Nuprl Lemma : Euclid-Prop19-lemma2_1
∀e:EuclideanPlane. ∀a,b,c,d,f:Point. (a # bc ⇒ abd ≅a cbd ⇒ a=d=c ⇒ a-f-c ⇒ cbf < abf ⇒ abd < abf)
Proof
Definitions occuring in Statement :
geo-lt-angle: abc < xyz,
geo-cong-angle: abc ≅a xyz,
euclidean-plane: EuclideanPlane,
geo-midpoint: a=m=b,
geo-strict-between: a-b-c,
geo-lsep: a # bc,
geo-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
basic-geometry: BasicGeometry,
and: P ∧ Q,
cand: A c∧ 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,
euclidean-plane: EuclideanPlane,
l_member: (x ∈ l),
nat: ℕ,
le: A ≤ B,
less_than': less_than'(a;b),
less_than: a < b,
squash: ↓T,
true: True,
ge: i ≥ j ,
geo-midpoint: a=m=b,
append: as @ bs,
so_lambda: so_lambda3,
so_apply: x[s1;s2;s3],
basic-geometry-: BasicGeometry-,
geo-eq: a ≡ b,
geo-strict-between: a-b-c,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
geo-sep: a # b,
geo-between: B(abc)
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a,b,c,d,f:Point.
(a \# bc {}\mRightarrow{} abd \mcong{}\msuba{} cbd {}\mRightarrow{} a=d=c {}\mRightarrow{} a-f-c {}\mRightarrow{} cbf < abf {}\mRightarrow{} abd < abf)
Date html generated:
2020_05_20-AM-10_38_57
Last ObjectModification:
2019_12_03-AM-09_48_35
Theory : euclidean!plane!geometry
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