Nuprl Lemma : rv-mul-rv-sub
∀[rv:RealVectorSpace]. ∀[a:ℝ]. ∀[x,y:Point]. a*x - y ≡ a*x - a*y
Proof
Definitions occuring in Statement :
rv-sub: x - y,
rv-mul: a*x,
real-vector-space: RealVectorSpace,
real: ℝ,
ss-eq: x ≡ y,
ss-point: Point,
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
rv-sub: x - y,
rv-minus: -x,
ss-eq: x ≡ y,
not: ¬A,
implies: P ⇒ Q,
false: False,
subtype_rel: A ⊆r B,
prop: ℙ,
all: ∀x:A. B[x],
uimplies: b supposing a,
uiff: uiff(P;Q),
and: P ∧ Q,
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
ss-sep_wf,
real-vector-space_subtype1,
rv-mul_wf,
rv-sub_wf,
ss-point_wf,
real_wf,
real-vector-space_wf,
rv-add_wf,
int-to-real_wf,
rmul_wf,
ss-eq_functionality,
ss-eq_weakening,
rv-add_functionality,
rv-mul-mul,
ss-eq_wf,
rminus_wf,
ss-eq_transitivity,
rv-mul-linear,
uiff_transitivity,
rv-mul_functionality,
rmul-minus,
req_inversion,
rminus-as-rmul,
rmul_over_rminus,
rminus_functionality,
rmul-one-both
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
sqequalHypSubstitution,
lambdaEquality,
dependent_functionElimination,
thin,
hypothesisEquality,
because_Cache,
extract_by_obid,
isectElimination,
applyEquality,
hypothesis,
isect_memberEquality,
voidElimination,
minusEquality,
natural_numberEquality,
independent_isectElimination,
independent_functionElimination,
productElimination
Latex:
\mforall{}[rv:RealVectorSpace]. \mforall{}[a:\mBbbR{}]. \mforall{}[x,y:Point]. a*x - y \mequiv{} a*x - a*y
Date html generated:
2017_10_04-PM-11_51_23
Last ObjectModification:
2017_03_09-PM-06_16_45
Theory : inner!product!spaces
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