Nuprl Lemma : cong-angle-out-exists1
∀e:BasicGeometry. ∀a,b,c,x,y,z:Point. (abc ≅a xyz ⇒ (∃x',z':Point. (out(y x'x) ∧ out(y z'z) ∧ Cong3(x'yz',abc))))
Proof
Definitions occuring in Statement :
geo-out: out(p ab),
geo-cong-tri: Cong3(abc,a'b'c'),
geo-cong-angle: abc ≅a xyz,
basic-geometry: BasicGeometry,
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,
cand: A c∧ B,
member: t ∈ T,
basic-geometry: BasicGeometry,
uall: ∀[x:A]. B[x],
prop: ℙ,
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
euclidean-plane: EuclideanPlane,
exists: ∃x:A. B[x],
basic-geometry-: BasicGeometry-,
iff: P ⇐⇒ Q,
geo-cong-tri: Cong3(abc,a'b'c'),
uiff: uiff(P;Q)
Lemmas referenced :
geo-sep-sym,
geo-cong-angle_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-proper-extend-exists,
geo-O_wf,
geo-X_wf,
geo-sep-O-X,
geo-strict-between-sep3,
geo-out_wf,
geo-cong-tri_wf,
geo-out-iff-between1,
geo-strict-between-implies-between,
subtype_rel_self,
basic-geometry-_wf,
geo-between-symmetry,
geo-out_inversion,
geo-congruent-iff-length,
geo-length-flip,
geo-sas2,
geo-cong-angle-symm2,
geo-out_weakening,
geo-eq_weakening,
out-preserves-angle-cong_1
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
sqequalHypSubstitution,
productElimination,
thin,
hypothesis,
independent_pairFormation,
introduction,
extract_by_obid,
dependent_functionElimination,
hypothesisEquality,
independent_functionElimination,
universeIsType,
isectElimination,
inhabitedIsType,
applyEquality,
instantiate,
independent_isectElimination,
sqequalRule,
setElimination,
rename,
because_Cache,
dependent_pairFormation_alt,
productIsType,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}e:BasicGeometry. \mforall{}a,b,c,x,y,z:Point.
(abc \mcong{}\msuba{} xyz {}\mRightarrow{} (\mexists{}x',z':Point. (out(y x'x) \mwedge{} out(y z'z) \mwedge{} Cong3(x'yz',abc))))
Date html generated:
2019_10_16-PM-01_31_01
Last ObjectModification:
2018_12_07-PM-03_41_13
Theory : euclidean!plane!geometry
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