Nuprl Lemma : ip-strict-between-iff
∀rv:InnerProductSpace. ∀a,b,c:Point. (a-b-c ⇐⇒ (∃t:ℝ. ((t ∈ (r0, r1)) ∧ b ≡ t*a + r1 - t*c)) ∧ a # c)
Proof
Definitions occuring in Statement :
ip-strict-between: a-b-c,
inner-product-space: InnerProductSpace,
rv-mul: a*x,
rv-add: x + y,
rooint: (l, u),
i-member: r ∈ I,
rsub: x - y,
int-to-real: r(n),
real: ℝ,
ss-eq: x ≡ y,
ss-sep: x # y,
ss-point: Point,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
ip-strict-between: a-b-c,
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
prop: ℙ,
rev_implies: P ⇐ Q,
so_lambda: λ2x.t[x],
guard: {T},
so_apply: x[s],
exists: ∃x:A. B[x],
cand: A c∧ B,
uiff: uiff(P;Q),
rsub: x - y,
top: Top,
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
ip-dist-between,
ip-between-iff,
ss-sep-symmetry,
ip-strict-between_wf,
exists_wf,
real_wf,
i-member_wf,
rooint_wf,
int-to-real_wf,
ss-eq_wf,
real-vector-space_subtype1,
inner-product-space_subtype,
subtype_rel_transitivity,
inner-product-space_wf,
real-vector-space_wf,
separation-space_wf,
rv-add_wf,
rv-mul_wf,
rsub_wf,
ss-sep_wf,
ss-point_wf,
ip-between-sep,
rv-norm-positive,
rv-sub_wf,
rv-sep-iff,
ip-dist-between-1,
rv-norm_wf,
req_wf,
rv-ip_wf,
rleq_wf,
rmul_wf,
rabs_wf,
req_functionality,
rv-norm_functionality,
rv-sub_functionality,
ss-eq_weakening,
ss-eq_inversion,
req_weakening,
rmul_preserves_rless,
radd_wf,
rminus_wf,
rless_functionality,
rmul-zero-both,
rmul_comm,
member_rooint_lemma,
radd-preserves-rleq,
rleq_weakening_rless,
radd-preserves-rless,
rless_wf,
rabs-of-nonneg,
uiff_transitivity,
rleq_functionality,
radd_comm,
radd-ac,
radd_functionality,
radd-rminus-both,
radd-zero-both,
rv-norm-positive-iff,
req_inversion,
ip-dist-between-2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
independent_pairFormation,
sqequalHypSubstitution,
productElimination,
thin,
cut,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
independent_isectElimination,
hypothesis,
dependent_functionElimination,
independent_functionElimination,
applyEquality,
because_Cache,
sqequalRule,
productEquality,
lambdaEquality,
natural_numberEquality,
instantiate,
setElimination,
rename,
setEquality,
promote_hyp,
isect_memberEquality,
voidElimination,
voidEquality,
addLevel,
levelHypothesis
Latex:
\mforall{}rv:InnerProductSpace. \mforall{}a,b,c:Point.
(a-b-c \mLeftarrow{}{}\mRightarrow{} (\mexists{}t:\mBbbR{}. ((t \mmember{} (r0, r1)) \mwedge{} b \mequiv{} t*a + r1 - t*c)) \mwedge{} a \# c)
Date html generated:
2017_10_05-AM-00_03_40
Last ObjectModification:
2017_03_12-PM-03_15_52
Theory : inner!product!spaces
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