Nuprl Lemma : not-rv-pos-angle-implies
∀[n:ℕ]. ∀a,b,c:ℝ^n. ((¬rv-pos-angle(n;a;b;c)) ⇒ (¬(a ≠ b ∧ b ≠ c ∧ c ≠ a ∧ (¬a-b-c) ∧ (¬b-c-a) ∧ (¬c-a-b))))
Proof
Definitions occuring in Statement :
rv-between: a-b-c,
real-vec-sep: a ≠ b,
rv-pos-angle: rv-pos-angle(n;a;b;c),
real-vec: ℝ^n,
nat: ℕ,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
implies: P ⇒ Q,
not: ¬A,
false: False,
and: P ∧ Q,
prop: ℙ,
rless: x < y,
sq_exists: ∃x:{A| B[x]},
real-vec-sep: a ≠ b,
iff: P ⇐⇒ Q,
exists: ∃x:A. B[x],
rev_implies: P ⇐ Q,
rneq: x ≠ y,
or: P ∨ Q,
rv-between: a-b-c,
rsub: x - y,
uimplies: b supposing a,
guard: {T},
rgt: x > y,
rge: x ≥ y,
true: True,
less_than': less_than'(a;b),
squash: ↓T,
less_than: a < b,
cand: A c∧ B,
real-vec-between: a-b-c,
top: Top,
rev_uimplies: rev_uimplies(P;Q),
uiff: uiff(P;Q),
nat: ℕ,
real-vec-add: X + Y,
real-vec-mul: a*X,
real-vec-sub: X - Y,
req-vec: req-vec(n;x;y),
subtype_rel: A ⊆r B,
real-vec: ℝ^n,
i-member: r ∈ I,
rooint: (l, u),
itermConstant: "const",
req_int_terms: t1 ≡ t2
Lemmas referenced :
real-vec-sep_wf,
not_wf,
rv-between_wf,
rv-pos-angle_wf,
real-vec_wf,
nat_wf,
not-rv-pos-angle,
real-vec-sep-symmetry,
rabs-positive-iff,
radd-assoc,
req_inversion,
radd_comm,
radd-ac,
req_weakening,
radd-rminus-both,
radd_functionality,
req_transitivity,
radd-rminus-assoc,
rless_functionality,
rless_wf,
rminus_wf,
radd_wf,
rsub_wf,
int-to-real_wf,
radd-preserves-rless,
rleq_weakening_rless,
rleq_weakening_equal,
rless_functionality_wrt_implies,
rless-int,
real-vec-mul_wf,
real-vec-add_wf,
req-vec_wf,
rooint_wf,
i-member_wf,
rdiv_wf,
rmul-one-both,
rmul-distrib2,
rmul-identity1,
rminus-as-rmul,
radd-int,
rmul_functionality,
rmul-rdiv-cancel2,
rmul_over_rminus,
rmul-distrib,
rmul-zero-both,
rminus_functionality,
rminus-zero,
radd-zero-both,
rmul_wf,
rmul_preserves_rless,
member_rooint_lemma,
real-vec-mul_functionality,
real-vec-add_functionality,
req-vec_weakening,
req-vec_functionality,
real-vec-sub_wf,
int_seg_wf,
iff_weakening_equal,
true_wf,
squash_wf,
rminus-rminus,
req_functionality,
uiff_transitivity,
req_wf,
equal_wf,
rmul_preserves_req,
real_wf,
rmul-ac,
rmul_comm,
rmul-assoc,
false_wf,
or_wf,
minimal-double-negation-hyp-elim,
minimal-not-not-excluded-middle,
rless-implies-rless,
real_term_polynomial,
itermSubtract_wf,
itermConstant_wf,
itermVar_wf,
real_term_value_const_lemma,
real_term_value_sub_lemma,
real_term_value_var_lemma,
req-iff-rsub-is-0,
trivial-rsub-rless,
rminus-radd,
rmul-rdiv-cancel,
rleq_antisymmetry,
not-rless,
real-vec-sep_functionality,
not-real-vec-sep-refl
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
thin,
sqequalHypSubstitution,
productElimination,
because_Cache,
hypothesis,
independent_functionElimination,
voidElimination,
productEquality,
extract_by_obid,
isectElimination,
hypothesisEquality,
sqequalRule,
lambdaEquality,
dependent_functionElimination,
independent_pairFormation,
unionElimination,
levelHypothesis,
independent_isectElimination,
addLevel,
natural_numberEquality,
baseClosed,
imageMemberEquality,
inrFormation,
dependent_pairFormation,
addEquality,
minusEquality,
voidEquality,
isect_memberEquality,
rename,
setElimination,
universeEquality,
imageElimination,
applyEquality,
equalitySymmetry,
equalityTransitivity,
functionEquality,
computeAll,
int_eqEquality,
intEquality
Latex:
\mforall{}[n:\mBbbN{}]
\mforall{}a,b,c:\mBbbR{}\^{}n.
((\mneg{}rv-pos-angle(n;a;b;c)) {}\mRightarrow{} (\mneg{}(a \mneq{} b \mwedge{} b \mneq{} c \mwedge{} c \mneq{} a \mwedge{} (\mneg{}a-b-c) \mwedge{} (\mneg{}b-c-a) \mwedge{} (\mneg{}c-a-b))))
Date html generated:
2017_10_03-AM-11_08_23
Last ObjectModification:
2017_07_28-AM-08_22_49
Theory : reals
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