Nuprl Lemma : p4-triangles
∀e:HeytingGeometry. ∀a,b,c,x,y,z:Point.
(a # bc ⇒ x # yz ⇒ ab ≅ xy ⇒ ac ≅ xz ⇒ bac ≅a yxz ⇒ ((bc ≅ yz ∧ Cong3(abc,xyz)) ∧ abc ≅a xyz ∧ bca ≅a yzx))
Proof
Definitions occuring in Statement :
geo-triangle: a # bc,
heyting-geometry: HeytingGeometry,
geo-cong-tri: Cong3(abc,a'b'c'),
geo-cong-angle: abc ≅a xyz,
geo-congruent: ab ≅ cd,
geo-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
cand: A c∧ B,
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
prop: ℙ,
geo-cong-angle: abc ≅a xyz,
exists: ∃x:A. B[x],
heyting-geometry: HeytingGeometry,
uiff: uiff(P;Q),
geo-cong-tri: Cong3(abc,a'b'c')
Lemmas referenced :
geo-cong-angle_wf,
euclidean-plane-subtype-basic,
heyting-geometry-subtype,
subtype_rel_transitivity,
heyting-geometry_wf,
euclidean-plane_wf,
basic-geometry_wf,
geo-congruent_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-triangle_wf,
geo-point_wf,
geo-inner-five-segment,
geo-congruent-iff-length,
geo-length-flip,
geo-inner-three-segment,
geo-between-symmetry,
geo-triangle-property,
geo-between-trivial,
geo-between_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
independent_pairFormation,
hypothesis,
sqequalHypSubstitution,
productElimination,
thin,
universeIsType,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
applyEquality,
instantiate,
independent_isectElimination,
sqequalRule,
because_Cache,
inhabitedIsType,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
dependent_pairFormation_alt,
productIsType
Latex:
\mforall{}e:HeytingGeometry. \mforall{}a,b,c,x,y,z:Point.
(a \# bc
{}\mRightarrow{} x \# yz
{}\mRightarrow{} ab \mcong{} xy
{}\mRightarrow{} ac \mcong{} xz
{}\mRightarrow{} bac \mcong{}\msuba{} yxz
{}\mRightarrow{} ((bc \mcong{} yz \mwedge{} Cong3(abc,xyz)) \mwedge{} abc \mcong{}\msuba{} xyz \mwedge{} bca \mcong{}\msuba{} yzx))
Date html generated:
2019_10_16-PM-02_09_49
Last ObjectModification:
2018_11_08-AM-11_30_05
Theory : euclidean!plane!geometry
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