Nuprl Lemma : angle-cong-preserves-straight-angle
∀g:EuclideanPlane. ∀a,b,c,x,y,z:Point. (x_y_z ⇒ abc ≅a xyz ⇒ a-b-c)
Proof
Definitions occuring in Statement :
geo-cong-angle: abc ≅a xyz,
euclidean-plane: EuclideanPlane,
geo-strict-between: a-b-c,
geo-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,
geo-strict-between: a-b-c,
and: P ∧ Q,
geo-cong-angle: abc ≅a xyz,
exists: ∃x:A. B[x],
member: t ∈ T,
basic-geometry: BasicGeometry,
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
basic-geometry-: BasicGeometry-,
cand: A c∧ B,
uiff: uiff(P;Q),
prop: ℙ,
subtype_rel: A ⊆r B,
guard: {T}
Lemmas referenced :
geo-congruent-preserves-strict-between,
geo-between-symmetry,
geo-between-outer-trans,
geo-sep-sym,
geo-between-exchange4,
euclidean-plane-axioms,
geo-between-sep,
geo-congruent-iff-length,
geo-length-flip,
geo-strict-between-implies-between,
geo-between-inner-trans,
geo-between-exchange3,
geo-cong-angle_wf,
geo-between_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-point_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
independent_pairFormation,
sqequalHypSubstitution,
productElimination,
thin,
cut,
introduction,
extract_by_obid,
dependent_functionElimination,
sqequalRule,
hypothesisEquality,
independent_functionElimination,
because_Cache,
isectElimination,
independent_isectElimination,
hypothesis,
equalityTransitivity,
equalitySymmetry,
universeIsType,
applyEquality,
instantiate,
inhabitedIsType
Latex:
\mforall{}g:EuclideanPlane. \mforall{}a,b,c,x,y,z:Point. (x\_y\_z {}\mRightarrow{} abc \mcong{}\msuba{} xyz {}\mRightarrow{} a-b-c)
Date html generated:
2019_10_16-PM-01_56_50
Last ObjectModification:
2019_08_30-AM-10_22_13
Theory : euclidean!plane!geometry
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