Nuprl Lemma : vs-subtract-self
∀[K:Rng]. ∀[vs:VectorSpace(K)]. ∀[x:Point(vs)]. ((x - x) = 0 ∈ Point(vs))
Proof
Definitions occuring in Statement :
vs-subtract: (x - y),
vs-0: 0,
vector-space: VectorSpace(K),
vs-point: Point(vs),
uall: ∀[x:A]. B[x],
equal: s = t ∈ T,
rng: Rng
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
rev_implies: P ⇐ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
guard: {T},
uimplies: b supposing a,
subtype_rel: A ⊆r B,
true: True,
rng: Rng,
prop: ℙ,
squash: ↓T,
member: t ∈ T,
uall: ∀[x:A]. B[x],
vs-subtract: (x - y)
Lemmas referenced :
vs-mul-zero,
rng_wf,
vector-space_wf,
vs-point_wf,
iff_weakening_equal,
vs-0_wf,
rng_minus_wf,
rng_one_wf,
vs-mul-add,
true_wf,
squash_wf,
equal_wf,
vs-mul_wf,
rng_car_wf,
rng_sig_wf,
rng_plus_inv,
vs-mul-one,
vs-add_wf
Rules used in proof :
dependent_functionElimination,
axiomEquality,
isect_memberEquality,
independent_functionElimination,
productElimination,
independent_isectElimination,
baseClosed,
imageMemberEquality,
sqequalRule,
natural_numberEquality,
rename,
setElimination,
because_Cache,
universeEquality,
equalitySymmetry,
hypothesis,
equalityTransitivity,
hypothesisEquality,
isectElimination,
extract_by_obid,
imageElimination,
sqequalHypSubstitution,
lambdaEquality,
thin,
applyEquality,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[K:Rng]. \mforall{}[vs:VectorSpace(K)]. \mforall{}[x:Point(vs)]. ((x - x) = 0)
Date html generated:
2018_05_22-PM-09_42_56
Last ObjectModification:
2018_01_09-PM-02_15_57
Theory : linear!algebra
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