Nuprl Lemma : geo-between-out-implies-out2
∀e:EuclideanPlane. ∀a,b,c,c':Point. (out(a cc') ⇒ ((a_b_c ∧ a ≠ b) ∨ (a_c_b ∧ a ≠ c)) ⇒ out(a c'b))
Proof
Definitions occuring in Statement :
geo-out: out(p ab),
euclidean-plane: EuclideanPlane,
geo-between: a_b_c,
geo-sep: a ≠ b,
geo-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q,
or: P ∨ Q,
and: P ∧ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
or: P ∨ Q,
and: P ∧ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
prop: ℙ,
basic-geometry: BasicGeometry,
geo-out: out(p ab)
Lemmas referenced :
geo-between_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-sep_wf,
geo-out_wf,
geo-point_wf,
geo-between-out,
geo-out_transitivity,
geo-out_inversion,
geo-between-sep
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
sqequalHypSubstitution,
unionElimination,
thin,
sqequalRule,
unionIsType,
productIsType,
universeIsType,
cut,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
applyEquality,
hypothesis,
instantiate,
independent_isectElimination,
because_Cache,
inhabitedIsType,
dependent_functionElimination,
productElimination,
independent_functionElimination
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a,b,c,c':Point.
(out(a cc') {}\mRightarrow{} ((a\_b\_c \mwedge{} a \mneq{} b) \mvee{} (a\_c\_b \mwedge{} a \mneq{} c)) {}\mRightarrow{} out(a c'b))
Date html generated:
2019_10_16-PM-01_24_00
Last ObjectModification:
2018_11_08-AM-11_44_15
Theory : euclidean!plane!geometry
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