Nuprl Lemma : geo-line-sep-symmetry
∀g:EuclideanParPlane. ∀l,m:Line. (((l # m) ∧ (∀L,M,N:Line. (L \/ M ⇒ (L \/ N ∨ M \/ N)))) ⇒ (m # l))
Proof
Definitions occuring in Statement :
euclidean-parallel-plane: EuclideanParPlane,
geo-intersect: L \/ M,
geo-line-sep: (l # m),
geo-line: Line,
all: ∀x:A. B[x],
implies: P ⇒ Q,
or: 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,
euclidean-parallel-plane: EuclideanParPlane,
or: P ∨ Q,
geo-line-sep: (l # m),
exists: ∃x:A. B[x],
geo-line: Line,
pi1: fst(t),
pi2: snd(t),
iff: P ⇐⇒ Q,
cand: A c∧ B,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
l_member: (x ∈ l),
nat: ℕ,
le: A ≤ B,
less_than': less_than'(a;b),
not: ¬A,
false: False,
select: L[n],
cons: [a / b],
subtract: n - m,
less_than: a < b,
squash: ↓T,
true: True,
ge: i ≥ j ,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
append: as @ bs,
so_lambda: so_lambda3,
so_apply: x[s1;s2;s3],
geo-colinear-set: geo-colinear-set(e; L),
l_all: (∀x∈L.P[x]),
int_seg: {i..j-},
lelt: i ≤ j < k
Latex:
\mforall{}g:EuclideanParPlane. \mforall{}l,m:Line.
(((l \# m) \mwedge{} (\mforall{}L,M,N:Line. (L \mbackslash{}/ M {}\mRightarrow{} (L \mbackslash{}/ N \mvee{} M \mbackslash{}/ N)))) {}\mRightarrow{} (m \# l))
Date html generated:
2020_05_20-AM-10_47_06
Last ObjectModification:
2020_01_13-PM-06_05_08
Theory : euclidean!plane!geometry
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