Nuprl Lemma : p8geo
∀e:BasicGeometry. ∀a,b,c,x,y,z:Point.
((Triangle(a;b;c) ∧ Triangle(x;y;z)) ⇒ Cong3(abc,xyz) ⇒ (abc ≅a xyz ∧ bac ≅a yxz ∧ bca ≅a yzx))
Proof
Definitions occuring in Statement :
geo-cong-tri: Cong3(abc,a'b'c'),
geo-cong-angle: abc ≅a xyz,
geo-tri: Triangle(a;b;c),
basic-geometry: BasicGeometry,
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,
geo-cong-tri: Cong3(abc,a'b'c'),
geo-tri: Triangle(a;b;c),
geo-cong-angle: abc ≅a xyz,
member: t ∈ T,
basic-geometry: BasicGeometry,
uall: ∀[x:A]. B[x],
prop: ℙ,
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
exists: ∃x:A. B[x],
uiff: uiff(P;Q)
Lemmas referenced :
geo-sep-sym,
geo-cong-tri_wf,
geo-tri_wf,
geo-point_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-between-trivial,
geo-congruent-iff-length,
geo-length-flip,
geo-between_wf,
geo-congruent_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
sqequalHypSubstitution,
productElimination,
thin,
cut,
independent_pairFormation,
hypothesis,
introduction,
extract_by_obid,
dependent_functionElimination,
hypothesisEquality,
independent_functionElimination,
because_Cache,
universeIsType,
isectElimination,
sqequalRule,
productIsType,
inhabitedIsType,
applyEquality,
instantiate,
independent_isectElimination,
dependent_pairFormation_alt,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}e:BasicGeometry. \mforall{}a,b,c,x,y,z:Point.
((Triangle(a;b;c) \mwedge{} Triangle(x;y;z)) {}\mRightarrow{} Cong3(abc,xyz) {}\mRightarrow{} (abc \mcong{}\msuba{} xyz \mwedge{} bac \mcong{}\msuba{} yxz \mwedge{} bca \mcong{}\msuba{} yzx))
Date html generated:
2019_10_16-PM-01_29_47
Last ObjectModification:
2018_11_08-AM-11_29_47
Theory : euclidean!plane!geometry
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