Nuprl Lemma : ip-weak-triangle-inequality
∀rv:InnerProductSpace. ∀a,b,x,p:Point. (ax=ab ⇒ a_x_p ⇒ bp ≥ xp)
Proof
Definitions occuring in Statement :
ip-ge: cd ≥ ab,
ip-between: a_b_c,
ip-congruent: ab=cd,
inner-product-space: InnerProductSpace,
ss-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
ip-congruent: ab=cd,
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
prop: ℙ,
subtype_rel: A ⊆r B,
guard: {T},
and: P ∧ Q,
uiff: uiff(P;Q)
Lemmas referenced :
ip-dist-between,
rv-norm-triangle-inequality2,
ip-between_wf,
ip-congruent_wf,
ss-point_wf,
real-vector-space_subtype1,
inner-product-space_subtype,
subtype_rel_transitivity,
inner-product-space_wf,
real-vector-space_wf,
separation-space_wf,
rv-norm_wf,
rv-sub_wf,
real_wf,
rleq_wf,
int-to-real_wf,
req_wf,
rmul_wf,
rv-ip_wf,
radd_wf,
radd-preserves-rleq,
rminus_wf,
rleq_functionality,
req_weakening,
radd_functionality,
uiff_transitivity,
req_transitivity,
rminus-as-rmul,
radd-assoc,
req_inversion,
rmul-identity1,
rmul-distrib2,
rmul_functionality,
radd-int,
rmul-zero-both,
radd-zero-both,
ip-ge-iff
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
cut,
introduction,
extract_by_obid,
isectElimination,
thin,
hypothesisEquality,
independent_isectElimination,
hypothesis,
applyEquality,
instantiate,
sqequalRule,
because_Cache,
lambdaEquality,
setElimination,
rename,
setEquality,
productEquality,
natural_numberEquality,
productElimination,
minusEquality,
addEquality,
independent_functionElimination
Latex:
\mforall{}rv:InnerProductSpace. \mforall{}a,b,x,p:Point. (ax=ab {}\mRightarrow{} a\_x\_p {}\mRightarrow{} bp \mgeq{} xp)
Date html generated:
2017_10_05-AM-00_12_31
Last ObjectModification:
2017_03_19-PM-11_35_38
Theory : inner!product!spaces
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