Nuprl Lemma : r2-left-or-right--pos-angle
∀a,b,c:ℝ^2. (r2-left(a;b;c) ∨ r2-left(a;c;b) ⇐⇒ rv-pos-angle(2;a;b;c))
Proof
Definitions occuring in Statement :
r2-left: r2-left(p;q;r),
rv-pos-angle: rv-pos-angle(n;a;b;c),
real-vec: ℝ^n,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
or: P ∨ Q,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
rev_implies: P ⇐ Q,
nat: ℕ,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
or: P ∨ Q,
r2-left: r2-left(p;q;r),
rneq: x ≠ y,
guard: {T},
uimplies: b supposing a,
uiff: uiff(P;Q),
req_int_terms: t1 ≡ t2,
top: Top
Lemmas referenced :
or_wf,
r2-left_wf,
rv-pos-angle_wf,
false_wf,
le_wf,
real-vec_wf,
r2-left-pos-angle,
rv-pos-angle-permute,
rv-pos-angle-symmetry,
r2-det-nonzero,
rless_wf,
int-to-real_wf,
r2-det_wf,
rminus_wf,
rless-implies-rless,
rsub_wf,
itermSubtract_wf,
itermConstant_wf,
itermMinus_wf,
itermVar_wf,
req-iff-rsub-is-0,
rless_functionality,
r2-det-antisymmetry,
req_weakening,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_const_lemma,
real_term_value_minus_lemma,
real_term_value_var_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
independent_pairFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
dependent_set_memberEquality,
natural_numberEquality,
sqequalRule,
unionElimination,
dependent_functionElimination,
independent_functionElimination,
because_Cache,
inlFormation,
inrFormation,
independent_isectElimination,
productElimination,
approximateComputation,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality
Latex:
\mforall{}a,b,c:\mBbbR{}\^{}2. (r2-left(a;b;c) \mvee{} r2-left(a;c;b) \mLeftarrow{}{}\mRightarrow{} rv-pos-angle(2;a;b;c))
Date html generated:
2017_10_03-AM-11_49_33
Last ObjectModification:
2017_08_11-PM-11_07_48
Theory : reals
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