Nuprl Lemma : 2opp-side-implies-same-side
∀e:BasicGeometry. ∀a,c,d,p,q:Point.
(((geo-tar-opp-side(e;a;d;p;q) ∧ geo-tar-opp-side(e;c;d;p;q)) ∧ a ≠ c) ⇒ geo-tar-same-side(e;a;c;p;q))
Proof
Definitions occuring in Statement :
geo-tar-same-side: geo-tar-same-side(e;a;b;p;q),
geo-tar-opp-side: geo-tar-opp-side(e;a;b;p;q),
basic-geometry: BasicGeometry,
geo-sep: a ≠ b,
geo-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
uimplies: b supposing a,
guard: {T},
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
prop: ℙ,
member: t ∈ T,
and: P ∧ Q,
implies: P ⇒ Q,
all: ∀x:A. B[x],
cand: A c∧ B,
exists: ∃x:A. B[x],
geo-tar-same-side: geo-tar-same-side(e;a;b;p;q)
Lemmas referenced :
geo-point_wf,
Error :basic-geo-primitives_wf,
Error :basic-geo-structure_wf,
basic-geometry_wf,
subtype_rel_transitivity,
basic-geometry-subtype,
geo-sep_wf,
geo-tar-opp-side_wf
Rules used in proof :
because_Cache,
sqequalRule,
independent_isectElimination,
instantiate,
applyEquality,
hypothesis,
hypothesisEquality,
isectElimination,
extract_by_obid,
introduction,
cut,
productEquality,
thin,
productElimination,
sqequalHypSubstitution,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
independent_pairFormation,
dependent_pairFormation
Latex:
\mforall{}e:BasicGeometry. \mforall{}a,c,d,p,q:Point.
(((geo-tar-opp-side(e;a;d;p;q) \mwedge{} geo-tar-opp-side(e;c;d;p;q)) \mwedge{} a \mneq{} c)
{}\mRightarrow{} geo-tar-same-side(e;a;c;p;q))
Date html generated:
2017_10_02-PM-06_23_31
Last ObjectModification:
2017_08_05-PM-04_17_40
Theory : euclidean!plane!geometry
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