Nuprl Lemma : lsep-all-sym
∀g:EuclideanPlane. ∀a,b,c:Point. (a # bc ⇒ {b # ca ∧ c # ab ∧ a # cb ∧ b # ac ∧ c # ba})
Proof
Definitions occuring in Statement :
euclidean-plane: EuclideanPlane,
geo-lsep: a # bc,
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,
member: t ∈ T,
cand: A c∧ B,
and: P ∧ Q,
implies: P ⇒ Q,
all: ∀x:A. B[x],
guard: {T}
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,
lsep-symmetry2,
lsep-symmetry
Rules used in proof :
independent_isectElimination,
instantiate,
applyEquality,
isectElimination,
because_Cache,
independent_pairFormation,
productElimination,
hypothesis,
independent_functionElimination,
hypothesisEquality,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation,
computationStep,
sqequalTransitivity,
sqequalReflexivity,
sqequalRule,
sqequalSubstitution
Latex:
\mforall{}g:EuclideanPlane. \mforall{}a,b,c:Point. (a \# bc {}\mRightarrow{} \{b \# ca \mwedge{} c \# ab \mwedge{} a \# cb \mwedge{} b \# ac \mwedge{} c \# ba\})
Date html generated:
2017_10_02-PM-03_29_37
Last ObjectModification:
2017_08_07-AM-10_52_11
Theory : euclidean!plane!geometry
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