Nuprl Lemma : geo-double-pasch-same-point
∀e:HeytingGeometry. ∀A,B,C,d,f,g,x,y:Point.
(A # BC ⇒ A-d-C ⇒ A-g-B ⇒ B-f-C ⇒ C-x-g ⇒ C-y-g ⇒ B-x-d ⇒ B-y-d ⇒ A-x-f ⇒ A-y-f ⇒ x ≡ y)
Proof
Definitions occuring in Statement :
geo-triangle: a # bc,
heyting-geometry: HeytingGeometry,
geo-strict-between: a-b-c,
geo-eq: a ≡ b,
geo-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
heyting-geometry: Error :heyting-geometry,
uimplies: b supposing a,
guard: {T},
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
prop: ℙ,
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,
and: P ∧ Q,
lelt: i ≤ j < k,
int_seg: {i..j-},
top: Top,
l_all: (∀x∈L.P[x]),
geo-colinear-set: geo-colinear-set(e; L),
so_apply: x[s1;s2;s3],
so_lambda: so_lambda(x,y,z.t[x; y; z]),
append: as @ bs,
so_apply: x[s],
so_lambda: λ2x.t[x],
or: P ∨ Q,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
cand: A c∧ B,
exists: ∃x:A. B[x]
Lemmas referenced :
geo-point_wf,
Error :geo-triangle_wf,
Error :basic-geo-primitives_wf,
Error :basic-geo-structure_wf,
basic-geometry_wf,
Error :heyting-geometry_wf,
subtype_rel_transitivity,
heyting-geometry-subtype,
basic-geometry-subtype,
geo-strict-between_wf,
lelt_wf,
false_wf,
length_of_nil_lemma,
length_of_cons_lemma,
geo-strict-between-implies-colinear,
geo-colinear-is-colinear-set,
geo-strict-between-sep1,
geo-intersection-unicity,
geo-colinear_wf,
list_ind_nil_lemma,
list_ind_cons_lemma,
exists_wf,
geo-sep_wf,
equal_wf,
l_member_wf,
cons_member,
geo-strict-between-sep3,
nil_wf,
cons_wf,
geo-colinear-append,
not-geo-triangle-iff-colinear
Rules used in proof :
rename,
setElimination,
because_Cache,
sqequalRule,
independent_isectElimination,
instantiate,
hypothesis,
applyEquality,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
baseClosed,
imageMemberEquality,
independent_pairFormation,
natural_numberEquality,
dependent_set_memberEquality,
voidEquality,
voidElimination,
isect_memberEquality,
independent_functionElimination,
dependent_functionElimination,
lambdaEquality,
productEquality,
inlFormation,
inrFormation,
productElimination,
dependent_pairFormation
Latex:
\mforall{}e:HeytingGeometry. \mforall{}A,B,C,d,f,g,x,y:Point.
(A \# BC {}\mRightarrow{} A-d-C {}\mRightarrow{} A-g-B {}\mRightarrow{} B-f-C {}\mRightarrow{} C-x-g {}\mRightarrow{} C-y-g {}\mRightarrow{} B-x-d {}\mRightarrow{} B-y-d {}\mRightarrow{} A-x-f {}\mRightarrow{} A-y-f {}\mRightarrow{} x \mequiv{} y)
Date html generated:
2017_10_02-PM-07_08_12
Last ObjectModification:
2017_08_08-PM-00_41_00
Theory : euclidean!plane!geometry
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