Nuprl Lemma : isosceles-mid-exists
∀e:HeytingGeometry. ∀a,b,c:Point. (a # bc ⇒ ab ≅ cb ⇒ (∃x:Point. a=x=c))
Proof
Definitions occuring in Statement :
geo-triangle: a # bc,
heyting-geometry: HeytingGeometry,
geo-midpoint: a=m=b,
geo-congruent: ab ≅ cd,
geo-point: Point,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q
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,
guard: {T},
uimplies: b supposing a,
prop: ℙ,
and: P ∧ Q,
heyting-geometry: HeytingGeometry,
cand: A c∧ B,
euclidean-plane: EuclideanPlane,
basic-geometry-: BasicGeometry-,
exists: ∃x:A. B[x],
subtract: n - m,
cons: [a / b],
select: L[n],
false: False,
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
or: P ∨ Q,
decidable: Dec(P),
lelt: i ≤ j < k,
int_seg: {i..j-},
l_all: (∀x∈L.P[x]),
geo-colinear-set: geo-colinear-set(e; L),
geo-midpoint: a=m=b,
uiff: uiff(P;Q),
stable: Stable{P},
geo-eq: a ≡ b,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
squash: ↓T,
true: True,
so_apply: x[s1;s2;s3],
so_lambda: so_lambda3,
append: as @ bs,
ge: i ≥ j ,
less_than: a < b,
less_than': less_than'(a;b),
le: A ≤ B,
nat: ℕ,
l_member: (x ∈ l)
Latex:
\mforall{}e:HeytingGeometry. \mforall{}a,b,c:Point. (a \# bc {}\mRightarrow{} ab \mcong{} cb {}\mRightarrow{} (\mexists{}x:Point. a=x=c))
Date html generated:
2020_05_20-AM-10_33_16
Last ObjectModification:
2020_01_27-PM-09_59_50
Theory : euclidean!plane!geometry
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