Nuprl Lemma : ip-between-iff
∀rv:InnerProductSpace. ∀a,b,c:Point. (a # b ⇒ c # b ⇒ (a_b_c ⇐⇒ ∃t:ℝ. ((t ∈ (r0, r1)) ∧ b ≡ t*a + r1 - t*c)))
Proof
Definitions occuring in Statement :
ip-between: a_b_c,
inner-product-space: InnerProductSpace,
rv-mul: a*x,
rv-add: x + y,
rooint: (l, u),
i-member: r ∈ I,
rsub: x - y,
int-to-real: r(n),
real: ℝ,
ss-eq: x ≡ y,
ss-sep: x # y,
ss-point: Point,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
and: P ∧ Q,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
rev_implies: P ⇐ Q,
exists: ∃x:A. B[x],
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
so_apply: x[s],
ip-between: a_b_c,
rev_uimplies: rev_uimplies(P;Q),
uiff: uiff(P;Q),
not: ¬A,
false: False,
less_than': less_than'(a;b),
le: A ≤ B,
nat: ℕ,
or: P ∨ Q,
rneq: x ≠ y,
rge: x ≥ y,
true: True,
squash: ↓T,
less_than: a < b,
cand: A c∧ B,
top: Top,
req_int_terms: t1 ≡ t2,
itermConstant: "const",
rdiv: (x/y),
rv-minus: -x,
rv-sub: x - y,
rsub: x - y,
label: ...$L... t,
ml-term-to-poly: ml-term-to-poly(t),
nil: [],
it: ⋅,
has-value: (a)↓
Lemmas referenced :
ip-between_wf,
exists_wf,
real_wf,
i-member_wf,
rooint_wf,
int-to-real_wf,
ss-eq_wf,
real-vector-space_subtype1,
inner-product-space_subtype,
subtype_rel_transitivity,
inner-product-space_wf,
real-vector-space_wf,
separation-space_wf,
rv-add_wf,
rv-mul_wf,
rsub_wf,
ss-sep_wf,
ss-point_wf,
rv-sep-iff,
rv-sub_wf,
rv-Cauchy-Schwarz-equality,
rnexp-rmul,
rv-norm-squared,
req_inversion,
rmul_functionality,
req_weakening,
req_functionality,
uiff_transitivity,
rleq_wf,
rv-norm_wf,
rmul_wf,
rv-ip_wf,
le_wf,
false_wf,
rnexp_wf,
req_wf,
equal_wf,
radd_wf,
rmul_comm,
rv-ip-symmetry,
rnexp_functionality,
radd-zero-both,
radd-rminus-assoc,
radd-ac,
radd_comm,
radd_functionality,
rminus_wf,
radd-preserves-req,
rminus-rminus,
rminus_functionality,
rmul_over_rminus,
req_transitivity,
rnexp2,
rv-norm_functionality,
ss-eq_weakening,
rv-ip_functionality,
rless_wf,
rv-norm-positive,
rv-norm-mul,
rv-ip-mul2,
rabs_wf,
rmul-ac,
rmul-zero-both,
rmul-distrib,
rmul_preserves_req,
rnexp-positive,
rleq_functionality,
radd-int,
rmul-distrib2,
rmul-identity1,
radd-assoc,
rminus-as-rmul,
zero-rleq-rabs,
radd-preserves-rleq,
rmul-is-positive,
rless_functionality,
rabs-positive-iff,
rless_transitivity1,
rless_transitivity2,
rless_irreflexivity,
rv-norm-nonneg,
rleq_weakening_equal,
rleq_weakening_rless,
rless_functionality_wrt_implies,
rv-mul-rv-sub,
ss-eq_functionality,
rless-int,
rdiv_wf,
req-iff-rsub-is-0,
real_term_value_var_lemma,
real_term_value_add_lemma,
real_term_value_sub_lemma,
real_term_value_const_lemma,
itermConstant_wf,
itermVar_wf,
itermAdd_wf,
itermSubtract_wf,
real_term_polynomial,
radd-preserves-rless,
rmul-rinv3,
real_term_value_mul_lemma,
itermMultiply_wf,
trivial-rless-radd,
rinv_wf2,
real_term_value_minus_lemma,
itermMinus_wf,
rless-implies-rless,
rmul_preserves_rless,
member_rooint_lemma,
rv-add-cancel-left,
ss-eq_inversion,
rv-add-0,
rv-mul0,
rv-mul_functionality,
rv-mul-1-add,
rv-add_functionality,
rv-add-swap,
rv-add-assoc,
rv-0_wf,
rv-minus_wf,
rv-mul-add,
rv-mul-add-alt,
rv-mul-mul,
rv-mul-linear,
radd-rminus-both,
iff_weakening_equal,
true_wf,
squash_wf,
rmul-assoc,
rmul-rdiv-cancel2,
rmul-one-both,
rminus-zero,
rdiv_functionality,
rmul-int,
rminus-radd,
rmul-minus,
rv-mul1,
ip-between_functionality,
real_polynomial_null,
evalall-sqequal,
rv-add-comm,
rv-mul-1-add-alt,
rv-ip-mul,
rabs-of-nonneg,
rabs-rminus
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
independent_pairFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
productElimination,
sqequalRule,
lambdaEquality,
productEquality,
natural_numberEquality,
applyEquality,
instantiate,
independent_isectElimination,
because_Cache,
independent_functionElimination,
dependent_functionElimination,
setEquality,
rename,
setElimination,
dependent_set_memberEquality,
equalitySymmetry,
equalityTransitivity,
levelHypothesis,
addLevel,
inrFormation,
addEquality,
minusEquality,
promote_hyp,
unionElimination,
voidElimination,
baseClosed,
imageMemberEquality,
dependent_pairFormation,
voidEquality,
isect_memberEquality,
intEquality,
int_eqEquality,
computeAll,
universeEquality,
imageElimination,
multiplyEquality,
sqleReflexivity,
mlComputation
Latex:
\mforall{}rv:InnerProductSpace. \mforall{}a,b,c:Point.
(a \# b {}\mRightarrow{} c \# b {}\mRightarrow{} (a\_b\_c \mLeftarrow{}{}\mRightarrow{} \mexists{}t:\mBbbR{}. ((t \mmember{} (r0, r1)) \mwedge{} b \mequiv{} t*a + r1 - t*c)))
Date html generated:
2017_10_04-PM-11_57_45
Last ObjectModification:
2017_07_28-AM-08_54_31
Theory : inner!product!spaces
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