Nuprl Lemma : geo-reflected-right-triangles-congruent
∀e:BasicGeometry. ∀a,b,c,d:Point. (a # bc ⇒ Rcba ⇒ a=b=d ⇒ Cong3(abc,dbc))
Proof
Definitions occuring in Statement :
geo-cong-tri: Cong3(abc,a'b'c'),
basic-geometry: BasicGeometry,
geo-lsep: a # bc,
right-angle: Rabc,
geo-midpoint: a=m=b,
geo-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
right-angle: Rabc,
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
prop: ℙ,
geo-midpoint: a=m=b,
and: P ∧ Q,
basic-geometry: BasicGeometry,
euclidean-plane: EuclideanPlane,
basic-geometry-: BasicGeometry-,
uiff: uiff(P;Q),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
cand: A c∧ B,
geo-cong-tri: Cong3(abc,a'b'c')
Lemmas referenced :
geo-midpoint-symmetry,
geo-midpoint_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,
right-angle_wf,
geo-lsep_wf,
geo-point_wf,
geo-construction-unicity,
subtype_rel_self,
basic-geometry-_wf,
midpoint-sep,
geo-between-sep,
lsep-implies-sep,
geo-between-symmetry,
geo-congruent-iff-length,
geo-length-flip,
geo-congruent_functionality,
geo-eq_weakening,
geo-congruent-refl,
geo-sas2,
geo-sep-sym,
geo-congruent-symmetry,
geo-congruent-sep,
geo-right-angles-congruent,
geo-cong-angle-symm3
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
sqequalHypSubstitution,
hypothesis,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
introduction,
extract_by_obid,
universeIsType,
isectElimination,
applyEquality,
instantiate,
independent_isectElimination,
sqequalRule,
because_Cache,
inhabitedIsType,
productElimination,
equalityTransitivity,
equalitySymmetry,
independent_pairFormation
Latex:
\mforall{}e:BasicGeometry. \mforall{}a,b,c,d:Point. (a \# bc {}\mRightarrow{} Rcba {}\mRightarrow{} a=b=d {}\mRightarrow{} Cong3(abc,dbc))
Date html generated:
2019_10_16-PM-01_53_18
Last ObjectModification:
2019_03_20-PM-01_20_16
Theory : euclidean!plane!geometry
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