Nuprl Lemma : geo-triangle-separation
∀e:HeytingGeometry. ∀a,b,c:Point.
(a # bc
⇒ {∀x,y:Point. (((Colinear(a;b;x) ∧ Colinear(c;b;y)) ∧ x ≠ b ∧ y ≠ b) ⇒ (x # by ∧ (∀m:Point. (x-m-y ⇒ m ≠ b))))})
Proof
Definitions occuring in Statement :
geo-triangle: a # bc,
heyting-geometry: HeytingGeometry,
geo-colinear: Colinear(a;b;c),
geo-strict-between: a-b-c,
geo-sep: a ≠ b,
geo-point: Point,
guard: {T},
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
heyting-geometry: Error :heyting-geometry,
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,
implies: P ⇒ Q,
all: ∀x:A. B[x],
subtract: n - m,
cons: [a / b],
select: L[n],
true: True,
squash: ↓T,
less_than: a < b,
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)
Lemmas referenced :
geo-point_wf,
Error :geo-triangle_wf,
Error :basic-geo-primitives_wf,
geo-sep_wf,
Error :basic-geo-structure_wf,
basic-geometry_wf,
Error :heyting-geometry_wf,
subtype_rel_transitivity,
heyting-geometry-subtype,
basic-geometry-subtype,
geo-colinear_wf,
geo-triangle-implies,
lelt_wf,
false_wf,
length_of_nil_lemma,
length_of_cons_lemma,
geo-colinear-is-colinear-set,
geo-triangle-symmetry,
geo-triangle-colinear,
geo-strict-between_wf,
geo-triangle-property,
geo-sep-sym,
geo-strict-between-implies-colinear,
geo-strict-between-sep3
Rules used in proof :
rename,
setElimination,
because_Cache,
sqequalRule,
independent_isectElimination,
instantiate,
applyEquality,
isectElimination,
productEquality,
productElimination,
hypothesis,
independent_functionElimination,
hypothesisEquality,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
baseClosed,
imageMemberEquality,
independent_pairFormation,
natural_numberEquality,
dependent_set_memberEquality,
voidEquality,
voidElimination,
isect_memberEquality
Latex:
\mforall{}e:HeytingGeometry. \mforall{}a,b,c:Point.
(a \# bc
{}\mRightarrow{} \{\mforall{}x,y:Point.
(((Colinear(a;b;x) \mwedge{} Colinear(c;b;y)) \mwedge{} x \mneq{} b \mwedge{} y \mneq{} b)
{}\mRightarrow{} (x \# by \mwedge{} (\mforall{}m:Point. (x-m-y {}\mRightarrow{} m \mneq{} b))))\})
Date html generated:
2017_10_02-PM-07_02_37
Last ObjectModification:
2017_08_08-PM-00_41_23
Theory : euclidean!plane!geometry
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