Nuprl Lemma : geo-col-between-cases
∀e:EuclideanPlane. ∀a,b,c,x:Point. (a-b-c ⇒ Colinear(a;x;c) ⇒ (¬¬(x_a_b ∨ a_x_b ∨ b_x_c ∨ b_c_x)))
Proof
Definitions occuring in Statement :
euclidean-plane: EuclideanPlane,
geo-colinear: Colinear(a;b;c),
geo-strict-between: a-b-c,
geo-between: a_b_c,
geo-point: Point,
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
or: P ∨ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
not: ¬A,
false: False,
member: t ∈ T,
uall: ∀[x:A]. B[x],
prop: ℙ,
or: P ∨ Q,
subtype_rel: A ⊆r B,
geo-eq: a ≡ b,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
basic-geometry: BasicGeometry,
stable: Stable{P},
basic-geometry-: BasicGeometry-
Lemmas referenced :
geo-colinear-cases,
not_wf,
geo-between_wf,
stable__not,
geo-eq_wf,
geo-sep-sym,
geo-strict-between-sep1,
istype-void,
geo-strict-between_wf,
geo-strict-between-implies-between,
geo-between-symmetry,
geo-between-inner-trans,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-colinear_wf,
geo-point_wf,
geo-between-trivial2,
geo-between_functionality,
geo-eq_weakening,
geo-between-trivial,
false_wf,
geo-sep_wf,
minimal-double-negation-hyp-elim,
geo-strict-between_functionality,
geo-colinear_functionality,
minimal-not-not-excluded-middle,
geo-between-middle-or,
geo-strict-between-sep2,
geo-strict-between-sep3,
geo-between-exchange3
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
because_Cache,
isectElimination,
hypothesisEquality,
sqequalRule,
unionEquality,
applyEquality,
hypothesis,
independent_functionElimination,
universeIsType,
functionIsType,
unionIsType,
inlFormation_alt,
independent_isectElimination,
voidElimination,
instantiate,
inhabitedIsType,
productElimination,
inrFormation_alt,
unionElimination,
functionEquality
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a,b,c,x:Point.
(a-b-c {}\mRightarrow{} Colinear(a;x;c) {}\mRightarrow{} (\mneg{}\mneg{}(x\_a\_b \mvee{} a\_x\_b \mvee{} b\_x\_c \mvee{} b\_c\_x)))
Date html generated:
2019_10_16-PM-01_24_59
Last ObjectModification:
2019_07_21-PM-11_23_46
Theory : euclidean!plane!geometry
Home
Index