Nuprl Lemma : sep-if-all-lsep
∀g:EuclideanPlane. ∀a,b,c:Point. (a # bc ⇒ (∀x:Point. (Colinear(x;b;c) ⇒ a ≠ x)))
Proof
Definitions occuring in Statement :
euclidean-plane: EuclideanPlane,
geo-lsep: a # bc,
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,
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
prop: ℙ,
oriented-plane: OrientedPlane
Lemmas referenced :
geo-colinear_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-lsep_wf,
geo-point_wf,
lsep-colinear-sep
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
universeIsType,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
applyEquality,
hypothesis,
instantiate,
independent_isectElimination,
sqequalRule,
inhabitedIsType,
because_Cache,
independent_functionElimination,
dependent_functionElimination
Latex:
\mforall{}g:EuclideanPlane. \mforall{}a,b,c:Point. (a \# bc {}\mRightarrow{} (\mforall{}x:Point. (Colinear(x;b;c) {}\mRightarrow{} a \mneq{} x)))
Date html generated:
2019_10_16-PM-01_43_05
Last ObjectModification:
2019_08_12-PM-02_57_28
Theory : euclidean!plane!geometry
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