Nuprl Lemma : geo-between-middle2
∀e:BasicGeometry. ∀a,b,c,d:Point. (a ≠ 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],
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,
and: P ∧ Q,
iff: P ⇐⇒ Q,
guard: {T},
uimplies: b supposing a
Lemmas referenced :
geo-between-middle,
iff_weakening_uiff,
not_wf,
geo-between_wf,
not_over_or,
istype-void,
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
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
hypothesisEquality,
because_Cache,
independent_functionElimination,
hypothesis,
isectElimination,
sqequalRule,
unionEquality,
applyEquality,
productEquality,
productElimination,
voidElimination,
functionIsType,
unionIsType,
universeIsType,
instantiate,
independent_isectElimination,
inhabitedIsType
Latex:
\mforall{}e:BasicGeometry. \mforall{}a,b,c,d:Point. (a \mneq{} d {}\mRightarrow{} a\_b\_d {}\mRightarrow{} a\_c\_d {}\mRightarrow{} (\mneg{}\mneg{}(b\_c\_d \mvee{} c\_b\_d)))
Date html generated:
2019_10_16-PM-01_20_33
Last ObjectModification:
2019_02_04-PM-08_01_38
Theory : euclidean!plane!geometry
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