Nuprl Lemma : cong-angle-out-exists-cong3
∀e:EuclideanPlane. ∀a,b,c,x,y,z:Point.
(abc ≅a xyz ⇒ (∃a',c':Point. (out(b a'a) ∧ out(b c'c) ∧ a'bc' ≅a xyz ∧ Cong3(a'bc',xyz))))
Proof
Definitions occuring in Statement :
geo-out: out(p ab),
geo-cong-tri: Cong3(abc,a'b'c'),
geo-cong-angle: abc ≅a xyz,
euclidean-plane: EuclideanPlane,
geo-point: Point,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
geo-cong-angle: abc ≅a xyz,
and: P ∧ Q,
member: t ∈ T,
basic-geometry: BasicGeometry,
exists: ∃x:A. B[x],
geo-midpoint: a=m=b,
guard: {T},
cand: A c∧ B,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
basic-geometry-: BasicGeometry-,
iff: P ⇐⇒ Q,
prop: ℙ,
geo-cong-tri: Cong3(abc,a'b'c'),
uiff: uiff(P;Q)
Lemmas referenced :
symmetric-point-construction,
geo-sep-sym,
geo-proper-extend-exists,
midpoint-sep,
geo-between-sep,
geo-out-iff-between1,
euclidean-plane-axioms,
geo-strict-between-sep3,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-strict-between-sep2,
geo-between-symmetry,
geo-strict-between-implies-between,
geo-out_inversion,
geo-out_wf,
geo-congruent_wf,
geo-cong-angle_wf,
geo-cong-tri_wf,
geo-point_wf,
geo-out_weakening,
geo-eq_weakening,
geo-congruent-iff-length,
geo-length-flip,
geo-sas2,
out-preserves-angle-cong_1
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
hypothesis,
sqequalHypSubstitution,
productElimination,
thin,
introduction,
extract_by_obid,
dependent_functionElimination,
sqequalRule,
hypothesisEquality,
independent_functionElimination,
because_Cache,
rename,
dependent_pairFormation_alt,
applyEquality,
instantiate,
isectElimination,
independent_isectElimination,
independent_pairFormation,
productIsType,
universeIsType,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a,b,c,x,y,z:Point.
(abc \mcong{}\msuba{} xyz {}\mRightarrow{} (\mexists{}a',c':Point. (out(b a'a) \mwedge{} out(b c'c) \mwedge{} a'bc' \mcong{}\msuba{} xyz \mwedge{} Cong3(a'bc',xyz))))
Date html generated:
2019_10_16-PM-01_50_12
Last ObjectModification:
2018_11_19-AM-10_46_45
Theory : euclidean!plane!geometry
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