Nuprl Lemma : ip-between-rless
∀rv:InnerProductSpace. ∀a,b,c:Point. (a_b_c ⇒ b # c ⇒ (||a - b|| < ||a - c||))
Proof
Definitions occuring in Statement :
ip-between: a_b_c,
rv-norm: ||x||,
rv-sub: x - y,
inner-product-space: InnerProductSpace,
rless: x < y,
ss-sep: x # y,
ss-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
prop: ℙ,
subtype_rel: A ⊆r B,
guard: {T},
and: P ∧ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
ip-dist-between,
ss-sep_wf,
real-vector-space_subtype1,
inner-product-space_subtype,
subtype_rel_transitivity,
inner-product-space_wf,
real-vector-space_wf,
separation-space_wf,
ip-between_wf,
ss-point_wf,
rv-norm_wf,
rv-sub_wf,
real_wf,
rleq_wf,
int-to-real_wf,
req_wf,
rmul_wf,
rv-ip_wf,
radd_wf,
rv-sep-iff-norm,
radd-preserves-rless,
rless_functionality,
req_weakening,
radd_comm,
radd-zero-both
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
independent_isectElimination,
hypothesis,
applyEquality,
instantiate,
sqequalRule,
because_Cache,
lambdaEquality,
setElimination,
rename,
setEquality,
productEquality,
natural_numberEquality,
dependent_functionElimination,
productElimination,
independent_functionElimination,
promote_hyp
Latex:
\mforall{}rv:InnerProductSpace. \mforall{}a,b,c:Point. (a\_b\_c {}\mRightarrow{} b \# c {}\mRightarrow{} (||a - b|| < ||a - c||))
Date html generated:
2017_10_05-AM-00_01_46
Last ObjectModification:
2017_03_13-PM-11_03_20
Theory : inner!product!spaces
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