Nuprl Lemma : cong-angle-out-exists
∀g:EuclideanPlane. ∀a,b,c,x,y,z,a',c',p:Point.
(abc ≅a xyz
⇒ a ≠ b
⇒ c ≠ b
⇒ p ≠ b
⇒ x ≠ y
⇒ z ≠ y
⇒ ((out(b aa') ∧ out(b cc')) ∧ a'-p-c')
⇒ (∃x',z',p':Point
(((x'-p'-z' ∧ Cong3(a'bc',x'yz')) ∧ out(y xx') ∧ out(y zz'))
∧ (x'p' ≅ a'p ∧ p'z' ≅ pc')
∧ bp ≅ yp'
∧ x'yp' ≅a a'bp)))
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-strict-between: a-b-c,
geo-congruent: ab ≅ cd,
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,
member: t ∈ T,
and: P ∧ Q,
basic-geometry: BasicGeometry,
euclidean-plane: EuclideanPlane,
exists: ∃x:A. B[x],
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
guard: {T},
uimplies: b supposing a,
geo-out: out(p ab),
prop: ℙ,
cand: A c∧ B,
uiff: uiff(P;Q),
basic-geometry-: BasicGeometry-,
iff: P ⇐⇒ Q,
geo-strict-between: a-b-c,
geo-cong-tri: Cong3(abc,a'b'c'),
geo-cong-angle: abc ≅a xyz
Lemmas referenced :
geo-proper-extend-exists,
geo-O_wf,
geo-X_wf,
geo-sep-O-X,
geo-sep-sym,
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-out_wf,
geo-strict-between_wf,
geo-sep_wf,
geo-cong-angle_wf,
geo-point_wf,
geo-sas2,
geo-congruent-iff-length,
out-preserves-angle-cong_1,
geo-out-iff-between1,
geo-strict-between-implies-between,
geo-between-symmetry,
geo-congruent-between-exists,
geo-strict-between-sep2,
geo-cong-tri_wf,
geo-congruent_wf,
geo-congruent-symmetry,
geo-congruent-sep,
geo-length-flip,
geo-inner-five-segment,
geo-between-trivial,
geo-between_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
productElimination,
sqequalRule,
hypothesisEquality,
setElimination,
rename,
hypothesis,
because_Cache,
independent_functionElimination,
applyEquality,
instantiate,
isectElimination,
independent_isectElimination,
productIsType,
universeIsType,
inhabitedIsType,
equalitySymmetry,
independent_pairFormation,
dependent_pairFormation_alt,
equalityTransitivity
Latex:
\mforall{}g:EuclideanPlane. \mforall{}a,b,c,x,y,z,a',c',p:Point.
(abc \mcong{}\msuba{} xyz
{}\mRightarrow{} a \mneq{} b
{}\mRightarrow{} c \mneq{} b
{}\mRightarrow{} p \mneq{} b
{}\mRightarrow{} x \mneq{} y
{}\mRightarrow{} z \mneq{} y
{}\mRightarrow{} ((out(b aa') \mwedge{} out(b cc')) \mwedge{} a'-p-c')
{}\mRightarrow{} (\mexists{}x',z',p':Point
(((x'-p'-z' \mwedge{} Cong3(a'bc',x'yz')) \mwedge{} out(y xx') \mwedge{} out(y zz'))
\mwedge{} (x'p' \mcong{} a'p \mwedge{} p'z' \mcong{} pc')
\mwedge{} bp \mcong{} yp'
\mwedge{} x'yp' \mcong{}\msuba{} a'bp)))
Date html generated:
2019_10_16-PM-01_31_14
Last ObjectModification:
2018_12_13-PM-04_02_36
Theory : euclidean!plane!geometry
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