Nuprl Lemma : geo-inner-pasch-ex2
∀e:HeytingGeometry. ∀a,b:Point. ∀c:{c:Point| c # ab} . ∀p,q:Point. (a-p-c ⇒ b-q-c ⇒ (∃x:Point. (b_x_p ∧ a_x_q)))
Proof
Definitions occuring in Statement :
geo-triangle: a # bc,
heyting-geometry: HeytingGeometry,
geo-strict-between: a-b-c,
geo-between: a_b_c,
geo-point: Point,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
set: {x:A| B[x]}
Definitions unfolded in proof :
so_apply: x[s],
so_lambda: λ2x.t[x],
guard: {T},
uimplies: b supposing a,
prop: ℙ,
uall: ∀[x:A]. B[x],
basic-geometry-: BasicGeometry-,
euclidean-plane: EuclideanPlane,
heyting-geometry: HeytingGeometry,
subtype_rel: A ⊆r B,
cand: A c∧ B,
and: P ∧ Q,
exists: ∃x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
all: ∀x:A. B[x]
Lemmas referenced :
geo-triangle_wf,
set_wf,
geo-primitives_wf,
euclidean-plane_wf,
heyting-geometry_wf,
subtype_rel_transitivity,
heyting-geometry-subtype,
euclidean-plane-subtype,
geo-point_wf,
geo-strict-between_wf,
geo-between_wf,
geo-between-symmetry,
geo-left-axioms_wf,
euclidean-plane-structure-subtype,
basic-geo-axioms_wf,
euclidean-plane-structure_wf,
subtype_rel_self,
geo-strict-between-implies-between,
geo-inner-pasch-ex
Rules used in proof :
lambdaEquality,
rename,
setElimination,
independent_pairFormation,
independent_isectElimination,
because_Cache,
cumulativity,
productEquality,
setEquality,
isectElimination,
instantiate,
sqequalRule,
applyEquality,
dependent_pairFormation,
productElimination,
independent_functionElimination,
hypothesisEquality,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
hypothesis,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
extract_by_obid,
introduction,
cut
Latex:
\mforall{}e:HeytingGeometry. \mforall{}a,b:Point. \mforall{}c:\{c:Point| c \# ab\} . \mforall{}p,q:Point.
(a-p-c {}\mRightarrow{} b-q-c {}\mRightarrow{} (\mexists{}x:Point. (b\_x\_p \mwedge{} a\_x\_q)))
Date html generated:
2017_10_02-PM-07_02_54
Last ObjectModification:
2017_08_06-PM-10_09_42
Theory : euclidean!plane!geometry
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