Nuprl Lemma : bnot-isleft
∀g:OrientedPlane. ∀a,b:Point. ∀c:{c:Point| a # bc} . ¬bisleft(a;b;c) = isleft(a;c;b)
Proof
Definitions occuring in Statement :
geo-isleft: isleft(a;b;c),
oriented-plane: OrientedPlane,
geo-lsep: a # bc,
geo-point: Point,
bnot: ¬bb,
bool: 𝔹,
all: ∀x:A. B[x],
set: {x:A| B[x]} ,
equal: s = t ∈ T
Definitions unfolded in proof :
not: ¬A,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
cand: A c∧ B,
and: P ∧ Q,
implies: P ⇒ Q,
so_apply: x[s],
oriented-plane: Error :oriented-plane,
so_lambda: λ2x.t[x],
uimplies: b supposing a,
guard: {T},
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
prop: ℙ,
member: t ∈ T,
all: ∀x:A. B[x],
false: False,
or: P ∨ Q,
geo-lsep: a # bc,
squash: ↓T,
sq_stable: SqStable(P)
Lemmas referenced :
assert-geo-isleft,
assert_of_bnot,
iff_weakening_uiff,
iff_transitivity,
iff_imp_equal_bool,
iff_wf,
geo-left_wf,
not_wf,
assert_wf,
lsep-all-sym,
geo-isleft_wf,
bnot_wf,
geo-lsep_wf,
Error :basic-geo-primitives_wf,
Error :basic-geo-structure_wf,
basic-geometry-_wf,
Error :oriented-plane_wf,
subtype_rel_transitivity,
Error :oriented-plane-subtype,
basic-geometry--subtype,
geo-point_wf,
set_wf,
geo-left-antisymmetry,
Error :sq_stable__geo-lsep
Rules used in proof :
impliesFunctionality,
independent_pairFormation,
addLevel,
productElimination,
independent_functionElimination,
dependent_functionElimination,
dependent_set_memberEquality,
because_Cache,
rename,
setElimination,
lambdaEquality,
sqequalRule,
independent_isectElimination,
instantiate,
applyEquality,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
hypothesis,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
voidElimination,
unionElimination,
imageElimination,
baseClosed,
imageMemberEquality
Latex:
\mforall{}g:OrientedPlane. \mforall{}a,b:Point. \mforall{}c:\{c:Point| a \# bc\} . \mneg{}\msubb{}isleft(a;b;c) = isleft(a;c;b)
Date html generated:
2017_10_02-PM-06_50_12
Last ObjectModification:
2017_08_06-PM-07_30_05
Theory : euclidean!plane!geometry
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