Nuprl Lemma : Euclid-erect-2perp
∀e:EuclideanPlane. ∀a:Point. ∀b:{b:Point| a ≠ b} . ∀c:{c:Point| Colinear(a;b;c)} .
∃q:Point. (∃p:Point [(ab ⊥c pc ∧ ab ⊥c qc ∧ p leftof ab ∧ q leftof ba)])
Proof
Definitions occuring in Statement :
geo-perp-in: ab ⊥x cd,
euclidean-plane: EuclideanPlane,
geo-colinear: Colinear(a;b;c),
geo-left: a leftof bc,
geo-sep: a ≠ b,
geo-point: Point,
all: ∀x:A. B[x],
sq_exists: ∃x:A [B[x]],
exists: ∃x:A. B[x],
and: P ∧ Q,
set: {x:A| B[x]}
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
so_lambda: λ2x.t[x],
so_apply: x[s],
euclidean-plane: EuclideanPlane,
sq_stable: SqStable(P),
implies: P ⇒ Q,
squash: ↓T,
basic-geometry: BasicGeometry,
or: P ∨ Q,
exists: ∃x:A. B[x],
geo-midpoint: a=m=b,
and: P ∧ Q,
geo-out: out(p ab),
cand: A c∧ B,
not: ¬A,
false: False,
basic-geometry-: BasicGeometry-,
stable: Stable{P},
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),
less_than: a < b,
true: True,
select: L[n],
cons: [a / b],
subtract: n - m,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
oriented-plane: OrientedPlane,
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
geo-lsep: a # bc,
sq_exists: ∃x:A [B[x]],
uiff: uiff(P;Q),
geo-perp-in: ab ⊥x cd
Lemmas referenced :
set_wf,
geo-point_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-colinear_wf,
geo-sep_wf,
sq_stable__geo-sep,
sq_stable__colinear,
geo-colinear-sep-cases,
symmetric-point-construction,
geo-sep-sym,
geo-between-sep,
midpoint-sep,
not_wf,
geo-between_wf,
stable__geo-between,
false_wf,
or_wf,
geo-between-symmetry,
geo-between-inner-trans,
geo-between-outer-trans,
minimal-double-negation-hyp-elim,
minimal-not-not-excluded-middle,
geo-between-exchange3,
geo-between-exchange4,
geo-between-same-side,
geo-colinear-is-colinear-set,
geo-between-implies-colinear,
length_of_cons_lemma,
length_of_nil_lemma,
lelt_wf,
geo-left_wf,
geo-midpoint_wf,
all_wf,
iff_wf,
exists_wf,
geo-left-out-1,
geo-left-out,
geo-congruent-symmetry,
geo-congruent-sep,
left-implies-sep,
oriented-colinear-append,
cons_wf,
nil_wf,
cons_member,
l_member_wf,
equal_wf,
list_ind_cons_lemma,
list_ind_nil_lemma,
geo-midpoint-symmetry,
lsep-colinear,
lsep-all-sym2,
not-left-and-right,
Euclid-Prop1-left,
sq_exists_wf,
geo-perp-in_wf,
implies-right-angle,
colinear-implies-midpoint,
geo-congruent-iff-length,
geo-length-flip,
geo-colinear-same,
right-angle-symmetry,
right-angle_wf,
geo-perp-in-iff,
colinear-lsep-cycle,
lsep-all-sym,
lsep-implies-sep
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
applyEquality,
hypothesis,
instantiate,
independent_isectElimination,
sqequalRule,
lambdaEquality,
because_Cache,
setElimination,
rename,
dependent_functionElimination,
independent_functionElimination,
imageMemberEquality,
baseClosed,
imageElimination,
unionElimination,
productElimination,
independent_pairFormation,
productEquality,
functionEquality,
voidElimination,
dependent_pairFormation,
isect_memberEquality,
voidEquality,
dependent_set_memberEquality,
natural_numberEquality,
inrFormation,
inlFormation,
dependent_set_memberFormation,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a:Point. \mforall{}b:\{b:Point| a \mneq{} b\} . \mforall{}c:\{c:Point| Colinear(a;b;c)\} .
\mexists{}q:Point. (\mexists{}p:Point [(ab \mbot{}c pc \mwedge{} ab \mbot{}c qc \mwedge{} p leftof ab \mwedge{} q leftof ba)])
Date html generated:
2018_05_22-PM-00_09_18
Last ObjectModification:
2018_04_04-PM-07_01_53
Theory : euclidean!plane!geometry
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