Nuprl Lemma : midpoint-of-equidistant-points-is-perp
∀e:EuclideanPlane. ∀u:Point. ∀v:{v:Point| u # v} . ∀c:{c:Point| c # uv} .  (cu ≅ cv 
⇒ (∀m:Point. (u=m=v 
⇒ uv  ⊥m mc)))
Proof
Definitions occuring in Statement : 
geo-perp-in: ab  ⊥x cd
, 
euclidean-plane: EuclideanPlane
, 
geo-midpoint: a=m=b
, 
geo-congruent: ab ≅ cd
, 
geo-lsep: a # bc
, 
geo-sep: a # b
, 
geo-point: Point
, 
all: ∀x:A. B[x]
, 
implies: P 
⇒ Q
, 
set: {x:A| B[x]} 
Definitions unfolded in proof : 
rev_implies: P 
⇐ Q
, 
subtract: n - m
, 
cons: [a / b]
, 
select: L[n]
, 
exists: ∃x:A. B[x]
, 
satisfiable_int_formula: satisfiable_int_formula(fmla)
, 
decidable: Dec(P)
, 
lelt: i ≤ j < k
, 
int_seg: {i..j-}
, 
top: Top
, 
l_all: (∀x∈L.P[x])
, 
geo-colinear-set: geo-colinear-set(e; L)
, 
cand: A c∧ B
, 
oriented-plane: OrientedPlane
, 
rev_uimplies: rev_uimplies(P;Q)
, 
uiff: uiff(P;Q)
, 
iff: P 
⇐⇒ Q
, 
geo-eq: a ≡ b
, 
stable: Stable{P}
, 
false: False
, 
not: ¬A
, 
or: P ∨ Q
, 
geo-perp-in: ab  ⊥x cd
, 
squash: ↓T
, 
sq_stable: SqStable(P)
, 
basic-geometry: BasicGeometry
, 
and: P ∧ Q
, 
geo-midpoint: a=m=b
, 
prop: ℙ
, 
uimplies: b supposing a
, 
guard: {T}
, 
subtype_rel: A ⊆r B
, 
uall: ∀[x:A]. B[x]
, 
member: t ∈ T
, 
implies: P 
⇒ Q
, 
all: ∀x:A. B[x]
Lemmas referenced : 
geo-colinear-permute, 
euclidean-plane-axioms, 
geo-sep-irrefl', 
geo-congruence-identity, 
right-angle-symmetry, 
geo-length-flip, 
geo-congruent-iff-length, 
colinear-implies-midpoint, 
implies-right-angle, 
geo-sep-sym, 
geo-perp-in-iff, 
geo-colinear-same, 
istype-less_than, 
istype-le, 
int_formula_prop_less_lemma, 
intformless_wf, 
decidable__lt, 
int_formula_prop_wf, 
int_term_value_constant_lemma, 
int_formula_prop_le_lemma, 
int_formula_prop_not_lemma, 
istype-int, 
itermConstant_wf, 
intformle_wf, 
intformnot_wf, 
full-omega-unsat, 
decidable__le, 
length_of_nil_lemma, 
length_of_cons_lemma, 
geo-between-implies-colinear, 
geo-colinear-is-colinear-set, 
geo-sep_functionality, 
geo-lsep_functionality, 
minimal-not-not-excluded-middle, 
geo-perp-in_functionality, 
geo-between_functionality, 
geo-eq_weakening, 
geo-congruent_functionality, 
minimal-double-negation-hyp-elim, 
istype-void, 
right-angle_wf, 
geo-colinear_wf, 
not_wf, 
false_wf, 
stable__geo-perp-in, 
sq_stable__geo-perp-in, 
geo-point_wf, 
geo-sep_wf, 
geo-lsep_wf, 
geo-congruent_wf, 
geo-primitives_wf, 
euclidean-plane-structure_wf, 
euclidean-plane_wf, 
subtype_rel_transitivity, 
euclidean-plane-subtype, 
euclidean-plane-structure-subtype, 
geo-midpoint_wf
Rules used in proof : 
equalitySymmetry, 
equalityTransitivity, 
lambdaEquality_alt, 
dependent_pairFormation_alt, 
approximateComputation, 
independent_pairFormation, 
natural_numberEquality, 
dependent_set_memberEquality_alt, 
isect_memberEquality_alt, 
promote_hyp, 
voidElimination, 
unionElimination, 
unionIsType, 
productIsType, 
functionIsType, 
productEquality, 
functionEquality, 
unionEquality, 
imageElimination, 
baseClosed, 
imageMemberEquality, 
independent_functionElimination, 
dependent_functionElimination, 
productElimination, 
setIsType, 
because_Cache, 
inhabitedIsType, 
rename, 
setElimination, 
sqequalRule, 
independent_isectElimination, 
instantiate, 
hypothesis, 
applyEquality, 
hypothesisEquality, 
thin, 
isectElimination, 
sqequalHypSubstitution, 
extract_by_obid, 
introduction, 
cut, 
universeIsType, 
lambdaFormation_alt, 
sqequalReflexivity, 
computationStep, 
sqequalTransitivity, 
sqequalSubstitution
Latex:
\mforall{}e:EuclideanPlane.  \mforall{}u:Point.  \mforall{}v:\{v:Point|  u  \#  v\}  .  \mforall{}c:\{c:Point|  c  \#  uv\}  .
    (cu  \mcong{}  cv  {}\mRightarrow{}  (\mforall{}m:Point.  (u=m=v  {}\mRightarrow{}  uv    \mbot{}m  mc)))
Date html generated:
2019_10_29-AM-09_17_12
Last ObjectModification:
2019_10_18-PM-03_15_32
Theory : euclidean!plane!geometry
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