Nuprl Lemma : right-angle-SAS
∀e:EuclideanPlane. ∀a,b,c,x,y,z:Point.
((Rabc ∧ a # b ∧ b # c) ⇒ (Rxyz ∧ x # y ∧ y # z) ⇒ ab ≅ xy ⇒ bc ≅ yz ⇒ ac ≅ xz)
Proof
Definitions occuring in Statement :
euclidean-plane: EuclideanPlane,
right-angle: Rabc,
geo-congruent: ab ≅ cd,
geo-sep: a # b,
geo-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
prop: ℙ,
basic-geometry: BasicGeometry,
right-angle: Rabc,
exists: ∃x:A. B[x],
geo-midpoint: a=m=b,
basic-geometry-: BasicGeometry-,
uiff: uiff(P;Q),
geo-equilateral: EQΔ(a;b;c),
squash: ↓T,
true: True,
cand: A c∧ B,
geo-colinear-set: geo-colinear-set(e; L),
l_all: (∀x∈L.P[x]),
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
select: L[n],
cons: [a / b],
subtract: n - m,
l_member: (x ∈ l),
nat: ℕ,
le: A ≤ B,
less_than': less_than'(a;b),
less_than: a < b,
ge: i ≥ j ,
append: as @ bs,
so_lambda: so_lambda3,
so_apply: x[s1;s2;s3],
euclidean-plane: EuclideanPlane,
geo-eq: a ≡ b,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
top: Top,
oriented-plane: OrientedPlane,
geo-strict-between: a-b-c,
geo-cong-angle: abc ≅a xyz,
geo-tri: Triangle(a;b;c)
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a,b,c,x,y,z:Point.
((Rabc \mwedge{} a \# b \mwedge{} b \# c) {}\mRightarrow{} (Rxyz \mwedge{} x \# y \mwedge{} y \# z) {}\mRightarrow{} ab \mcong{} xy {}\mRightarrow{} bc \mcong{} yz {}\mRightarrow{} ac \mcong{} xz)
Date html generated:
2020_05_20-AM-10_06_49
Last ObjectModification:
2019_12_31-PM-06_12_36
Theory : euclidean!plane!geometry
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