Nuprl Lemma : double-pasch-exists
∀e:HeytingGeometry. ∀a,b,c,a',b',p:Point. (a-b-c ⇒ a'-b'-c ⇒ a-p-a' ⇒ c # aa' ⇒ (∃q:Point. (p-q-c ∧ b-q-b')))
Proof
Definitions occuring in Statement :
geo-triangle: a # bc,
heyting-geometry: HeytingGeometry,
geo-strict-between: a-b-c,
geo-point: Point,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
uimplies: b supposing a,
guard: {T},
subtype_rel: A ⊆r B,
heyting-geometry: Error :heyting-geometry,
uall: ∀[x:A]. B[x],
prop: ℙ,
and: P ∧ Q,
exists: ∃x:A. B[x],
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),
cand: A c∧ B
Lemmas referenced :
geo-point_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,
Error :geo-triangle_wf,
geo-strict-between-sym,
geo-inner-pasch-ex,
geo-triangle-property,
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-sep2,
geo-sep-sym,
geo-triangle-symmetry,
geo-triangle-colinear,
geo-strict-between-sep3,
geo-triangle-colinear2,
geo-triangle-colinear3,
geo-colinear-same,
geo-strict-between-sep1,
geo-strict-between-trans3,
geo-strict-between-trans2,
geo-strict-between-trans
Rules used in proof :
sqequalRule,
independent_isectElimination,
instantiate,
applyEquality,
rename,
setElimination,
isectElimination,
productElimination,
hypothesis,
independent_functionElimination,
hypothesisEquality,
because_Cache,
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,
productEquality,
dependent_pairFormation
Latex:
\mforall{}e:HeytingGeometry. \mforall{}a,b,c,a',b',p:Point.
(a-b-c {}\mRightarrow{} a'-b'-c {}\mRightarrow{} a-p-a' {}\mRightarrow{} c \# aa' {}\mRightarrow{} (\mexists{}q:Point. (p-q-c \mwedge{} b-q-b')))
Date html generated:
2017_10_02-PM-07_04_33
Last ObjectModification:
2017_08_08-PM-00_38_41
Theory : euclidean!plane!geometry
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