Nuprl Lemma : not-parallel-implies
∀eu:EuclideanParPlane. ∀l,m,n:Line. ((¬l || m) ⇒ (¬(l || n ∧ m || n)))
Proof
Definitions occuring in Statement :
euclidean-parallel-plane: EuclideanParPlane,
geo-Aparallel: l || m,
geo-line: Line,
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
not: ¬A,
false: False,
and: P ∧ Q,
member: t ∈ T,
guard: {T},
subtype_rel: A ⊆r B,
prop: ℙ,
uall: ∀[x:A]. B[x],
uimplies: b supposing a
Lemmas referenced :
geo-Aparallel_inversion,
euclidean-planes-subtype,
geo-Aparallel_wf,
euclidean-plane-structure-subtype,
subtype_rel_transitivity,
euclidean-parallel-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
not_wf,
geo-line_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
thin,
sqequalHypSubstitution,
productElimination,
hypothesis,
independent_functionElimination,
introduction,
extract_by_obid,
dependent_functionElimination,
hypothesisEquality,
applyEquality,
sqequalRule,
voidElimination,
productEquality,
isectElimination,
instantiate,
independent_isectElimination,
because_Cache
Latex:
\mforall{}eu:EuclideanParPlane. \mforall{}l,m,n:Line. ((\mneg{}l || m) {}\mRightarrow{} (\mneg{}(l || n \mwedge{} m || n)))
Date html generated:
2018_05_22-PM-01_15_23
Last ObjectModification:
2018_04_16-AM-10_59_58
Theory : euclidean!plane!geometry
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