Nuprl Lemma : geo-between-implies-out2
∀e:BasicGeometry. ∀p,a,b,c:Point. (p ≠ c ⇒ p_c_a ⇒ p_c_b ⇒ out(p ab))
Proof
Definitions occuring in Statement :
geo-out: out(p ab),
basic-geometry: BasicGeometry,
geo-between: a_b_c,
geo-sep: a ≠ b,
geo-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
geo-out: out(p ab),
and: P ∧ Q,
basic-geometry: BasicGeometry,
cand: A c∧ B,
prop: ℙ,
subtype_rel: A ⊆r B,
guard: {T}
Lemmas referenced :
geo-between-same-side,
geo-between-sep,
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
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
isectElimination,
independent_isectElimination,
hypothesis,
independent_pairFormation,
because_Cache,
independent_functionElimination,
applyEquality,
instantiate,
sqequalRule
Latex:
\mforall{}e:BasicGeometry. \mforall{}p,a,b,c:Point. (p \mneq{} c {}\mRightarrow{} p\_c\_a {}\mRightarrow{} p\_c\_b {}\mRightarrow{} out(p ab))
Date html generated:
2018_05_22-AM-11_58_28
Last ObjectModification:
2018_04_18-PM-05_08_47
Theory : euclidean!plane!geometry
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