Nuprl Lemma : rectangle-sides-cong
∀g:EuclideanPlane. ∀a,b,c,d,e,f:Point.
(e # ac
⇒ f # eb
⇒ ac ⊥b be
⇒ df ⊥e eb
⇒ d-e-f
⇒ a-b-c
⇒ ab ≅ eb
⇒ bc ≅ eb
⇒ de ≅ eb
⇒ ef ≅ eb
⇒ {ad ≅ fc ∧ dc ≅ fa})
Proof
Definitions occuring in Statement :
geo-perp-in: ab ⊥x cd,
euclidean-plane: EuclideanPlane,
geo-strict-between: a-b-c,
geo-congruent: ab ≅ cd,
geo-lsep: a # bc,
geo-point: Point,
guard: {T},
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: 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,
basic-geometry: BasicGeometry,
and: P ∧ Q,
cand: A c∧ B,
geo-perp-in: ab ⊥x cd,
uiff: uiff(P;Q),
geo-lsep: a # bc,
subtract: n - m,
cons: [a / b],
select: L[n],
false: False,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
or: P ∨ Q,
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),
euclidean-plane: EuclideanPlane,
basic-geometry-: BasicGeometry-,
oriented-plane: OrientedPlane,
geo-eq: a ≡ b,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
so_apply: x[s1;s2;s3],
so_lambda: so_lambda3,
append: as @ bs,
ge: i ≥ j ,
true: True,
squash: ↓T,
less_than: a < b,
less_than': less_than'(a;b),
le: A ≤ B,
nat: ℕ,
l_member: (x ∈ l),
sq_exists: ∃x:A [B[x]],
stable: Stable{P},
geo-cong-angle: abc ≅a xyz,
geo-strict-between: a-b-c,
geo-out: out(p ab),
geo-tri: Triangle(a;b;c)
Latex:
\mforall{}g:EuclideanPlane. \mforall{}a,b,c,d,e,f:Point.
(e \# ac
{}\mRightarrow{} f \# eb
{}\mRightarrow{} ac \mbot{}b be
{}\mRightarrow{} df \mbot{}e eb
{}\mRightarrow{} d-e-f
{}\mRightarrow{} a-b-c
{}\mRightarrow{} ab \mcong{} eb
{}\mRightarrow{} bc \mcong{} eb
{}\mRightarrow{} de \mcong{} eb
{}\mRightarrow{} ef \mcong{} eb
{}\mRightarrow{} \{ad \mcong{} fc \mwedge{} dc \mcong{} fa\})
Date html generated:
2020_05_20-AM-10_42_52
Last ObjectModification:
2020_01_27-PM-09_58_43
Theory : euclidean!plane!geometry
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