Nuprl Lemma : isosc-bisectors-between_1
∀e:HeytingGeometry. ∀a,b,c,m,a',b',m':Point.
(c # ab ⇒ ac ≅ bc ⇒ (c-a'-a ∧ c-b'-b) ⇒ a=m=b ⇒ a'=m'=b' ⇒ aa' ≅ bb' ⇒ c-m'-m)
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-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,
guard: {T},
cand: A c∧ B,
heyting-geometry: HeytingGeometry,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
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],
prop: ℙ,
false: False,
select: L[n],
cons: [a / b],
subtract: n - m,
geo-midpoint: a=m=b,
geo-strict-between: a-b-c,
uiff: uiff(P;Q),
euclidean-plane: EuclideanPlane,
basic-geometry-: BasicGeometry-,
iff: P ⇐⇒ Q,
geo-triangle: a # bc,
rev_implies: P ⇐ Q,
geo-colinear: Colinear(a;b;c)
Latex:
\mforall{}e:HeytingGeometry. \mforall{}a,b,c,m,a',b',m':Point.
(c \# ab {}\mRightarrow{} ac \mcong{} bc {}\mRightarrow{} (c-a'-a \mwedge{} c-b'-b) {}\mRightarrow{} a=m=b {}\mRightarrow{} a'=m'=b' {}\mRightarrow{} aa' \mcong{} bb' {}\mRightarrow{} c-m'-m)
Date html generated:
2020_05_20-AM-10_33_33
Last ObjectModification:
2019_12_03-AM-09_50_54
Theory : euclidean!plane!geometry
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