Nuprl Lemma : ip-dist-between
∀[rv:InnerProductSpace]. ∀[a,b,c:Point]. ||a - c|| = (||a - b|| + ||b - c||) supposing a_b_c
Proof
Definitions occuring in Statement :
ip-between: a_b_c,
rv-norm: ||x||,
rv-sub: x - y,
inner-product-space: InnerProductSpace,
req: x = y,
radd: a + b,
ss-point: Point,
uimplies: b supposing a,
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
and: P ∧ Q,
prop: ℙ,
implies: P ⇒ Q,
guard: {T},
stable: Stable{P},
not: ¬A,
or: P ∨ Q,
false: False,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
exists: ∃x:A. B[x],
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
top: Top,
rsub: x - y,
ss-eq: x ≡ y,
rv-sub: x - y,
rv-minus: -x
Lemmas referenced :
req_witness,
rv-norm_wf,
rv-sub_wf,
inner-product-space_subtype,
real_wf,
rleq_wf,
int-to-real_wf,
req_wf,
rmul_wf,
rv-ip_wf,
radd_wf,
ip-between_wf,
ss-point_wf,
real-vector-space_subtype1,
subtype_rel_transitivity,
inner-product-space_wf,
real-vector-space_wf,
separation-space_wf,
stable_req,
false_wf,
or_wf,
ss-sep_wf,
not_wf,
minimal-double-negation-hyp-elim,
minimal-not-not-excluded-middle,
ip-between-iff,
ss-sep-symmetry,
rv-add_wf,
rv-mul_wf,
rsub_wf,
rabs_wf,
uiff_transitivity,
req_functionality,
req_weakening,
radd_functionality,
rv-norm_functionality,
rv-sub_functionality,
ss-eq_weakening,
ip-dist-between-2,
ip-dist-between-1,
member_rooint_lemma,
rleq_weakening_rless,
radd-preserves-rleq,
rminus_wf,
rmul_functionality,
rabs-of-nonneg,
rleq_functionality,
radd_comm,
radd-ac,
radd-rminus-both,
radd-zero-both,
req_transitivity,
rmul-distrib,
rmul_over_rminus,
rmul-one-both,
rmul_comm,
rminus_functionality,
req_inversion,
radd-assoc,
ss-eq_wf,
rv-0_wf,
ss-eq_functionality,
rv-mul-1-add,
rv-mul_functionality,
radd-int,
rv-mul0,
rv-norm0
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
applyEquality,
hypothesis,
sqequalRule,
lambdaEquality,
setElimination,
rename,
setEquality,
productEquality,
natural_numberEquality,
because_Cache,
independent_functionElimination,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
instantiate,
independent_isectElimination,
functionEquality,
lambdaFormation,
unionElimination,
voidElimination,
dependent_functionElimination,
productElimination,
voidEquality,
minusEquality
Latex:
\mforall{}[rv:InnerProductSpace]. \mforall{}[a,b,c:Point]. ||a - c|| = (||a - b|| + ||b - c||) supposing a\_b\_c
Date html generated:
2017_10_05-AM-00_01_37
Last ObjectModification:
2017_03_11-PM-06_07_11
Theory : inner!product!spaces
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