Nuprl Lemma : geo-intersect-iff2
∀e:EuclideanPlane. ∀p,l:LINE.
(p \/ l
⇐⇒ ∃a,b,c,d,v:Point
(a-v-b ∧ c-v-d ∧ a I p ∧ b I p ∧ c I l ∧ d I l ∧ a leftof cd ∧ b leftof dc ∧ c leftof ba ∧ d leftof ab))
Proof
Definitions occuring in Statement :
geo-intersect: L \/ M,
geo-incident: p I L,
geoline: LINE,
euclidean-plane: EuclideanPlane,
geo-strict-between: a-b-c,
geo-left: a leftof bc,
geo-point: Point,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
exists: ∃x:A. B[x],
member: t ∈ T,
cand: A c∧ B,
prop: ℙ,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
guard: {T},
uimplies: b supposing a,
rev_implies: P ⇐ Q,
geo-strict-between: a-b-c,
basic-geometry-: BasicGeometry-,
or: P ∨ Q
Lemmas referenced :
geo-strict-between_wf,
geo-incident_wf,
geo-left_wf,
exists_wf,
geo-point_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-intersect-iff,
geo-intersect_wf,
iff_wf,
geoline_wf,
left-between-triangle2,
all_wf,
left-all-symmetry,
geo-strict-between-sym,
left-convex,
or_wf,
geo-between_wf,
geo-sep_wf,
geo-between-symmetry
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
independent_pairFormation,
sqequalHypSubstitution,
productElimination,
thin,
dependent_pairFormation,
hypothesisEquality,
hypothesis,
productEquality,
introduction,
extract_by_obid,
isectElimination,
applyEquality,
because_Cache,
sqequalRule,
lambdaEquality,
instantiate,
independent_isectElimination,
addLevel,
impliesFunctionality,
dependent_functionElimination,
independent_functionElimination,
functionEquality,
inlFormation
Latex:
\mforall{}e:EuclideanPlane. \mforall{}p,l:LINE.
(p \mbackslash{}/ l
\mLeftarrow{}{}\mRightarrow{} \mexists{}a,b,c,d,v:Point
(a-v-b
\mwedge{} c-v-d
\mwedge{} a I p
\mwedge{} b I p
\mwedge{} c I l
\mwedge{} d I l
\mwedge{} a leftof cd
\mwedge{} b leftof dc
\mwedge{} c leftof ba
\mwedge{} d leftof ab))
Date html generated:
2018_05_22-PM-01_05_44
Last ObjectModification:
2018_05_10-PM-06_01_28
Theory : euclidean!plane!geometry
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