Nuprl Lemma : cong-angle-between-exists
∀e:BasicGeometry. ∀a,b,c,x,y,z:Point.
(abc ≅a xyz
⇒ b ≠ a
⇒ b ≠ c
⇒ y ≠ x
⇒ y ≠ z
⇒ (∃a',c',x',z':Point. (b_a_a' ∧ b_c_c' ∧ y_x_x' ∧ y_z_z' ∧ ((ba ≅ xx' ∧ aa' ≅ yx) ∧ bc ≅ zz') ∧ cc' ≅ yz)))
Proof
Definitions occuring in Statement :
geo-cong-angle: abc ≅a xyz,
basic-geometry: BasicGeometry,
geo-congruent: ab ≅ cd,
geo-between: a_b_c,
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,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
prop: ℙ,
cand: A c∧ B,
basic-geometry: BasicGeometry,
euclidean-plane: EuclideanPlane,
basic-geometry-: BasicGeometry-,
uiff: uiff(P;Q)
Lemmas referenced :
geo-proper-extend-exists,
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-between-symmetry,
geo-strict-between-implies-between,
subtype_rel_self,
basic-geometry-_wf,
geo-congruent-iff-length,
geo-between_wf,
geo-congruent_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
sqequalHypSubstitution,
productElimination,
thin,
cut,
introduction,
extract_by_obid,
dependent_functionElimination,
hypothesisEquality,
independent_functionElimination,
hypothesis,
rename,
because_Cache,
universeIsType,
isectElimination,
applyEquality,
instantiate,
independent_isectElimination,
sqequalRule,
inhabitedIsType,
dependent_pairFormation_alt,
independent_pairFormation,
equalitySymmetry,
productIsType
Latex:
\mforall{}e:BasicGeometry. \mforall{}a,b,c,x,y,z:Point.
(abc \mcong{}\msuba{} xyz
{}\mRightarrow{} b \mneq{} a
{}\mRightarrow{} b \mneq{} c
{}\mRightarrow{} y \mneq{} x
{}\mRightarrow{} y \mneq{} z
{}\mRightarrow{} (\mexists{}a',c',x',z':Point
(b\_a\_a' \mwedge{} b\_c\_c' \mwedge{} y\_x\_x' \mwedge{} y\_z\_z' \mwedge{} ((ba \mcong{} xx' \mwedge{} aa' \mcong{} yx) \mwedge{} bc \mcong{} zz') \mwedge{} cc' \mcong{} yz)))
Date html generated:
2019_10_16-PM-01_26_50
Last ObjectModification:
2018_11_07-PM-00_55_26
Theory : euclidean!plane!geometry
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