Nuprl Lemma : isosceles-mid-between
∀e:HeytingGeometry. ∀a,b,c,p,q,r,x:Point.
(a # bc ⇒ a_p_b ⇒ c_q_b ⇒ p=x=q ⇒ a=r=c ⇒ a=p=b ⇒ c=q=b ⇒ pb ≅ qb ⇒ b-x-r)
Proof
Definitions occuring in Statement :
geo-triangle: a # bc,
heyting-geometry: HeytingGeometry,
geo-midpoint: a=m=b,
geo-strict-between: a-b-c,
geo-congruent: ab ≅ cd,
geo-between: a_b_c,
geo-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
heyting-geometry: Error :heyting-geometry,
subtract: n - m,
cons: [a / b],
select: L[n],
true: True,
squash: ↓T,
less_than: a < b,
prop: ℙ,
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),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
cand: A c∧ B,
and: P ∧ Q,
guard: {T},
subtype_rel: A ⊆r B,
member: t ∈ T,
implies: P ⇒ Q,
all: ∀x:A. B[x],
geo-midpoint: a=m=b,
exists: ∃x:A. B[x],
geo-strict-between: a-b-c,
uiff: uiff(P;Q),
iff: P ⇐⇒ Q
Lemmas referenced :
geo-point_wf,
Error :geo-triangle_wf,
geo-between_wf,
geo-midpoint_wf,
Error :basic-geo-primitives_wf,
Error :basic-geo-structure_wf,
basic-geometry_wf,
Error :heyting-geometry_wf,
subtype_rel_transitivity,
basic-geometry-subtype,
geo-congruent_wf,
geo-triangle-symmetry,
lelt_wf,
false_wf,
length_of_nil_lemma,
length_of_cons_lemma,
geo-between-implies-colinear,
geo-colinear-is-colinear-set,
geo-triangle-colinear,
geo-triangle-property,
geo-sep-sym,
heyting-geometry-subtype,
midpoint-sep,
geo-between-symmetry,
double-pasch-exists,
geo-inner-five-segment,
geo-congruent-refl,
geo-length-type_wf,
true_wf,
squash_wf,
geo-add-length_wf,
geo-add-length-between,
geo-length-flip,
geo-congruent-iff-length,
geo-strict-between-implies-between,
geo-strict-between_functionality,
geo-congruent_functionality,
geo-eq_weakening,
geo-midpoint_functionality,
geo-strict-between-sym,
at-most-one-midpoint
Rules used in proof :
rename,
setElimination,
instantiate,
baseClosed,
imageMemberEquality,
independent_pairFormation,
natural_numberEquality,
dependent_set_memberEquality,
voidEquality,
voidElimination,
isect_memberEquality,
independent_isectElimination,
isectElimination,
productElimination,
because_Cache,
independent_functionElimination,
sqequalRule,
hypothesis,
applyEquality,
hypothesisEquality,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
imageElimination,
lambdaEquality,
equalitySymmetry,
equalityTransitivity
Latex:
\mforall{}e:HeytingGeometry. \mforall{}a,b,c,p,q,r,x:Point.
(a \# bc {}\mRightarrow{} a\_p\_b {}\mRightarrow{} c\_q\_b {}\mRightarrow{} p=x=q {}\mRightarrow{} a=r=c {}\mRightarrow{} a=p=b {}\mRightarrow{} c=q=b {}\mRightarrow{} pb \00D0 qb {}\mRightarrow{} b-x-r)
Date html generated:
2017_10_02-PM-07_05_28
Last ObjectModification:
2017_08_08-PM-00_37_21
Theory : euclidean!plane!geometry
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