Nuprl Lemma : geo-tar-same-side-invariant
∀e:BasicGeometry. ∀A,B,P,Q,C,D:Point.
(C ≠ D ⇒ Colinear(C;P;Q) ⇒ Colinear(D;P;Q) ⇒ geo-tar-same-side(e;A;B;P;Q) ⇒ geo-tar-same-side(e;A;B;C;D))
Proof
Definitions occuring in Statement :
geo-tar-same-side: geo-tar-same-side(e;a;b;p;q),
basic-geometry: BasicGeometry,
geo-colinear: Colinear(a;b;c),
geo-sep: a ≠ b,
geo-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
geo-tar-same-side: geo-tar-same-side(e;a;b;p;q),
exists: ∃x:A. B[x],
member: t ∈ T,
and: P ∧ Q,
cand: A c∧ B,
uall: ∀[x:A]. B[x],
prop: ℙ,
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a
Lemmas referenced :
geo-tar-opp-side-invariant,
geo-tar-opp-side_wf,
geo-tar-same-side_wf,
geo-colinear_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
basic-geometry-subtype,
subtype_rel_transitivity,
basic-geometry_wf,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-sep_wf,
geo-point_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
sqequalHypSubstitution,
productElimination,
thin,
dependent_pairFormation_alt,
hypothesisEquality,
independent_pairFormation,
cut,
hypothesis,
introduction,
extract_by_obid,
dependent_functionElimination,
independent_functionElimination,
because_Cache,
sqequalRule,
functionIsType,
universeIsType,
isectElimination,
productIsType,
applyEquality,
instantiate,
independent_isectElimination,
inhabitedIsType
Latex:
\mforall{}e:BasicGeometry. \mforall{}A,B,P,Q,C,D:Point.
(C \mneq{} D
{}\mRightarrow{} Colinear(C;P;Q)
{}\mRightarrow{} Colinear(D;P;Q)
{}\mRightarrow{} geo-tar-same-side(e;A;B;P;Q)
{}\mRightarrow{} geo-tar-same-side(e;A;B;C;D))
Date html generated:
2019_10_16-PM-01_21_45
Last ObjectModification:
2018_12_11-PM-00_22_19
Theory : euclidean!plane!geometry
Home
Index