Nuprl Lemma : geo-perp-all-symmetry
∀e:BasicGeometry. ∀[a,b,c,d:Point]. (ab ⊥ cd ⇒ {ba ⊥ cd ∧ ab ⊥ dc ∧ ba ⊥ dc ∧ cd ⊥ ab ∧ dc ⊥ ab ∧ cd ⊥ ba ∧ dc ⊥ ba})
Proof
Definitions occuring in Statement :
geo-perp: ab ⊥ cd,
basic-geometry: BasicGeometry,
geo-point: Point,
uall: ∀[x:A]. B[x],
guard: {T},
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
guard: {T},
and: P ∧ Q,
geo-perp: ab ⊥ cd,
exists: ∃x:A. B[x],
member: t ∈ T,
prop: ℙ,
subtype_rel: A ⊆r B,
uimplies: b supposing a
Lemmas referenced :
geo-perp-in-symmetry,
geo-perp-in_wf,
geo-perp_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
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
isect_memberFormation,
independent_pairFormation,
sqequalHypSubstitution,
productElimination,
thin,
dependent_pairFormation,
hypothesisEquality,
cut,
introduction,
extract_by_obid,
dependent_functionElimination,
because_Cache,
isectElimination,
independent_functionElimination,
hypothesis,
applyEquality,
instantiate,
independent_isectElimination,
sqequalRule
Latex:
\mforall{}e:BasicGeometry
\mforall{}[a,b,c,d:Point].
(ab \mbot{} cd {}\mRightarrow{} \{ba \mbot{} cd \mwedge{} ab \mbot{} dc \mwedge{} ba \mbot{} dc \mwedge{} cd \mbot{} ab \mwedge{} dc \mbot{} ab \mwedge{} cd \mbot{} ba \mwedge{} dc \mbot{} ba\})
Date html generated:
2018_05_22-PM-00_04_52
Last ObjectModification:
2018_04_19-AM-00_20_53
Theory : euclidean!plane!geometry
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