Nuprl Lemma : not-ip-triangle
∀rv:InnerProductSpace. ∀a,b,c:Point(rv). (a # b ⇒ c # b ⇒ (¬Δ(a;b;c)) ⇒ (∃t:ℝ. ((r0 < |t|) ∧ c ≡ b + t*a - b)))
Proof
Definitions occuring in Statement :
ip-triangle: Δ(a;b;c),
rv-sub: x - y,
inner-product-space: InnerProductSpace,
rv-mul: a*x,
rv-add: x + y,
rless: x < y,
rabs: |x|,
int-to-real: r(n),
real: ℝ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
and: P ∧ Q,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
ip-triangle: Δ(a;b;c),
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
not: ¬A,
prop: ℙ,
false: False,
guard: {T},
uiff: uiff(P;Q),
and: P ∧ Q,
rev_uimplies: rev_uimplies(P;Q),
req_int_terms: t1 ≡ t2,
top: Top,
iff: P ⇐⇒ Q,
exists: ∃x:A. B[x],
cand: A c∧ B,
or: P ∨ Q,
rless: x < y,
sq_exists: ∃x:A [B[x]],
nat_plus: ℕ+,
less_than: a < b,
squash: ↓T,
satisfiable_int_formula: satisfiable_int_formula(fmla),
rv-sub: x - y,
rv-minus: -x,
rev_implies: P ⇐ Q
Lemmas referenced :
not-rless,
rabs_wf,
rv-ip_wf,
rv-sub_wf,
inner-product-space_subtype,
rmul_wf,
rv-norm_wf,
ip-triangle_wf,
istype-void,
Error :ss-sep_wf,
real-vector-space_subtype1,
subtype_rel_transitivity,
inner-product-space_wf,
real-vector-space_wf,
Error :separation-space_wf,
Error :ss-point_wf,
rv-Cauchy-Schwarz',
rleq_antisymmetry,
rv-Cauchy-Schwarz-equality',
rv-ip-symmetry,
req_wf,
itermSubtract_wf,
itermMultiply_wf,
itermVar_wf,
req-iff-rsub-is-0,
uiff_transitivity,
req_functionality,
rabs_functionality,
req_weakening,
real_polynomial_null,
int-to-real_wf,
istype-int,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
rv-sep-iff,
rless_wf,
Error :ss-eq_wf,
rv-add_wf,
rv-mul_wf,
rv-sep-iff-norm,
rless_functionality,
rv-norm_functionality,
rv-norm-mul,
rmul-is-positive,
zero-rleq-rabs,
rless_transitivity2,
rless_transitivity1,
nat_plus_properties,
full-omega-unsat,
intformless_wf,
itermAdd_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,
rv-add-cancel-left,
Error :ss-eq_functionality,
Error :ss-eq_weakening,
Error :ss-eq_inversion,
radd_wf,
rv-minus_wf,
rv-0_wf,
iff_weakening_uiff,
rv-add-assoc,
rv-add-swap,
rv-add_functionality,
rv-mul-1-add,
rv-mul_functionality,
rv-mul0,
rv-add-0,
real_term_value_add_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
sqequalHypSubstitution,
cut,
introduction,
extract_by_obid,
isectElimination,
thin,
hypothesisEquality,
applyEquality,
hypothesis,
sqequalRule,
because_Cache,
lambdaEquality_alt,
setElimination,
rename,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
functionIsType,
universeIsType,
instantiate,
dependent_functionElimination,
independent_functionElimination,
natural_numberEquality,
productElimination,
approximateComputation,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
dependent_pairFormation_alt,
independent_pairFormation,
productIsType,
unionElimination,
imageElimination,
Error :memTop,
minusEquality
Latex:
\mforall{}rv:InnerProductSpace. \mforall{}a,b,c:Point(rv).
(a \# b {}\mRightarrow{} c \# b {}\mRightarrow{} (\mneg{}\mDelta{}(a;b;c)) {}\mRightarrow{} (\mexists{}t:\mBbbR{}. ((r0 < |t|) \mwedge{} c \mequiv{} b + t*a - b)))
Date html generated:
2020_05_20-PM-01_13_31
Last ObjectModification:
2019_12_09-PM-11_41_09
Theory : inner!product!spaces
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