Nuprl Lemma : lsep-implies-sep
∀g:EuclideanPlane. ∀a,b,c:Point. (a # bc ⇒ {a ≠ b ∧ a ≠ c ∧ b ≠ c})
Proof
Definitions occuring in Statement :
euclidean-plane: EuclideanPlane,
geo-lsep: a # bc,
geo-sep: a ≠ b,
geo-point: Point,
guard: {T},
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
uimplies: b supposing a,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
prop: ℙ,
cand: A c∧ B,
and: P ∧ Q,
guard: {T},
member: t ∈ T,
or: P ∨ Q,
geo-lsep: a # bc,
implies: P ⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
geo-point_wf,
geo-primitives_wf,
euclidean-plane-structure_wf,
euclidean-plane_wf,
subtype_rel_transitivity,
euclidean-plane-subtype,
euclidean-plane-structure-subtype,
geo-lsep_wf,
geo-sep-sym,
left-implies-sep
Rules used in proof :
sqequalRule,
independent_isectElimination,
instantiate,
applyEquality,
isectElimination,
because_Cache,
productElimination,
independent_pairFormation,
hypothesis,
independent_functionElimination,
hypothesisEquality,
dependent_functionElimination,
extract_by_obid,
introduction,
cut,
thin,
unionElimination,
sqequalHypSubstitution,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}g:EuclideanPlane. \mforall{}a,b,c:Point. (a \# bc {}\mRightarrow{} \{a \mneq{} b \mwedge{} a \mneq{} c \mwedge{} b \mneq{} c\})
Date html generated:
2017_10_02-PM-03_29_28
Last ObjectModification:
2017_08_07-AM-10_50_16
Theory : euclidean!plane!geometry
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