈ T,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
guard: {T},
uimplies: b supposing a,
subtype_rel: A ⊆r B,
true: True,
squash: ↓T,
prop: ℙ
Lemmas referenced :
rng_wf,
vector-space_wf,
vs-map_wf,
vs-0_wf,
rng_zero_wf,
iff_weakening_equal,
vs-mul-zero,
vs-point_wf,
true_wf,
squash_wf,
equal_wf
Rules used in proof :
axiomEquality,
isect_memberEquality,
sqequalRule,
hypothesisEquality,
because_Cache,
isectElimination,
extract_by_obid,
dependent_functionElimination,
hypothesis,
productElimination,
rename,
thin,
setElimination,
sqequalHypSubstitution,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
independent_functionElimination,
independent_isectElimination,
baseClosed,
imageMemberEquality,
natural_numberEquality,
functionExtensionality,
universeEquality,
equalityTransitivity,
imageElimination,
lambdaEquality,
applyEquality,
equalitySymmetry,
hyp_replacement
Latex:
\mforall{}[K:Rng]. \mforall{}[A,B:VectorSpace(K)]. \mforall{}[f:A {}\mrightarrow{} B]. ((f 0) = 0)
Date html generated:
2018_05_22-PM-09_42_50
Last ObjectModification:
2018_01_09-PM-01_50_09
Theory : linear!algebra
Home
Index