Nuprl Lemma : Euclid-erect-perp
∀e:EuclideanPlane. ∀a:Point. ∀b:{b:Point| a ≠ b} . ∀c:{c:Point| Colinear(a;b;c)} . (∃p:Point [(ab ⊥c pc ∧ p # ab)])
Proof
Definitions occuring in Statement :
geo-perp-in: ab ⊥x cd,
euclidean-plane: EuclideanPlane,
geo-lsep: a # bc,
geo-colinear: Colinear(a;b;c),
geo-sep: a ≠ b,
geo-point: Point,
all: ∀x:A. B[x],
sq_exists: ∃x:A [B[x]],
and: P ∧ Q,
set: {x:A| B[x]}
Definitions unfolded in proof :
so_apply: x[s1;s2;s3],
so_lambda: so_lambda(x,y,z.t[x; y; z]),
append: as @ bs,
oriented-plane: OrientedPlane,
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),
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
geo-equilateral: EQΔ(a;b;c),
so_apply: x[s],
so_lambda: λ2x.t[x],
uimplies: b supposing a,
guard: {T},
sq_exists: ∃x:A [B[x]],
geo-midpoint: a=m=b,
subtype_rel: A ⊆r B,
prop: ℙ,
implies: P ⇒ Q,
basic-geometry: BasicGeometry,
uall: ∀[x:A]. B[x],
cand: A c∧ B,
and: P ∧ Q,
exists: ∃x:A. B[x],
or: P ∨ Q,
euclidean-plane: EuclideanPlane,
member: t ∈ T,
all: ∀x:A. B[x]
Lemmas referenced :
list_ind_nil_lemma,
list_ind_cons_lemma,
equal_wf,
l_member_wf,
cons_member,
nil_wf,
cons_wf,
oriented-colinear-append,
colinear-lsep-general,
lsep-implies-sep,
lelt_wf,
false_wf,
length_of_nil_lemma,
length_of_cons_lemma,
geo-between-implies-colinear,
geo-colinear-is-colinear-set,
lsep-all-sym,
colinear-lsep-cycle,
geo-perp-in-iff,
exists_wf,
right-angle_wf,
right-angle-symmetry,
euclidean-plane-axioms,
geo-midpoint-symmetry,
implies-right-angle,
geo-primitives_wf,
euclidean-plane-structure_wf,
euclidean-plane_wf,
subtype_rel_transitivity,
euclidean-plane-subtype,
euclidean-plane-structure-subtype,
geo-point_wf,
set_wf,
geo-lsep_wf,
geo-perp-in_wf,
Euclid-Prop1,
geo-between-sep,
symmetric-point-construction,
geo-sep_wf,
geo-colinear_wf,
geo-sep-sym,
geo-colinear-same,
geo-sep-or
Rules used in proof :
inlFormation,
inrFormation,
baseClosed,
imageMemberEquality,
natural_numberEquality,
dependent_set_memberEquality,
voidEquality,
voidElimination,
isect_memberEquality,
lambdaEquality,
independent_isectElimination,
instantiate,
dependent_set_memberFormation,
applyEquality,
productEquality,
independent_functionElimination,
independent_pairFormation,
productElimination,
sqequalRule,
isectElimination,
because_Cache,
dependent_pairFormation,
unionElimination,
hypothesis,
hypothesisEquality,
rename,
setElimination,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a:Point. \mforall{}b:\{b:Point| a \mneq{} b\} . \mforall{}c:\{c:Point| Colinear(a;b;c)\} .
(\mexists{}p:Point [(ab \mbot{}c pc \mwedge{} p \# ab)])
Date html generated:
2018_05_22-PM-00_09_36
Last ObjectModification:
2018_05_21-AM-01_18_40
Theory : euclidean!plane!geometry
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