Nuprl Lemma : perp-aux-general-construction
∀e:HeytingGeometry. ∀a,b,c,x:Point.
(((c # ba ∧ ab ⊥x cx) ∧ a ≠ x)
⇒ (∃c1,c',p:Point. (((c=a=c1 ∧ c=x=c') ∧ c'a ≅ ca) ∧ c' # c1a ∧ ((a # cc' ∧ c1=p=c') ∧ ab ⊥a pa) ∧ p # ab)))
Proof
Definitions occuring in Statement :
geo-triangle: a # bc,
heyting-geometry: HeytingGeometry,
geo-perp-in: ab ⊥x cd,
geo-midpoint: a=m=b,
geo-congruent: ab ≅ cd,
geo-sep: a ≠ b,
geo-point: Point,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
geo-perp-in: ab ⊥x cd,
member: t ∈ T,
uall: ∀[x:A]. B[x],
prop: ℙ,
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
heyting-geometry: HeytingGeometry,
euclidean-plane: EuclideanPlane,
basic-geometry: BasicGeometry,
oriented-plane: OrientedPlane,
geo-lsep: a # bc,
or: P ∨ Q,
geo-triangle: a # bc,
geo-colinear-set: geo-colinear-set(e; L),
l_all: (∀x∈L.P[x]),
top: Top,
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
less_than: a < b,
squash: ↓T,
true: True,
select: L[n],
cons: [a / b],
subtract: n - m,
exists: ∃x:A. B[x],
right-angle: Rabc,
geo-midpoint: a=m=b,
basic-geometry-: BasicGeometry-,
uiff: uiff(P;Q),
cand: A c∧ B,
so_lambda: λ2x.t[x],
so_apply: x[s],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
geo-strict-between: a-b-c
Lemmas referenced :
geo-colinear-same,
geo-triangle_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
heyting-geometry-subtype,
subtype_rel_transitivity,
heyting-geometry_wf,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-perp-in_wf,
subtype_rel_self,
basic-geo-axioms_wf,
geo-left-axioms_wf,
geo-sep_wf,
geo-point_wf,
lsep-colinear-sep,
geo-colinear-is-colinear-set,
length_of_cons_lemma,
length_of_nil_lemma,
false_wf,
lelt_wf,
geo-proper-extend-exists,
geo-strict-between-implies-between,
geo-between-symmetry,
geo-congruent-iff-length,
geo-length-flip,
geo-sep-sym,
geo-triangle-property,
geo-triangle-symmetry,
geo-triangle-colinear,
geo-strict-between-sep1,
geo-strict-between-implies-colinear,
geo-triangle-colinear2,
geo-strict-between-sep3,
geo-congruent-mid-exists,
geo-midpoint_wf,
geo-congruent_wf,
exists_wf,
geo-midpoint-symmetry,
geo-perp-midsegments,
midpoint-sep,
geo-between-implies-colinear,
geo-colinear_wf,
right-angle_wf,
geo-perp-in-iff,
geo-between-sep,
perp-aux2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
cut,
hypothesis,
dependent_functionElimination,
hypothesisEquality,
independent_functionElimination,
introduction,
extract_by_obid,
isectElimination,
because_Cache,
productEquality,
applyEquality,
instantiate,
independent_isectElimination,
sqequalRule,
setEquality,
cumulativity,
isect_memberEquality,
voidElimination,
voidEquality,
dependent_set_memberEquality,
natural_numberEquality,
independent_pairFormation,
imageMemberEquality,
baseClosed,
rename,
equalityTransitivity,
equalitySymmetry,
dependent_pairFormation,
lambdaEquality
Latex:
\mforall{}e:HeytingGeometry. \mforall{}a,b,c,x:Point.
(((c \# ba \mwedge{} ab \mbot{}x cx) \mwedge{} a \mneq{} x)
{}\mRightarrow{} (\mexists{}c1,c',p:Point
(((c=a=c1 \mwedge{} c=x=c') \mwedge{} c'a \00D0 ca) \mwedge{} c' \# c1a \mwedge{} ((a \# cc' \mwedge{} c1=p=c') \mwedge{} ab \mbot{}a pa) \mwedge{} p \# ab)))
Date html generated:
2017_10_02-PM-07_10_12
Last ObjectModification:
2017_08_10-PM-05_08_48
Theory : euclidean!plane!geometry
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