Nuprl Lemma : tarski-erect-perp-same-side
∀e:HeytingGeometry. ∀a,b,c:Point.
(c # ba ⇒ (∃p,t,d:Point. (((ab ⊥ pa ∧ Colinear(a;b;t)) ∧ p-t-d) ∧ geo-tar-same-side(e;c;d;a;b))))
Proof
Definitions occuring in Statement :
geo-triangle: a # bc,
heyting-geometry: HeytingGeometry,
geo-perp: ab ⊥ cd,
geo-tar-same-side: geo-tar-same-side(e;a;b;p;q),
geo-colinear: Colinear(a;b;c),
geo-strict-between: a-b-c,
geo-point: Point,
all: ∀x:A. B[x],
exists: ∃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,
guard: {T},
and: P ∧ Q,
cand: A c∧ B,
exists: ∃x:A. B[x],
heyting-geometry: HeytingGeometry,
euclidean-plane: EuclideanPlane,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
prop: ℙ,
or: P ∨ Q,
uimplies: b supposing a,
geo-midpoint: a=m=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),
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
select: L[n],
cons: [a / b],
subtract: n - m,
geo-strict-between: a-b-c,
geo-perp: ab ⊥ cd,
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 ,
append: as @ bs,
so_lambda: so_lambda3,
so_apply: x[s1;s2;s3],
geo-tar-same-side: geo-tar-same-side(e;a;b;p;q),
geo-tar-opp-side: geo-tar-opp-side(e;a;b;p;q),
geo-triangle: a # bc,
basic-geometry-: BasicGeometry-,
geo-perp-in: ab ⊥x cd,
uiff: uiff(P;Q),
iff: P ⇐⇒ Q
Latex:
\mforall{}e:HeytingGeometry. \mforall{}a,b,c:Point.
(c \# ba {}\mRightarrow{} (\mexists{}p,t,d:Point. (((ab \mbot{} pa \mwedge{} Colinear(a;b;t)) \mwedge{} p-t-d) \mwedge{} geo-tar-same-side(e;c;d;a;b))))
Date html generated:
2020_05_20-AM-10_34_28
Last ObjectModification:
2020_01_13-PM-04_34_22
Theory : euclidean!plane!geometry
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