Nuprl Lemma : vs-grp_inverse
∀[K:Rng]. ∀[vs:VectorSpace(K)]. ∀[x:Point(vs)]. ((x + -(x) = 0 ∈ Point(vs)) ∧ (-(x) + x = 0 ∈ Point(vs)))
Proof
Definitions occuring in Statement :
vs-neg: -(x),
vs-add: x + y,
vs-0: 0,
vector-space: VectorSpace(K),
vs-point: Point(vs),
uall: ∀[x:A]. B[x],
and: P ∧ Q,
equal: s = t ∈ T,
rng: Rng
Definitions unfolded in proof :
all: ∀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,
rng: Rng,
prop: ℙ,
squash: ↓T,
cand: A c∧ B,
and: P ∧ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
rng_wf,
vector-space_wf,
vs-neg_wf,
vs-add-comm,
iff_weakening_equal,
vs-0_wf,
vs-add-neg,
vs-point_wf,
true_wf,
squash_wf,
equal_wf
Rules used in proof :
dependent_functionElimination,
isect_memberEquality,
axiomEquality,
independent_pairEquality,
independent_pairFormation,
independent_functionElimination,
productElimination,
independent_isectElimination,
baseClosed,
imageMemberEquality,
sqequalRule,
natural_numberEquality,
because_Cache,
rename,
setElimination,
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) \mwedge{} (-(x) + x = 0))
Date html generated:
2018_05_22-PM-09_41_08
Last ObjectModification:
2018_01_09-AM-10_32_55
Theory : linear!algebra
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