Nuprl Lemma : tarski-perp-in-exists
∀e:HeytingGeometry. ∀a,b,c:Point. (a # bc ⇒ (∃x:Point. (Colinear(a;b;x) ∧ ab ⊥x cx)))
Proof
Definitions occuring in Statement :
geo-triangle: a # bc,
heyting-geometry: HeytingGeometry,
geo-perp-in: ab ⊥x cd,
geo-colinear: Colinear(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 :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
guard: {T},
and: P ∧ Q,
cand: A c∧ B,
prop: ℙ,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
heyting-geometry: HeytingGeometry,
exists: ∃x:A. B[x],
euclidean-plane: EuclideanPlane,
basic-geometry-: BasicGeometry-,
basic-geometry: BasicGeometry,
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,
oriented-plane: OrientedPlane,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
or: P ∨ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
geo-midpoint: a=m=b,
uiff: uiff(P;Q),
geo-cong-tri: Cong3(abc,a'b'c'),
right-angle: Rabc
Lemmas referenced :
geo-triangle-property,
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-point_wf,
geo-sep-sym,
geo-proper-extend-exists,
geo-triangle-colinear,
geo-strict-between-sep1,
subtype_rel_self,
basic-geo-axioms_wf,
geo-left-axioms_wf,
geo-colinear-is-colinear-set,
geo-strict-between-implies-colinear,
length_of_cons_lemma,
length_of_nil_lemma,
false_wf,
lelt_wf,
geo-strict-between-sep3,
geo-congruent-mid-exists,
geo-triangle-symmetry,
geo-midpoint_wf,
implies-right-angle,
geo-midpoint-symmetry,
midpoint-sep,
geo-strict-between-sep2,
geo-strict-between-sym,
geo-strict-between-trans3,
geo-triangle-colinear',
oriented-colinear-append,
cons_wf,
nil_wf,
cons_member,
l_member_wf,
equal_wf,
geo-sep_wf,
exists_wf,
list_ind_cons_lemma,
list_ind_nil_lemma,
symmetric-point-construction,
geo-between-sep,
geo-between-implies-colinear,
euclidean-plane-axioms,
geo-between-symmetry,
geo-strict-between-implies-between,
geo-between-outer-trans,
geo-congruent-symmetry,
geo-congruent-iff-length,
geo-length-flip,
geo-congruent-flip,
geo-five-segment,
geo-between-exchange3,
congruence-preserves-right-angle,
geo-krippen-lemma,
right-angle-symmetry,
geo-colinear_wf,
geo-perp-in_wf,
geo-perp-in-iff,
geo-colinear-same,
right-angle_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
hypothesis,
productElimination,
isectElimination,
applyEquality,
instantiate,
independent_isectElimination,
sqequalRule,
because_Cache,
rename,
setEquality,
productEquality,
cumulativity,
isect_memberEquality,
voidElimination,
voidEquality,
dependent_set_memberEquality,
natural_numberEquality,
independent_pairFormation,
imageMemberEquality,
baseClosed,
dependent_pairFormation,
inrFormation,
inlFormation,
lambdaEquality,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}e:HeytingGeometry. \mforall{}a,b,c:Point. (a \# bc {}\mRightarrow{} (\mexists{}x:Point. (Colinear(a;b;x) \mwedge{} ab \mbot{}x cx)))
Date html generated:
2017_10_02-PM-07_10_00
Last ObjectModification:
2017_08_16-PM-00_16_47
Theory : euclidean!plane!geometry
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