Nuprl Lemma : ip-triangle-implies-separated
∀rv:InnerProductSpace. ∀a,b,c:Point. (Δ(a;b;c) ⇒ a # c)
Proof
Definitions occuring in Statement :
ip-triangle: Δ(a;b;c),
inner-product-space: InnerProductSpace,
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,
ip-triangle: Δ(a;b;c),
member: t ∈ T,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
prop: ℙ,
guard: {T},
uimplies: b supposing a,
rv-sub: x - y,
rv-minus: -x,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
nat: ℕ,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
nat_plus: ℕ+,
less_than: a < b,
squash: ↓T,
true: True,
cand: A c∧ B,
itermConstant: "const",
req_int_terms: t1 ≡ t2,
top: Top
Lemmas referenced :
rv-sep-iff,
rv-norm-positive-iff,
rv-sub_wf,
inner-product-space_subtype,
ip-triangle_wf,
ss-point_wf,
real-vector-space_subtype1,
subtype_rel_transitivity,
inner-product-space_wf,
real-vector-space_wf,
separation-space_wf,
ss-eq_wf,
rv-add_wf,
rv-mul_wf,
int-to-real_wf,
radd_wf,
rmul_wf,
rv-minus_wf,
rv-0_wf,
rv-norm_wf,
real_wf,
rleq_wf,
req_wf,
rv-ip_wf,
rless_wf,
rabs_wf,
equal_wf,
uiff_transitivity,
ss-eq_functionality,
ss-eq_weakening,
rv-add_functionality,
rv-mul-linear,
rv-add-assoc,
rv-mul-mul,
rv-add-swap,
rv-mul-add-alt,
rv-mul_functionality,
req_transitivity,
radd_functionality,
rmul-int,
req_weakening,
radd-int,
rv-mul0,
rv-0-add,
rless_functionality,
rv-norm_functionality,
square-rless-implies,
rv-norm-nonneg,
rnexp_wf,
false_wf,
le_wf,
less_than_wf,
rsub_wf,
rnexp-rless,
zero-rleq-rabs,
rnexp0,
rv-norm-squared,
rv-ip-sub-squared,
req_inversion,
rsub_functionality,
rnexp2-nonneg,
rabs-rnexp,
rnexp-rmul,
rabs-of-nonneg,
radd-preserves-rleq,
rleq_functionality,
real_term_polynomial,
itermSubtract_wf,
itermAdd_wf,
itermMultiply_wf,
itermConstant_wf,
itermVar_wf,
real_term_value_const_lemma,
real_term_value_sub_lemma,
real_term_value_add_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
req-iff-rsub-is-0,
rmul_preserves_rless,
rless-int,
rnexp2,
rless_transitivity1,
radd-preserves-rless,
radd-non-neg,
rmul_functionality,
rless-implies-rless
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
cut,
introduction,
extract_by_obid,
dependent_functionElimination,
thin,
hypothesisEquality,
productElimination,
independent_functionElimination,
because_Cache,
isectElimination,
applyEquality,
hypothesis,
sqequalRule,
instantiate,
independent_isectElimination,
minusEquality,
natural_numberEquality,
lambdaEquality,
setElimination,
rename,
setEquality,
productEquality,
equalityTransitivity,
equalitySymmetry,
dependent_set_memberEquality,
independent_pairFormation,
imageMemberEquality,
baseClosed,
computeAll,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality
Latex:
\mforall{}rv:InnerProductSpace. \mforall{}a,b,c:Point. (\mDelta{}(a;b;c) {}\mRightarrow{} a \# c)
Date html generated:
2017_10_04-PM-11_58_43
Last ObjectModification:
2017_07_28-AM-08_54_37
Theory : inner!product!spaces
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