Nuprl Lemma : geo-perp-in-unicity2
∀e:BasicGeometry. ∀a,b,c:Point.
((¬Colinear(a;b;c))
⇒ (∀d,x:Point. (c ≠ d ⇒ ab ⊥x cd ⇒ (∀y:Point. (Colinear(a;b;y) ⇒ Colinear(c;d;y) ⇒ x ≡ y)))))
Proof
Definitions occuring in Statement :
geo-perp-in: ab ⊥x cd,
basic-geometry: BasicGeometry,
geo-colinear: Colinear(a;b;c),
geo-eq: a ≡ b,
geo-sep: a ≠ b,
geo-point: Point,
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q
Definitions unfolded in proof :
uimplies: b supposing a,
guard: {T},
subtype_rel: A ⊆r B,
basic-geometry: BasicGeometry,
uall: ∀[x:A]. B[x],
prop: ℙ,
member: t ∈ T,
and: P ∧ Q,
geo-perp-in: ab ⊥x cd,
implies: P ⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
not_wf,
geo-sep_wf,
geo-perp-in_wf,
Error :basic-geo-primitives_wf,
Error :basic-geo-structure_wf,
basic-geometry_wf,
subtype_rel_transitivity,
basic-geometry-subtype,
geo-point_wf,
geo-colinear_wf,
geo-intersection-unicity
Rules used in proof :
because_Cache,
sqequalRule,
independent_isectElimination,
instantiate,
applyEquality,
rename,
setElimination,
isectElimination,
hypothesis,
independent_functionElimination,
hypothesisEquality,
dependent_functionElimination,
extract_by_obid,
introduction,
cut,
thin,
productElimination,
sqequalHypSubstitution,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}e:BasicGeometry. \mforall{}a,b,c:Point.
((\mneg{}Colinear(a;b;c))
{}\mRightarrow{} (\mforall{}d,x:Point.
(c \mneq{} d {}\mRightarrow{} ab \mbot{}x cd {}\mRightarrow{} (\mforall{}y:Point. (Colinear(a;b;y) {}\mRightarrow{} Colinear(c;d;y) {}\mRightarrow{} x \mequiv{} y)))))
Date html generated:
2017_10_02-PM-06_43_29
Last ObjectModification:
2017_08_05-PM-04_49_25
Theory : euclidean!plane!geometry
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