Nuprl Lemma : r2-not-left-right
∀a,b,c:ℝ^2. ((¬r2-left(a;b;c)) ⇒ (¬r2-left(a;c;b)) ⇒ (¬((¬rv-T(2;a;b;c)) ∧ (¬rv-T(2;b;c;a)) ∧ (¬rv-T(2;c;a;b)))))
Proof
Definitions occuring in Statement :
r2-left: r2-left(p;q;r),
rv-T: rv-T(n;a;b;c),
real-vec: ℝ^n,
all: ∀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,
not: ¬A,
false: False,
member: t ∈ T,
rneq: x ≠ y,
or: P ∨ Q,
r2-left: r2-left(p;q;r),
rless: x < y,
sq_exists: ∃x:{A| B[x]},
prop: ℙ,
uall: ∀[x:A]. B[x],
nat: ℕ,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
uimplies: b supposing a,
uiff: uiff(P;Q),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
req_int_terms: t1 ≡ t2,
top: Top
Lemmas referenced :
r2-det-nonzero,
rv-pos-angle_wf,
false_wf,
le_wf,
not_wf,
r2-left_wf,
real-vec_wf,
not-rv-pos-angle-implies2,
int-to-real_wf,
r2-det_wf,
rminus_wf,
rless-implies-rless,
rsub_wf,
itermSubtract_wf,
itermConstant_wf,
itermVar_wf,
itermMinus_wf,
req-iff-rsub-is-0,
rless_functionality,
req_weakening,
r2-det-antisymmetry,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_const_lemma,
real_term_value_var_lemma,
real_term_value_minus_lemma
Rules used in proof :
cut,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
hypothesisEquality,
independent_functionElimination,
hypothesis,
unionElimination,
voidElimination,
because_Cache,
isectElimination,
dependent_set_memberEquality,
natural_numberEquality,
sqequalRule,
independent_pairFormation,
independent_isectElimination,
productElimination,
approximateComputation,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidEquality
Latex:
\mforall{}a,b,c:\mBbbR{}\^{}2.
((\mneg{}r2-left(a;b;c))
{}\mRightarrow{} (\mneg{}r2-left(a;c;b))
{}\mRightarrow{} (\mneg{}((\mneg{}rv-T(2;a;b;c)) \mwedge{} (\mneg{}rv-T(2;b;c;a)) \mwedge{} (\mneg{}rv-T(2;c;a;b)))))
Date html generated:
2017_10_03-AM-11_57_53
Last ObjectModification:
2017_06_14-PM-06_16_11
Theory : reals
Home
Index