Nuprl Lemma : cong-tri-implies-cong-angle2
∀e:BasicGeometry. ∀a,b,c,x,y,z:Point. (a ≠ b ⇒ b ≠ c ⇒ ab ≅ xy ⇒ bc ≅ yz ⇒ ca ≅ zx ⇒ abc ≅a xyz)
Proof
Definitions occuring in Statement :
geo-cong-angle: abc ≅a xyz,
basic-geometry: BasicGeometry,
geo-congruent: ab ≅ cd,
geo-sep: a ≠ b,
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,
geo-cong-tri: Cong3(abc,a'b'c'),
and: P ∧ Q,
cand: A c∧ B,
basic-geometry: BasicGeometry,
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
subtype_rel: A ⊆r B,
guard: {T},
prop: ℙ
Lemmas referenced :
cong-tri-implies-cong-angle,
geo-congruent-symmetry,
geo-congruent-sep,
geo-congruent_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
basic-geometry-subtype,
subtype_rel_transitivity,
basic-geometry_wf,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-sep_wf,
geo-point_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
independent_pairFormation,
hypothesis,
isectElimination,
because_Cache,
independent_isectElimination,
universeIsType,
applyEquality,
instantiate,
sqequalRule,
inhabitedIsType
Latex:
\mforall{}e:BasicGeometry. \mforall{}a,b,c,x,y,z:Point.
(a \mneq{} b {}\mRightarrow{} b \mneq{} c {}\mRightarrow{} ab \mcong{} xy {}\mRightarrow{} bc \mcong{} yz {}\mRightarrow{} ca \mcong{} zx {}\mRightarrow{} abc \mcong{}\msuba{} xyz)
Date html generated:
2019_10_16-PM-01_22_50
Last ObjectModification:
2018_12_15-PM-10_03_16
Theory : euclidean!plane!geometry
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