Nuprl Lemma : ip-dist-between-2
∀[rv:InnerProductSpace]. ∀[t:ℝ]. ∀[a,c:Point]. (||t*a + r1 - t*c - c|| = (|t| * ||a - c||))
Proof
Definitions occuring in Statement :
rv-norm: ||x||,
rv-sub: x - y,
inner-product-space: InnerProductSpace,
rv-mul: a*x,
rv-add: x + y,
rabs: |x|,
rsub: x - y,
req: x = y,
rmul: a * b,
int-to-real: r(n),
real: ℝ,
ss-point: Point,
uall: ∀[x:A]. B[x],
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
and: P ∧ Q,
prop: ℙ,
implies: P ⇒ Q,
guard: {T},
uimplies: b supposing a,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
rv-sub: x - y,
all: ∀x:A. B[x],
rv-minus: -x,
rsub: x - y
Lemmas referenced :
req_witness,
rv-norm_wf,
rv-sub_wf,
inner-product-space_subtype,
rv-add_wf,
rv-mul_wf,
rsub_wf,
int-to-real_wf,
req_wf,
rv-ip_wf,
rmul_wf,
rabs_wf,
real_wf,
rleq_wf,
ss-point_wf,
real-vector-space_subtype1,
subtype_rel_transitivity,
inner-product-space_wf,
real-vector-space_wf,
separation-space_wf,
rv-norm_functionality,
ss-eq_wf,
radd_wf,
rminus_wf,
rv-minus_wf,
req_weakening,
req_functionality,
req_inversion,
rv-norm-mul,
uiff_transitivity,
ss-eq_functionality,
ss-eq_weakening,
rv-mul-linear,
rv-mul-add-alt,
rv-add_functionality,
rv-mul-mul,
rv-mul_functionality,
req_transitivity,
radd-assoc,
radd_functionality,
radd_comm,
radd-ac,
rmul-minus,
rmul_over_rminus,
rminus_functionality,
rmul-one-both,
radd-int,
radd-zero-both
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
applyEquality,
hypothesis,
sqequalRule,
because_Cache,
natural_numberEquality,
lambdaEquality,
setElimination,
rename,
setEquality,
productEquality,
independent_functionElimination,
instantiate,
independent_isectElimination,
isect_memberEquality,
minusEquality,
addEquality,
productElimination,
dependent_functionElimination
Latex:
\mforall{}[rv:InnerProductSpace]. \mforall{}[t:\mBbbR{}]. \mforall{}[a,c:Point]. (||t*a + r1 - t*c - c|| = (|t| * ||a - c||))
Date html generated:
2017_10_05-AM-00_01_30
Last ObjectModification:
2017_03_11-PM-05_59_11
Theory : inner!product!spaces
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