Nuprl Lemma : straight-angle-sum1
∀e:EuclideanPlane. ∀a,b,c,x,y,z,i,j,k:Point. (abc + xyz ≅ ijk ⇒ a-b-c ⇒ out(y xz))
Proof
Definitions occuring in Statement :
hp-angle-sum: abc + xyz ≅ def,
geo-out: out(p ab),
euclidean-plane: EuclideanPlane,
geo-strict-between: a-b-c,
geo-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
hp-angle-sum: abc + xyz ≅ def,
exists: ∃x:A. B[x],
and: P ∧ Q,
geo-cong-angle: abc ≅a xyz,
cand: A c∧ B,
member: t ∈ T,
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
basic-geometry-: BasicGeometry-,
basic-geometry: BasicGeometry,
subtype_rel: A ⊆r B,
guard: {T},
prop: ℙ,
geo-out: out(p ab),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
geo-sep-sym,
angle-cong-preserves-straight-angle,
geo-between-symmetry,
geo-strict-between-implies-between,
geo-cong-angle-symm2,
geo-strict-between_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
hp-angle-sum_wf,
geo-point_wf,
angle-cong-preserves-zero-angle,
geo-out_transitivity,
geo-out_inversion,
geo-out-iff-between1,
geo-strict-between-sep3,
extended-out-preserves-between,
geo-between-inner-trans,
geo-between-exchange3,
geo-between-exchange4
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
sqequalHypSubstitution,
productElimination,
thin,
cut,
introduction,
extract_by_obid,
dependent_functionElimination,
hypothesisEquality,
independent_functionElimination,
hypothesis,
independent_pairFormation,
because_Cache,
isectElimination,
independent_isectElimination,
sqequalRule,
universeIsType,
applyEquality,
instantiate,
inhabitedIsType
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a,b,c,x,y,z,i,j,k:Point. (abc + xyz \mcong{} ijk {}\mRightarrow{} a-b-c {}\mRightarrow{} out(y xz))
Date html generated:
2019_10_16-PM-02_04_36
Last ObjectModification:
2019_06_21-PM-09_45_18
Theory : euclidean!plane!geometry
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