Nuprl Lemma : tarski-erect-perp-or
∀e:HeytingGeometry. ∀a,b,c:Point. (c # ba ⇒ (∃p,t:Point. (((ab ⊥ pa ∨ ab ⊥ pb) ∧ Colinear(a;b;t)) ∧ p-t-c)))
Proof
Definitions occuring in Statement :
geo-triangle: a # bc,
heyting-geometry: HeytingGeometry,
geo-perp: ab ⊥ cd,
geo-colinear: Colinear(a;b;c),
geo-strict-between: a-b-c,
geo-point: Point,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
or: P ∨ Q,
and: P ∧ Q
Definitions unfolded in proof :
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],
basic-geometry: BasicGeometry,
euclidean-plane: EuclideanPlane,
heyting-geometry: HeytingGeometry,
geo-perp-in: ab ⊥x cd,
exists: ∃x:A. B[x],
cand: A c∧ B,
and: 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),
geo-triangle: a # bc,
or: P ∨ Q,
geo-lsep: a # bc,
oriented-plane: OrientedPlane,
uiff: uiff(P;Q),
basic-geometry-: BasicGeometry-,
geo-midpoint: a=m=b,
right-angle: Rabc,
geo-strict-between: a-b-c,
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],
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
geo-perp: ab ⊥ cd
Lemmas referenced :
geo-point_wf,
geo-primitives_wf,
euclidean-plane-structure_wf,
euclidean-plane_wf,
heyting-geometry_wf,
subtype_rel_transitivity,
heyting-geometry-subtype,
euclidean-plane-subtype,
euclidean-plane-structure-subtype,
geo-triangle_wf,
geo-left-axioms_wf,
basic-geo-axioms_wf,
subtype_rel_self,
geo-colinear-same,
geo-triangle-symmetry,
tarski-perp-in-exists,
geo-proper-extend-exists,
lelt_wf,
false_wf,
length_of_nil_lemma,
length_of_cons_lemma,
geo-colinear-is-colinear-set,
lsep-colinear-sep,
geo-length-flip,
geo-congruent-iff-length,
geo-between-symmetry,
geo-strict-between-implies-between,
geo-sep_wf,
geo-triangle-property,
geo-sep-sym,
geo-sep-or,
geo-strict-between-sep3,
geo-triangle-colinear2,
geo-strict-between-implies-colinear,
geo-strict-between-sep1,
geo-triangle-colinear,
geo-congruent-mid-exists,
geo-perp-midsegments,
geo-between-implies-colinear,
midpoint-sep,
geo-strict-between-sym,
double-pasch-exists,
geo-perp-in-iff,
geo-strict-between_wf,
geo-colinear_wf,
geo-perp_wf,
or_wf,
list_ind_nil_lemma,
list_ind_cons_lemma,
exists_wf,
equal_wf,
l_member_wf,
cons_member,
nil_wf,
cons_wf,
oriented-colinear-append,
right-angle_wf,
geo-perp-in_wf,
geo-perp-symmetry2
Rules used in proof :
because_Cache,
sqequalRule,
independent_isectElimination,
instantiate,
hypothesis,
applyEquality,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
cumulativity,
productEquality,
setEquality,
productElimination,
independent_functionElimination,
dependent_functionElimination,
rename,
baseClosed,
imageMemberEquality,
independent_pairFormation,
natural_numberEquality,
dependent_set_memberEquality,
voidEquality,
voidElimination,
isect_memberEquality,
equalitySymmetry,
equalityTransitivity,
unionElimination,
setElimination,
lambdaEquality,
inlFormation,
inrFormation,
dependent_pairFormation,
universeEquality
Latex:
\mforall{}e:HeytingGeometry. \mforall{}a,b,c:Point.
(c \# ba {}\mRightarrow{} (\mexists{}p,t:Point. (((ab \mbot{} pa \mvee{} ab \mbot{} pb) \mwedge{} Colinear(a;b;t)) \mwedge{} p-t-c)))
Date html generated:
2018_05_22-PM-00_21_17
Last ObjectModification:
2017_10_26-PM-00_37_35
Theory : euclidean!plane!geometry
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