Nuprl Lemma : geo-intersect-iff3
∀e:EuclideanPlane. ∀p,l:LINE.
(p \/ l
⇐⇒ ∃a,b,c,d,v:Point
∃sab:a ≠ b
∃scd:c ≠ d
((p = <a, b, sab> ∈ LINE)
∧ (l = <c, d, scd> ∈ LINE)
∧ a-v-b
∧ c-v-d
∧ a leftof cd
∧ b leftof dc
∧ c leftof ba
∧ d leftof ab))
Proof
Definitions occuring in Statement :
geo-intersect: L \/ M,
geoline: LINE,
euclidean-plane: EuclideanPlane,
geo-strict-between: a-b-c,
geo-left: a leftof bc,
geo-sep: a ≠ b,
geo-point: Point,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
pair: <a, b>,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
exists: ∃x:A. B[x],
cand: A c∧ B,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
prop: ℙ,
geo-line: Line,
respects-equality: respects-equality(S;T),
rev_implies: P ⇐ Q,
basic-geometry-: BasicGeometry-,
uiff: uiff(P;Q),
pi1: fst(t),
pi2: snd(t),
squash: ↓T,
true: True,
rev_uimplies: rev_uimplies(P;Q),
basic-geometry: BasicGeometry
Lemmas referenced :
geo-intersect-iff2,
geo-sep_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
subtype-respects-equality,
geoline_wf,
geo-point_wf,
geoline-subtype1,
geo-strict-between_wf,
geo-left_wf,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-intersect_wf,
geo-incident_wf,
geo-strict-between-sep1,
geo-line-eq-geoline,
squash_wf,
true_wf,
subtype_rel_self,
iff_weakening_equal,
geo-incident-line,
geo-colinear-same
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
productElimination,
independent_pairFormation,
independent_functionElimination,
dependent_pairFormation_alt,
sqequalRule,
productIsType,
universeIsType,
isectElimination,
applyEquality,
instantiate,
because_Cache,
independent_isectElimination,
equalityIstype,
inhabitedIsType,
dependent_pairEquality_alt,
productEquality,
rename,
equalitySymmetry,
lambdaEquality_alt,
imageElimination,
equalityTransitivity,
natural_numberEquality,
imageMemberEquality,
baseClosed,
universeEquality
Latex:
\mforall{}e:EuclideanPlane. \mforall{}p,l:LINE.
(p \mbackslash{}/ l
\mLeftarrow{}{}\mRightarrow{} \mexists{}a,b,c,d,v:Point
\mexists{}sab:a \mneq{} b
\mexists{}scd:c \mneq{} d
((p = <a, b, sab>)
\mwedge{} (l = <c, d, scd>)
\mwedge{} a-v-b
\mwedge{} c-v-d
\mwedge{} a leftof cd
\mwedge{} b leftof dc
\mwedge{} c leftof ba
\mwedge{} d leftof ab))
Date html generated:
2019_10_16-PM-02_40_56
Last ObjectModification:
2018_12_12-PM-04_41_44
Theory : euclidean!plane!geometry
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