Nuprl Lemma : ip-triangle-not-between
∀[rv:InnerProductSpace]. ∀[a,b,c:Point]. ¬a_b_c supposing Δ(a;b;c)
Proof
Definitions occuring in Statement :
ip-triangle: Δ(a;b;c),
ip-between: a_b_c,
inner-product-space: InnerProductSpace,
ss-point: Point,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
not: ¬A
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
not: ¬A,
implies: P ⇒ Q,
false: False,
ip-between: a_b_c,
ip-triangle: Δ(a;b;c),
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
prop: ℙ,
and: P ∧ Q,
guard: {T},
uiff: uiff(P;Q),
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
squash: ↓T,
rless: x < y,
sq_exists: ∃x:{A| B[x]},
nat_plus: ℕ+,
less_than: a < b,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top
Lemmas referenced :
rv-sub_wf,
inner-product-space_subtype,
req_wf,
radd_wf,
rmul_wf,
rv-norm_wf,
real_wf,
rleq_wf,
int-to-real_wf,
rv-ip_wf,
rless_wf,
rabs_wf,
equal_wf,
ip-between_wf,
ip-triangle_wf,
ss-point_wf,
real-vector-space_subtype1,
subtype_rel_transitivity,
inner-product-space_wf,
real-vector-space_wf,
separation-space_wf,
radd-preserves-req,
rminus_wf,
uiff_transitivity,
req_functionality,
req_transitivity,
radd_functionality,
req_weakening,
rminus-as-rmul,
radd-assoc,
req_inversion,
rmul-identity1,
rmul-distrib2,
rmul_functionality,
radd-int,
rmul-zero-both,
radd_comm,
radd-zero-both,
iff_transitivity,
rless_functionality,
rabs_functionality,
squash_wf,
true_wf,
rabs-rminus,
iff_weakening_equal,
rmul-nonneg-case1,
rv-norm-nonneg,
nat_plus_properties,
satisfiable-full-omega-tt,
intformless_wf,
itermAdd_wf,
itermVar_wf,
itermConstant_wf,
int_formula_prop_less_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
rabs-of-nonneg
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
thin,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
applyEquality,
hypothesis,
sqequalRule,
because_Cache,
independent_functionElimination,
voidElimination,
lambdaEquality,
setElimination,
rename,
setEquality,
productEquality,
natural_numberEquality,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
isect_memberEquality,
instantiate,
independent_isectElimination,
productElimination,
minusEquality,
addEquality,
independent_pairFormation,
imageElimination,
imageMemberEquality,
baseClosed,
universeEquality,
dependent_pairFormation,
int_eqEquality,
intEquality,
voidEquality,
computeAll
Latex:
\mforall{}[rv:InnerProductSpace]. \mforall{}[a,b,c:Point]. \mneg{}a\_b\_c supposing \mDelta{}(a;b;c)
Date html generated:
2017_10_04-PM-11_59_02
Last ObjectModification:
2017_03_10-PM-04_41_06
Theory : inner!product!spaces
Home
Index