Nuprl Lemma : geo-out-iff-exists
∀e:BasicGeometry. ∀p,a,b:Point. (out(p ab) ⇐⇒ (∃c:Point. (p ≠ c ∧ a_p_c ∧ b_p_c)) ∧ p ≠ a ∧ p ≠ b)
Proof
Definitions occuring in Statement :
geo-out: out(p ab),
basic-geometry: BasicGeometry,
geo-between: a_b_c,
geo-sep: a ≠ b,
geo-point: Point,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
exists: ∃x:A. B[x],
so_apply: x[s],
so_lambda: λ2x.t[x],
uimplies: b supposing a,
guard: {T},
subtype_rel: A ⊆r B,
rev_implies: P ⇐ Q,
uall: ∀[x:A]. B[x],
prop: ℙ,
member: t ∈ T,
geo-out: out(p ab),
implies: P ⇒ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
all: ∀x:A. B[x],
not: ¬A,
stable: Stable{P},
basic-geometry: BasicGeometry,
false: False,
cand: A c∧ B
Lemmas referenced :
geo-between_wf,
geo-sep_wf,
Error :basic-geo-primitives_wf,
Error :basic-geo-structure_wf,
basic-geometry_wf,
subtype_rel_transitivity,
basic-geometry-subtype,
geo-point_wf,
exists_wf,
geo-out_wf,
geo-extend-exists,
geo-sep-sym,
geo-congruent-sep,
not_wf,
stable__geo-between,
geo-between-outer-trans,
geo-between-symmetry,
geo-between-exchange3,
geo-between-inner-trans,
geo-between-same-side2
Rules used in proof :
because_Cache,
lambdaEquality,
sqequalRule,
independent_isectElimination,
instantiate,
applyEquality,
productEquality,
hypothesisEquality,
isectElimination,
extract_by_obid,
introduction,
cut,
hypothesis,
thin,
productElimination,
sqequalHypSubstitution,
independent_pairFormation,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
rename,
independent_functionElimination,
dependent_functionElimination,
dependent_pairFormation,
setElimination,
voidElimination
Latex:
\mforall{}e:BasicGeometry. \mforall{}p,a,b:Point. (out(p ab) \mLeftarrow{}{}\mRightarrow{} (\mexists{}c:Point. (p \mneq{} c \mwedge{} a\_p\_c \mwedge{} b\_p\_c)) \mwedge{} p \mneq{} a \mwedge{} p \mneq{} b)
Date html generated:
2017_10_02-PM-06_27_08
Last ObjectModification:
2017_08_05-PM-04_20_36
Theory : euclidean!plane!geometry
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