Nuprl Lemma : p4geo
∀e:BasicGeometry. ∀a,b,c,A,B,C:Point.
((Triangle(a;b;c) ∧ Triangle(A;B;C))
⇒ (ab ≅ AB ∧ ac ≅ AC ∧ bac ≅a BAC)
⇒ (bc ≅ BC ∧ abc ≅a ABC ∧ bca ≅a BCA ∧ Cong3(abc,ABC)))
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-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-tri: Triangle(a;b;c),
geo-cong-angle: abc ≅a xyz,
exists: ∃x:A. B[x],
basic-geometry: BasicGeometry,
uiff: uiff(P;Q),
geo-cong-tri: Cong3(abc,a'b'c')
Lemmas referenced :
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-cong-angle_wf,
geo-tri_wf,
geo-point_wf,
geo-sas,
geo-between_wf,
geo-between-trivial,
geo-congruent-iff-length,
geo-length-flip
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
sqequalHypSubstitution,
productElimination,
thin,
cut,
independent_pairFormation,
hypothesis,
sqequalRule,
productIsType,
universeIsType,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
applyEquality,
instantiate,
independent_isectElimination,
because_Cache,
dependent_functionElimination,
inhabitedIsType,
dependent_pairFormation_alt,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}e:BasicGeometry. \mforall{}a,b,c,A,B,C:Point.
((Triangle(a;b;c) \mwedge{} Triangle(A;B;C))
{}\mRightarrow{} (ab \mcong{} AB \mwedge{} ac \mcong{} AC \mwedge{} bac \mcong{}\msuba{} BAC)
{}\mRightarrow{} (bc \mcong{} BC \mwedge{} abc \mcong{}\msuba{} ABC \mwedge{} bca \mcong{}\msuba{} BCA \mwedge{} Cong3(abc,ABC)))
Date html generated:
2019_10_16-PM-01_30_48
Last ObjectModification:
2018_11_07-PM-00_59_12
Theory : euclidean!plane!geometry
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