Nuprl Lemma : geo-out-iff-colinear
∀e:BasicGeometry. ∀p,a,b:Point. (out(p ab) ⇐⇒ (Colinear(p;a;b) ∧ (¬a_p_b)) ∧ p ≠ a ∧ p ≠ b)
Proof
Definitions occuring in Statement :
geo-out: out(p ab),
basic-geometry: BasicGeometry,
geo-colinear: Colinear(a;b;c),
geo-between: a_b_c,
geo-sep: a ≠ b,
geo-point: Point,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
not: ¬A,
and: P ∧ Q
Definitions unfolded in proof :
cand: A c∧ B,
basic-geometry: BasicGeometry,
rev_implies: P ⇐ Q,
geo-out: out(p ab),
uimplies: b supposing a,
guard: {T},
subtype_rel: A ⊆r B,
subtract: n - m,
cons: [a / b],
select: L[n],
uall: ∀[x:A]. B[x],
true: True,
squash: ↓T,
less_than: a < b,
prop: ℙ,
not: ¬A,
false: False,
less_than': less_than'(a;b),
le: A ≤ B,
lelt: i ≤ j < k,
int_seg: {i..j-},
top: Top,
l_all: (∀x∈L.P[x]),
geo-colinear-set: geo-colinear-set(e; L),
member: t ∈ T,
implies: P ⇒ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
geo-point_wf,
geo-sep_wf,
not_wf,
geo-colinear_wf,
geo-out_wf,
Error :basic-geo-primitives_wf,
Error :basic-geo-structure_wf,
basic-geometry_wf,
subtype_rel_transitivity,
basic-geometry-subtype,
geo-between_wf,
lelt_wf,
false_wf,
length_of_nil_lemma,
length_of_cons_lemma,
geo-out-colinear,
geo-colinear-is-colinear-set,
geo-sep-irrefl',
geo-sep_functionality,
geo-eq_weakening,
geo-between_functionality,
geo-between-same,
geo-between-inner-trans,
geo-between-symmetry,
geo-sep-sym,
geo-between-outer-trans,
geo-between-exchange4,
geo-colinear-implies
Rules used in proof :
rename,
setElimination,
productEquality,
productElimination,
independent_isectElimination,
instantiate,
applyEquality,
isectElimination,
baseClosed,
imageMemberEquality,
natural_numberEquality,
dependent_set_memberEquality,
voidEquality,
voidElimination,
isect_memberEquality,
sqequalRule,
hypothesis,
because_Cache,
independent_functionElimination,
hypothesisEquality,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
independent_pairFormation,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
promote_hyp
Latex:
\mforall{}e:BasicGeometry. \mforall{}p,a,b:Point. (out(p ab) \mLeftarrow{}{}\mRightarrow{} (Colinear(p;a;b) \mwedge{} (\mneg{}a\_p\_b)) \mwedge{} p \mneq{} a \mwedge{} p \mneq{} b)
Date html generated:
2017_10_02-PM-06_27_00
Last ObjectModification:
2017_08_05-PM-04_20_31
Theory : euclidean!plane!geometry
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