Nuprl Lemma : isosceles-sep-implies-lsep
∀e:EuclideanPlane. ∀a,b,x:Point. (xa ≅ xb ⇒ a ≠ b ⇒ (∀m:{m:Point| a=m=b} . m ≠ x) ⇒ x # ab)
Proof
Definitions occuring in Statement :
euclidean-plane: EuclideanPlane,
geo-lsep: a # bc,
geo-midpoint: a=m=b,
geo-congruent: ab ≅ cd,
geo-sep: a ≠ b,
geo-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]}
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
prop: ℙ,
sq_exists: ∃x:A [B[x]],
euclidean-plane: EuclideanPlane,
sq_stable: SqStable(P),
squash: ↓T,
basic-geometry: BasicGeometry,
guard: {T},
and: P ∧ Q,
uimplies: b supposing a,
geo-midpoint: a=m=b,
geo-colinear-set: geo-colinear-set(e; L),
l_all: (∀x∈L.P[x]),
top: Top,
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
select: L[n],
cons: [a / b],
subtract: n - m,
geo-perp-in: ab ⊥x cd,
right-angle: Rabc,
iff: P ⇐⇒ Q,
cand: A c∧ B,
rev_implies: P ⇐ Q,
oriented-plane: OrientedPlane
Lemmas referenced :
Euclid-midpoint,
geo-sep_wf,
sq_stable__midpoint,
midpoint-sep,
geo-midpoint_wf,
Euclid-erect-2perp,
geo-colinear-is-colinear-set,
geo-between-implies-colinear,
length_of_cons_lemma,
istype-void,
length_of_nil_lemma,
decidable__le,
full-omega-unsat,
intformnot_wf,
intformle_wf,
itermConstant_wf,
istype-int,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
istype-le,
istype-less_than,
geo-colinear_wf,
sq_stable__and,
geo-perp-in_wf,
geo-left_wf,
sq_stable__geo-perp-in,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
sq_stable__geo-left,
upper-dimension-axiom,
geo-congruent-left-comm,
geo-congruent_wf,
geo-point_wf,
geo-colinear-same,
right-angle-symmetry,
geo-midpoint-symmetry,
lsep-iff-all-sep,
lsep-all-sym2,
between-preserves-left-2,
geo-between-symmetry,
between-preserves-left-4,
geo-midpoint-implies-between,
colinear-lsep2,
geo-sep-sym,
lsep-all-sym
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
dependent_set_memberEquality_alt,
hypothesis,
universeIsType,
isectElimination,
applyEquality,
because_Cache,
sqequalRule,
setElimination,
rename,
independent_functionElimination,
imageMemberEquality,
baseClosed,
imageElimination,
productElimination,
independent_isectElimination,
isect_memberEquality_alt,
voidElimination,
natural_numberEquality,
independent_pairFormation,
unionElimination,
approximateComputation,
dependent_pairFormation_alt,
lambdaEquality_alt,
productIsType,
productEquality,
instantiate,
functionIsType,
setIsType,
inhabitedIsType
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a,b,x:Point. (xa \mcong{} xb {}\mRightarrow{} a \mneq{} b {}\mRightarrow{} (\mforall{}m:\{m:Point| a=m=b\} . m \mneq{} x) {}\mRightarrow{} x \# ab)
Date html generated:
2019_10_16-PM-01_43_44
Last ObjectModification:
2019_08_07-PM-01_08_20
Theory : euclidean!plane!geometry
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