Nuprl Lemma : rv-sub-is-zero
∀[rv:RealVectorSpace]. ∀[x,y:Point]. uiff(x - y ≡ 0;x ≡ y)
Proof
Definitions occuring in Statement :
rv-sub: x - y,
rv-0: 0,
real-vector-space: RealVectorSpace,
ss-eq: x ≡ y,
ss-point: Point,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
ss-eq: x ≡ y,
not: ¬A,
implies: P ⇒ Q,
false: False,
subtype_rel: A ⊆r B,
prop: ℙ,
all: ∀x:A. B[x],
rv-sub: x - y,
rev_uimplies: rev_uimplies(P;Q),
rv-minus: -x,
req_int_terms: t1 ≡ t2,
top: Top
Lemmas referenced :
ss-sep_wf,
real-vector-space_subtype1,
ss-eq_wf,
rv-sub_wf,
rv-0_wf,
ss-point_wf,
real-vector-space_wf,
rv-add_functionality,
ss-eq_weakening,
rv-add-0,
ss-eq_functionality,
rv-add_wf,
rv-0-add,
rv-add-minus,
rv-add-assoc,
ss-eq_inversion,
ss-eq_transitivity,
uiff_transitivity,
rv-minus_wf,
rv-mul_wf,
int-to-real_wf,
radd_wf,
itermSubtract_wf,
itermAdd_wf,
itermConstant_wf,
req-iff-rsub-is-0,
rv-sub_functionality,
rv-mul-1-add,
rv-mul_functionality,
rv-mul0,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_add_lemma,
real_term_value_const_lemma
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,
independent_isectElimination,
independent_functionElimination,
natural_numberEquality,
minusEquality,
approximateComputation,
intEquality,
voidEquality
Latex:
\mforall{}[rv:RealVectorSpace]. \mforall{}[x,y:Point]. uiff(x - y \mequiv{} 0;x \mequiv{} y)
Date html generated:
2017_10_04-PM-11_51_10
Last ObjectModification:
2017_07_28-AM-08_53_52
Theory : inner!product!spaces
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