Nuprl Lemma : geo-between-middle-or
∀e:BasicGeometry. ∀a,b,c,d:Point. (a ≠ b ⇒ b ≠ c ⇒ b ≠ d ⇒ a_b_d ⇒ a_c_d ⇒ (b_c_d ∨ c_b_d))
Proof
Definitions occuring in Statement :
basic-geometry: BasicGeometry,
geo-between: a_b_c,
geo-sep: a ≠ b,
geo-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q,
or: P ∨ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
prop: ℙ,
and: P ∧ Q,
not: ¬A,
false: False,
or: P ∨ Q,
iff: P ⇐⇒ Q,
basic-geometry: BasicGeometry,
euclidean-plane: EuclideanPlane,
basic-geometry-: BasicGeometry-,
exists: ∃x:A. B[x],
stable: Stable{P},
geo-eq: a ≡ b,
cand: A c∧ B
Lemmas referenced :
geo-between_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-sep_wf,
geo-point_wf,
geo-sep-sym,
geo-strict-between-sep3,
iff_weakening_uiff,
not_wf,
not_over_or,
istype-void,
geo-between-middle,
geo-between-sep,
geo-sep-or,
geo-strict-between-sep1,
subtype_rel_self,
basic-geometry-_wf,
geo-proper-extend-exists,
stable__geo-between,
double-negation-hyp-elim,
geo-construction-unicity,
geo-between-symmetry,
geo-strict-between-implies-between,
geo-between-inner-trans,
extended-out-preserves-between,
geo-out-iff-between1,
euclidean-plane-axioms
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
universeIsType,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
applyEquality,
hypothesis,
instantiate,
independent_isectElimination,
sqequalRule,
because_Cache,
inhabitedIsType,
productElimination,
dependent_functionElimination,
independent_functionElimination,
unionEquality,
productEquality,
voidElimination,
functionIsType,
unionIsType,
productIsType,
setElimination,
rename,
dependent_set_memberEquality_alt,
unionElimination,
inrFormation_alt,
inlFormation_alt,
independent_pairFormation
Latex:
\mforall{}e:BasicGeometry. \mforall{}a,b,c,d:Point. (a \mneq{} b {}\mRightarrow{} b \mneq{} c {}\mRightarrow{} b \mneq{} d {}\mRightarrow{} a\_b\_d {}\mRightarrow{} a\_c\_d {}\mRightarrow{} (b\_c\_d \mvee{} c\_b\_d))
Date html generated:
2019_10_16-PM-01_24_49
Last ObjectModification:
2019_01_13-PM-00_56_38
Theory : euclidean!plane!geometry
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