Nuprl Lemma : not-perp-point-construction
∀e:HeytingGeometry. ∀a,b,c:Point.
(a # bc ⇒ (∃c',b':Point. ((c=c'=b' ∧ ab' ≅ ac) ∧ Colinear(a;b';b) ∧ a # c'b' ∧ Rac'b')))
Proof
Definitions occuring in Statement :
geo-triangle: a # bc,
heyting-geometry: HeytingGeometry,
right-angle: Rabc,
geo-midpoint: a=m=b,
geo-colinear: Colinear(a;b;c),
geo-congruent: ab ≅ cd,
geo-point: Point,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
so_apply: x[s],
so_lambda: λ2x.t[x],
exists: ∃x:A. B[x],
uiff: uiff(P;Q),
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,
and: P ∧ Q,
uimplies: b supposing a,
guard: {T},
subtype_rel: A ⊆r B,
heyting-geometry: Error :heyting-geometry,
uall: ∀[x:A]. B[x],
prop: ℙ,
member: t ∈ T,
implies: P ⇒ Q,
all: ∀x:A. B[x],
geo-midpoint: a=m=b,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
right-angle: Rabc
Lemmas referenced :
geo-proper-extend-exists,
exists_wf,
right-angle_wf,
geo-colinear_wf,
geo-congruent_wf,
geo-midpoint_wf,
geo-length-flip,
geo-congruent-iff-length,
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-sep3,
geo-triangle-symmetry,
geo-triangle-colinear,
isosceles-mid-exists,
geo-triangle-property,
geo-sep-sym,
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-point_wf,
Error :geo-triangle_wf,
geo-between-implies-colinear,
midpoint-sep,
geo-congruent_functionality,
geo-eq_weakening,
geo-midpoint_functionality,
geo-midpoint-symmetry,
symmetric-point-unicity
Rules used in proof :
lambdaEquality,
productEquality,
dependent_pairFormation,
equalitySymmetry,
equalityTransitivity,
baseClosed,
imageMemberEquality,
independent_pairFormation,
natural_numberEquality,
dependent_set_memberEquality,
voidEquality,
voidElimination,
isect_memberEquality,
productElimination,
independent_functionElimination,
dependent_functionElimination,
because_Cache,
sqequalRule,
independent_isectElimination,
instantiate,
applyEquality,
hypothesis,
hypothesisEquality,
rename,
setElimination,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}e:HeytingGeometry. \mforall{}a,b,c:Point.
(a \# bc {}\mRightarrow{} (\mexists{}c',b':Point. ((c=c'=b' \mwedge{} ab' \00D0 ac) \mwedge{} Colinear(a;b';b) \mwedge{} a \# c'b' \mwedge{} Rac'b')))
Date html generated:
2017_10_02-PM-07_07_19
Last ObjectModification:
2017_08_08-PM-00_37_28
Theory : euclidean!plane!geometry
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