Nuprl Lemma : cong-angle-out-exists3
∀e:BasicGeometry. ∀a,b,c,x,y,z:Point.
(abc ≅a xyz
⇒ a ≠ b
⇒ c ≠ b
⇒ x ≠ y
⇒ z ≠ y
⇒ (∃a',c',x',z':Point. ((out(b a'a) ∧ out(b c'c) ∧ out(y x'x) ∧ out(y z'z) ∧ a'bc' ≅a x'yz') ∧ Cong3(a'bc',x'yz'))))
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-sep: a ≠ b,
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,
exists: ∃x:A. B[x],
member: t ∈ T,
basic-geometry: BasicGeometry,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
prop: ℙ,
cand: A c∧ B,
uiff: uiff(P;Q),
geo-cong-tri: Cong3(abc,a'b'c')
Lemmas referenced :
geo-between-out,
geo-sep-sym,
geo-between-sep,
geo-sep_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-point_wf,
geo-out_inversion,
cong-tri-implies-cong-angle2,
geo-congruent-iff-length,
geo-length-flip,
geo-out_wf,
geo-cong-tri_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
hypothesis,
sqequalHypSubstitution,
productElimination,
thin,
introduction,
extract_by_obid,
dependent_functionElimination,
hypothesisEquality,
independent_functionElimination,
because_Cache,
universeIsType,
isectElimination,
applyEquality,
instantiate,
independent_isectElimination,
sqequalRule,
inhabitedIsType,
dependent_pairFormation_alt,
independent_pairFormation,
equalityTransitivity,
equalitySymmetry,
productIsType
Latex:
\mforall{}e:BasicGeometry. \mforall{}a,b,c,x,y,z:Point.
(abc \mcong{}\msuba{} xyz
{}\mRightarrow{} a \mneq{} b
{}\mRightarrow{} c \mneq{} b
{}\mRightarrow{} x \mneq{} y
{}\mRightarrow{} z \mneq{} y
{}\mRightarrow{} (\mexists{}a',c',x',z':Point
((out(b a'a) \mwedge{} out(b c'c) \mwedge{} out(y x'x) \mwedge{} out(y z'z) \mwedge{} a'bc' \mcong{}\msuba{} x'yz') \mwedge{} Cong3(a'bc',x'yz'))))
Date html generated:
2019_10_16-PM-01_26_30
Last ObjectModification:
2018_11_07-PM-00_55_32
Theory : euclidean!plane!geometry
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