Nuprl Lemma : vs-map-subtract
∀[K:Rng]. ∀[A,B:VectorSpace(K)]. ∀[f:A ⟶ B]. ∀[x,y:Point(A)]. ((f (x - y)) = (f x - f y) ∈ Point(B))
Proof
Definitions occuring in Statement :
vs-subtract: (x - y),
vs-map: A ⟶ B,
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 :
true: True,
all: ∀x:A. B[x],
rng: Rng,
vs-subtract: (x - y),
and: P ∧ Q,
vs-map: A ⟶ B,
member: t ∈ T,
uall: ∀[x:A]. B[x],
squash: ↓T,
prop: ℙ,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
vs-add_wf,
rng_one_wf,
rng_minus_wf,
vs-mul_wf,
rng_wf,
vector-space_wf,
vs-map_wf,
vs-point_wf,
equal_wf,
squash_wf,
true_wf,
rng_sig_wf,
iff_weakening_equal
Rules used in proof :
natural_numberEquality,
functionExtensionality,
applyEquality,
dependent_functionElimination,
because_Cache,
axiomEquality,
isect_memberEquality,
sqequalRule,
hypothesisEquality,
isectElimination,
extract_by_obid,
hypothesis,
productElimination,
rename,
thin,
setElimination,
sqequalHypSubstitution,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
lambdaEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
imageMemberEquality,
baseClosed,
independent_isectElimination,
independent_functionElimination
Latex:
\mforall{}[K:Rng]. \mforall{}[A,B:VectorSpace(K)]. \mforall{}[f:A {}\mrightarrow{} B]. \mforall{}[x,y:Point(A)]. ((f (x - y)) = (f x - f y))
Date html generated:
2018_05_22-PM-09_43_01
Last ObjectModification:
2018_01_09-PM-02_22_29
Theory : linear!algebra
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