Nuprl Lemma : unique-angles-in-half-plane-better2
∀e:EuclideanPlane. ∀a,b,c,q:Point. (a leftof bc ⇒ q leftof bc ⇒ qcb ≅a acb ⇒ out(c aq))
Proof
Definitions occuring in Statement :
geo-out: out(p ab),
geo-cong-angle: abc ≅a xyz,
euclidean-plane: EuclideanPlane,
geo-left: a leftof bc,
geo-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
basic-geometry: BasicGeometry,
euclidean-plane: EuclideanPlane,
guard: {T},
and: P ∧ Q,
exists: ∃x:A. B[x],
uall: ∀[x:A]. B[x],
prop: ℙ,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
uiff: uiff(P;Q),
oriented-plane: OrientedPlane,
basic-geometry-: BasicGeometry-,
cand: A c∧ B,
iff: P ⇐⇒ Q
Lemmas referenced :
geo-proper-extend-exists,
geo-O_wf,
geo-X_wf,
left-implies-sep,
geo-sep-O-X,
geo-cong-angle_wf,
geo-left_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-point_wf,
geo-sep-sym,
geo-strict-between-sep3,
Euclid-Prop7,
geo-congruent-iff-length,
left-between-implies-right1,
geo-strict-between-implies-between,
left-between-implies-right2,
geo-congruent-refl,
geo-sas2,
geo-cong-angle-symmetry,
geo-cong-angle-transitivity,
geo-cong-angle-refl,
geo-out_weakening,
geo-eq_weakening,
out-preserves-angle-cong_1,
geo-out-if-between,
geo-strict-between-sym,
geo-eq_inversion,
geo-out_functionality
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
sqequalRule,
hypothesisEquality,
setElimination,
rename,
hypothesis,
because_Cache,
independent_functionElimination,
productElimination,
universeIsType,
isectElimination,
applyEquality,
instantiate,
independent_isectElimination,
inhabitedIsType,
dependent_set_memberEquality_alt,
equalitySymmetry,
independent_pairFormation
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a,b,c,q:Point. (a leftof bc {}\mRightarrow{} q leftof bc {}\mRightarrow{} qcb \mcong{}\msuba{} acb {}\mRightarrow{} out(c aq))
Date html generated:
2019_10_16-PM-01_54_07
Last ObjectModification:
2019_08_12-PM-02_29_37
Theory : euclidean!plane!geometry
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