Nuprl Lemma : rv-mul-cancel
∀[rv:RealVectorSpace]. ∀[a:ℝ]. ∀[x,y:Point]. uiff(a*x ≡ a*y;x ≡ y) supposing a ≠ r0
Proof
Definitions occuring in Statement :
rv-mul: a*x,
real-vector-space: RealVectorSpace,
rneq: x ≠ y,
int-to-real: r(n),
real: ℝ,
ss-eq: x ≡ y,
ss-point: Point,
uiff: uiff(P;Q),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
uiff: uiff(P;Q),
and: P ∧ Q,
ss-eq: x ≡ y,
not: ¬A,
implies: P ⇒ Q,
false: False,
subtype_rel: A ⊆r B,
prop: ℙ,
all: ∀x:A. B[x],
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
ss-sep_wf,
real-vector-space_subtype1,
ss-eq_wf,
rv-mul_wf,
rneq_wf,
int-to-real_wf,
ss-point_wf,
real_wf,
real-vector-space_wf,
rdiv_wf,
req_weakening,
rv-mul_functionality,
rmul_wf,
ss-eq_functionality,
rv-mul-mul,
rmul-rdiv-cancel2,
ss-eq_weakening,
rv-mul1
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
sqequalRule,
sqequalHypSubstitution,
lambdaEquality,
dependent_functionElimination,
thin,
hypothesisEquality,
because_Cache,
extract_by_obid,
isectElimination,
applyEquality,
hypothesis,
voidElimination,
productElimination,
independent_pairEquality,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
natural_numberEquality,
independent_isectElimination,
independent_functionElimination
Latex:
\mforall{}[rv:RealVectorSpace]. \mforall{}[a:\mBbbR{}]. \mforall{}[x,y:Point]. uiff(a*x \mequiv{} a*y;x \mequiv{} y) supposing a \mneq{} r0
Date html generated:
2017_10_04-PM-11_50_41
Last ObjectModification:
2017_06_22-PM-06_44_45
Theory : inner!product!spaces
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