Nuprl Lemma : geo-segments-cross
∀e:HeytingGeometry. ∀p,b,q,a:Point. ((∃c:Point. (a-p-c ∧ b-q-c ∧ c # ab)) ⇒ (∃x:Point. (p-x-b ∧ q-x-a)))
Proof
Definitions occuring in Statement :
geo-triangle: a # bc,
heyting-geometry: HeytingGeometry,
geo-strict-between: a-b-c,
geo-point: Point,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
so_apply: x[s],
so_lambda: λ2x.t[x],
uimplies: b supposing a,
guard: {T},
cand: A c∧ B,
prop: ℙ,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
member: t ∈ T,
and: P ∧ Q,
exists: ∃x:A. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
geo-primitives_wf,
euclidean-plane-structure_wf,
euclidean-plane_wf,
heyting-geometry_wf,
subtype_rel_transitivity,
heyting-geometry-subtype,
euclidean-plane-subtype,
euclidean-plane-structure-subtype,
geo-point_wf,
exists_wf,
geo-strict-between_wf,
geo-strict-between-sym,
geo-triangle_wf,
geo-inner-pasch-ex
Rules used in proof :
lambdaEquality,
independent_isectElimination,
instantiate,
productEquality,
independent_pairFormation,
dependent_pairFormation,
independent_functionElimination,
sqequalRule,
applyEquality,
isectElimination,
hypothesis,
dependent_set_memberEquality,
because_Cache,
hypothesisEquality,
dependent_functionElimination,
extract_by_obid,
introduction,
cut,
thin,
productElimination,
sqequalHypSubstitution,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}e:HeytingGeometry. \mforall{}p,b,q,a:Point.
((\mexists{}c:Point. (a-p-c \mwedge{} b-q-c \mwedge{} c \# ab)) {}\mRightarrow{} (\mexists{}x:Point. (p-x-b \mwedge{} q-x-a)))
Date html generated:
2017_10_02-PM-07_03_02
Last ObjectModification:
2017_08_06-PM-10_18_51
Theory : euclidean!plane!geometry
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