Nuprl Lemma : geo-Aparallel-trans-lines
∀e:EuclideanParPlane. ∀l,m,n:Line. (l || m ⇒ m || n ⇒ l || n)
Proof
Definitions occuring in Statement :
euclidean-parallel-plane: EuclideanParPlane,
geo-Aparallel: l || m,
geo-line: Line,
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
geo-Aparallel: l || m,
not: ¬A,
member: t ∈ T,
iff: P ⇐⇒ Q,
and: P ∧ Q,
exists: ∃x:A. B[x],
cand: A c∧ B,
prop: ℙ,
uall: ∀[x:A]. B[x],
euclidean-parallel-plane: EuclideanParPlane,
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
geoline: LINE,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
false: False
Lemmas referenced :
geo-intersect-iff2,
geo-playfair-axiom,
geo-Aparallel_sym,
geo-intersect_wf,
geo-Aparallel_wf,
geoline-subtype1,
geo-line_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
euclidean-planes-subtype,
subtype_rel_transitivity,
euclidean-parallel-plane_wf,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-strict-between-incident,
quotient-member-eq,
geo-line-eq_wf,
geo-line-eq-equiv,
geo-intersect-irreflexive,
and_wf,
equal_wf,
geoline_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
hypothesis,
addLevel,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
because_Cache,
productElimination,
independent_functionElimination,
hypothesisEquality,
independent_pairFormation,
levelHypothesis,
isectElimination,
setElimination,
rename,
applyEquality,
sqequalRule,
instantiate,
independent_isectElimination,
lambdaEquality,
equalityTransitivity,
equalitySymmetry,
hyp_replacement,
dependent_set_memberEquality,
applyLambdaEquality,
voidElimination
Latex:
\mforall{}e:EuclideanParPlane. \mforall{}l,m,n:Line. (l || m {}\mRightarrow{} m || n {}\mRightarrow{} l || n)
Date html generated:
2018_05_22-PM-01_10_43
Last ObjectModification:
2018_05_11-PM-11_14_10
Theory : euclidean!plane!geometry
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