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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