Nuprl Lemma : r2-not-left-right-iff
∀a,b,c:ℝ^2. (¬(r2-left(a;b;c) ∨ r2-left(a;c;b)) ⇐⇒ ¬((¬a_b_c) ∧ (¬b_c_a) ∧ (¬c_a_b)))
Proof
Definitions occuring in Statement :
r2-left: r2-left(p;q;r),
rv-be: a_b_c,
real-vec: ℝ^n,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
not: ¬A,
or: P ∨ Q,
and: P ∧ Q,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
implies: P ⇒ Q,
not: ¬A,
and: P ∧ Q,
nat: ℕ,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
prop: ℙ,
uall: ∀[x:A]. B[x],
iff: P ⇐⇒ Q,
rv-be: a_b_c,
or: P ∨ Q,
guard: {T},
cand: A c∧ B,
rev_implies: P ⇐ Q,
uimplies: b supposing a
Lemmas referenced :
r2-not-left-right,
rv-T-iff,
false_wf,
le_wf,
rv-T_wf,
not_wf,
real-vec-sep_wf,
rv-between_wf,
r2-left_wf,
rv-be_wf,
or_wf,
real-vec_wf,
r2-left-pos-angle,
rv-pos-angle-permute,
rv-pos-angle-symmetry,
rv-pos-angle-not-be
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
productElimination,
independent_pairFormation,
impliesFunctionality,
dependent_set_memberEquality,
natural_numberEquality,
sqequalRule,
isectElimination,
productEquality,
because_Cache,
promote_hyp,
inlFormation,
voidElimination,
inrFormation,
unionElimination,
independent_isectElimination
Latex:
\mforall{}a,b,c:\mBbbR{}\^{}2. (\mneg{}(r2-left(a;b;c) \mvee{} r2-left(a;c;b)) \mLeftarrow{}{}\mRightarrow{} \mneg{}((\mneg{}a\_b\_c) \mwedge{} (\mneg{}b\_c\_a) \mwedge{} (\mneg{}c\_a\_b)))
Date html generated:
2017_10_03-AM-11_58_38
Last ObjectModification:
2017_08_11-PM-10_59_04
Theory : reals
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