Nuprl Lemma : geo-intersect-all-parallel2
∀e:EuclideanPlane. ∀L,M,N:Line. ((L \/ N ∧ (∀l,m:LINE. (l \/ m ⇒ (∀n:LINE. (l \/ n ∨ m \/ n))))) ⇒ L || M ⇒ M \/ N)
Proof
Definitions occuring in Statement :
geo-Aparallel: l || m,
geo-intersect: L \/ M,
geoline: LINE,
geo-line: Line,
euclidean-plane: EuclideanPlane,
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,
prop: ℙ,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
or: P ∨ Q,
guard: {T},
uimplies: b supposing a,
geo-Aparallel: l || m,
not: ¬A,
false: False
Lemmas referenced :
geo-Aparallel_wf,
geoline-subtype1,
geo-intersect_wf,
all_wf,
geoline_wf,
or_wf,
geo-line_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-intersect-symmetry
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
cut,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
applyEquality,
hypothesis,
sqequalRule,
productEquality,
lambdaEquality,
functionEquality,
dependent_functionElimination,
instantiate,
independent_isectElimination,
because_Cache,
independent_functionElimination,
unionElimination,
voidElimination
Latex:
\mforall{}e:EuclideanPlane. \mforall{}L,M,N:Line.
((L \mbackslash{}/ N \mwedge{} (\mforall{}l,m:LINE. (l \mbackslash{}/ m {}\mRightarrow{} (\mforall{}n:LINE. (l \mbackslash{}/ n \mvee{} m \mbackslash{}/ n))))) {}\mRightarrow{} L || M {}\mRightarrow{} M \mbackslash{}/ N)
Date html generated:
2018_05_22-PM-01_12_45
Last ObjectModification:
2018_05_16-PM-02_27_00
Theory : euclidean!plane!geometry
Home
Index