Nuprl Lemma : geo-perp-in-not-eq
∀e:BasicGeometry. ∀x:Point. ∀[a,b,c,d:Point]. (ab ⊥x cd ⇒ (¬a ≡ b))
Proof
Definitions occuring in Statement :
geo-perp-in: ab ⊥x cd,
basic-geometry: BasicGeometry,
geo-eq: a ≡ b,
geo-point: Point,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
not: ¬A,
false: False,
geo-perp-in: ab ⊥x cd,
and: P ∧ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
basic-geometry: BasicGeometry,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
euclidean-plane: EuclideanPlane,
cand: A c∧ B,
basic-geometry-: BasicGeometry-,
geo-eq: a ≡ b
Lemmas referenced :
geo-eq_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-perp-in_wf,
geo-point_wf,
right-angle-legs-same,
geo-colinear_wf,
geo-colinear-same,
geo-colinear_functionality,
geo-eq_weakening,
geo-eq_inversion,
geo-sep-O-X,
geo-O_wf,
geo-X_wf,
geo-eq_transitivity,
subtype_rel_self,
basic-geometry-_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
isect_memberFormation,
introduction,
cut,
thin,
sqequalHypSubstitution,
productElimination,
extract_by_obid,
isectElimination,
hypothesisEquality,
applyEquality,
because_Cache,
hypothesis,
sqequalRule,
independent_functionElimination,
voidElimination,
instantiate,
independent_isectElimination,
lambdaEquality,
dependent_functionElimination,
isect_memberEquality,
allFunctionality,
promote_hyp,
setElimination,
rename,
independent_pairFormation
Latex:
\mforall{}e:BasicGeometry. \mforall{}x:Point. \mforall{}[a,b,c,d:Point]. (ab \mbot{}x cd {}\mRightarrow{} (\mneg{}a \mequiv{} b))
Date html generated:
2018_05_22-PM-00_04_45
Last ObjectModification:
2018_04_20-AM-10_20_57
Theory : euclidean!plane!geometry
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