Nuprl Lemma : assert-geo-isleft
∀g:OrientedPlane. ∀a,b:Point. ∀c:{c:Point| a # bc} . (↑isleft(a;b;c) ⇐⇒ a leftof bc)
Proof
Definitions occuring in Statement :
geo-isleft: isleft(a;b;c),
oriented-plane: OrientedPlane,
geo-lsep: a # bc,
geo-left: a leftof bc,
geo-point: Point,
assert: ↑b,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
set: {x:A| B[x]}
Definitions unfolded in proof :
so_apply: x[s],
so_lambda: λ2x.t[x],
guard: {T},
subtype_rel: A ⊆r B,
bfalse: ff,
false: False,
not: ¬A,
uimplies: b supposing a,
true: True,
btrue: tt,
isl: isl(x),
ifthenelse: if b then t else f fi ,
assert: ↑b,
rev_implies: P ⇐ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
or: P ∨ Q,
implies: P ⇒ Q,
prop: ℙ,
oriented-plane: Error :oriented-plane,
uall: ∀[x:A]. B[x],
member: t ∈ T,
geo-isleft: isleft(a;b;c),
all: ∀x:A. B[x]
Lemmas referenced :
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,
equal_wf,
btrue_neq_bfalse,
bfalse_wf,
assert_elim,
isl_wf,
assert_wf,
geo-left_wf,
or_wf,
geo-lsep_wf,
geo-orientation_wf,
not-left-and-right
Rules used in proof :
lambdaEquality,
instantiate,
applyEquality,
dependent_functionElimination,
voidElimination,
independent_functionElimination,
equalitySymmetry,
equalityTransitivity,
sqequalRule,
levelHypothesis,
independent_isectElimination,
inrEquality,
addLevel,
natural_numberEquality,
inlEquality,
voidEquality,
independent_pairFormation,
unionElimination,
because_Cache,
dependent_set_memberEquality,
hypothesis,
hypothesisEquality,
rename,
setElimination,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}g:OrientedPlane. \mforall{}a,b:Point. \mforall{}c:\{c:Point| a \# bc\} . (\muparrow{}isleft(a;b;c) \mLeftarrow{}{}\mRightarrow{} a leftof bc)
Date html generated:
2017_10_02-PM-06_50_05
Last ObjectModification:
2017_08_06-PM-07_29_51
Theory : euclidean!plane!geometry
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