Nuprl Lemma : hp-right-angles-out
∀e:EuclideanPlane. ∀a,b,p1,p2:Point.
(((Rbap1 ∧ Rbap2) ∧ p1 # ab ∧ (p1 leftof ab ⇐⇒ p2 leftof ab) ∧ (p1 leftof ba ⇐⇒ p2 leftof ba)) ⇒ out(a p1p2))
Proof
Definitions occuring in Statement :
geo-out: out(p ab),
euclidean-plane: EuclideanPlane,
geo-lsep: a # bc,
right-angle: Rabc,
geo-left: a leftof bc,
geo-point: Point,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
geo-out: out(p ab),
member: t ∈ T,
guard: {T},
not: ¬A,
false: False,
prop: ℙ,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
rev_implies: P ⇐ Q,
geo-lsep: a # bc,
or: P ∨ Q,
cand: A c∧ B,
so_lambda: λ2x.t[x],
so_apply: x[s],
basic-geometry: BasicGeometry,
basic-geometry-: BasicGeometry-,
oriented-plane: OrientedPlane
Lemmas referenced :
geo-sep-sym,
lsep-implies-sep,
not_wf,
geo-between_wf,
right-angle_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-lsep_wf,
iff_wf,
geo-left_wf,
geo-point_wf,
left-implies-sep,
all_wf,
geo-sep_wf,
lsep-all-sym,
right-angle-symmetry,
adjacent-right-angles,
geo-colinear_wf,
geo-colinear-symmetry,
geo-simple-colinear-cases,
stable__not,
geo-between-symmetry,
left-between,
left-all-symmetry,
not-left-and-right
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
independent_pairFormation,
cut,
introduction,
extract_by_obid,
dependent_functionElimination,
because_Cache,
independent_functionElimination,
hypothesisEquality,
hypothesis,
voidElimination,
productEquality,
isectElimination,
applyEquality,
sqequalRule,
instantiate,
independent_isectElimination,
unionElimination,
lambdaEquality,
functionEquality,
addLevel,
levelHypothesis
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a,b,p1,p2:Point.
(((Rbap1 \mwedge{} Rbap2) \mwedge{} p1 \# ab \mwedge{} (p1 leftof ab \mLeftarrow{}{}\mRightarrow{} p2 leftof ab) \mwedge{} (p1 leftof ba \mLeftarrow{}{}\mRightarrow{} p2 leftof ba))
{}\mRightarrow{} out(a p1p2))
Date html generated:
2018_05_22-PM-00_18_25
Last ObjectModification:
2018_04_20-PM-06_31_00
Theory : euclidean!plane!geometry
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