Nuprl Lemma : vs-cancel-add
∀[K:Rng]. ∀[vs:VectorSpace(K)]. ∀[x,y,z:Point(vs)]. (z + x = z + y ∈ Point(vs) ⇐⇒ x = y ∈ Point(vs))
Proof
Definitions occuring in Statement :
vs-add: x + y,
vector-space: VectorSpace(K),
vs-point: Point(vs),
uall: ∀[x:A]. B[x],
iff: P ⇐⇒ Q,
equal: s = t ∈ T,
rng: Rng
Definitions unfolded in proof :
all: ∀x:A. B[x],
rev_implies: P ⇐ Q,
rng: Rng,
prop: ℙ,
implies: P ⇒ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
true: True,
squash: ↓T,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T}
Lemmas referenced :
rng_wf,
vector-space_wf,
and_wf,
vs-add_wf,
vs-point_wf,
equal_wf,
vs-neg_wf,
squash_wf,
true_wf,
vs-add-assoc,
rng_sig_wf,
vs-neg-add,
vs-zero-add,
iff_weakening_equal
Rules used in proof :
isect_memberEquality,
axiomEquality,
dependent_functionElimination,
lambdaEquality,
independent_pairEquality,
sqequalRule,
productElimination,
applyLambdaEquality,
dependent_set_memberEquality,
equalitySymmetry,
because_Cache,
hypothesisEquality,
rename,
setElimination,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
hypothesis,
lambdaFormation,
independent_pairFormation,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
natural_numberEquality,
levelHypothesis,
equalityUniverse,
applyEquality,
imageElimination,
equalityTransitivity,
universeEquality,
imageMemberEquality,
baseClosed,
independent_isectElimination,
independent_functionElimination
Latex:
\mforall{}[K:Rng]. \mforall{}[vs:VectorSpace(K)]. \mforall{}[x,y,z:Point(vs)]. (z + x = z + y \mLeftarrow{}{}\mRightarrow{} x = y)
Date html generated:
2018_05_22-PM-09_41_12
Last ObjectModification:
2018_01_09-PM-01_04_25
Theory : linear!algebra
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