Nuprl Lemma : ip-triangle-linearity
∀rv:InnerProductSpace. ∀a,b,c:Point. (Δ(a;b;c) ⇒ (∀t:ℝ. ((r0 < |t|) ⇒ Δ(b + t*a - b;b;c))))
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,
ss-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
ip-triangle: Δ(a;b;c),
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
and: P ∧ Q,
rv-sub: x - y,
rv-minus: -x,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
itermConstant: "const",
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
top: Top
Lemmas referenced :
rless_wf,
int-to-real_wf,
rabs_wf,
real_wf,
ip-triangle_wf,
ss-point_wf,
real-vector-space_subtype1,
inner-product-space_subtype,
subtype_rel_transitivity,
inner-product-space_wf,
real-vector-space_wf,
separation-space_wf,
ss-eq_wf,
rv-sub_wf,
rv-add_wf,
rv-mul_wf,
radd_wf,
rmul_wf,
rminus_wf,
rv-minus_wf,
req_weakening,
rv-ip_wf,
rv-norm_wf,
rleq_wf,
req_wf,
uiff_transitivity,
ss-eq_functionality,
ss-eq_weakening,
rv-mul-linear,
rv-add_functionality,
rv-add-assoc,
rv-mul-mul,
rv-mul-add-1-alt,
rv-add-comm,
rv-mul-1-add-alt,
rv-mul_functionality,
req_transitivity,
radd_functionality,
rmul_functionality,
radd_comm,
rmul-minus,
rmul_over_rminus,
rminus_functionality,
rmul-distrib,
rmul-one-both,
rminus-radd,
rmul-int,
req_inversion,
rminus-as-rmul,
radd-ac,
req_functionality,
radd-assoc,
radd-int,
radd-zero-both,
rless_functionality,
rabs_functionality,
rv-ip-mul,
rabs-rmul,
rv-norm-mul,
rv-ip_functionality,
rv-norm_functionality,
rmul_preserves_rless,
rless-implies-rless,
real_term_polynomial,
itermSubtract_wf,
itermMultiply_wf,
itermVar_wf,
real_term_value_const_lemma,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
req-iff-rsub-is-0,
rsub_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
cut,
introduction,
extract_by_obid,
isectElimination,
thin,
natural_numberEquality,
hypothesis,
hypothesisEquality,
applyEquality,
instantiate,
independent_isectElimination,
sqequalRule,
because_Cache,
minusEquality,
multiplyEquality,
addEquality,
lambdaEquality,
setElimination,
rename,
setEquality,
productEquality,
independent_functionElimination,
dependent_functionElimination,
productElimination,
addLevel,
computeAll,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality
Latex:
\mforall{}rv:InnerProductSpace. \mforall{}a,b,c:Point. (\mDelta{}(a;b;c) {}\mRightarrow{} (\mforall{}t:\mBbbR{}. ((r0 < |t|) {}\mRightarrow{} \mDelta{}(b + t*a - b;b;c))))
Date html generated:
2017_10_04-PM-11_58_36
Last ObjectModification:
2017_07_28-AM-08_54_33
Theory : inner!product!spaces
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