Nuprl Lemma : geo-perp-in-symmetry
∀e:BasicGeometry. ∀x:Point.
∀[a,b,c,d:Point]. (ab ⊥x cd ⇒ {ba ⊥x cd ∧ ab ⊥x dc ∧ ba ⊥x dc ∧ cd ⊥x ab ∧ dc ⊥x ab ∧ cd ⊥x ba ∧ dc ⊥x ba})
Proof
Definitions occuring in Statement :
geo-perp-in: ab ⊥x 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,
member: t ∈ T,
guard: {T},
and: P ∧ Q,
cand: A c∧ B,
prop: ℙ,
subtype_rel: A ⊆r B,
uimplies: b supposing a
Lemmas referenced :
geo-perp-in-symmetry1,
geo-perp-in-symmetry2,
geo-perp-in_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,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
because_Cache,
independent_pairFormation,
hypothesis,
isectElimination,
applyEquality,
instantiate,
independent_isectElimination,
sqequalRule,
independent_functionElimination
Latex:
\mforall{}e:BasicGeometry. \mforall{}x:Point.
\mforall{}[a,b,c,d:Point].
(ab \mbot{}x cd
{}\mRightarrow{} \{ba \mbot{}x cd \mwedge{} ab \mbot{}x dc \mwedge{} ba \mbot{}x dc \mwedge{} cd \mbot{}x ab \mwedge{} dc \mbot{}x ab \mwedge{} cd \mbot{}x ba \mwedge{} dc \mbot{}x ba\})
Date html generated:
2018_05_22-PM-00_04_32
Last ObjectModification:
2018_04_18-PM-09_59_25
Theory : euclidean!plane!geometry
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