Nuprl Lemma : geo-Aparallel_weakening2
∀g:EuclideanParPlane. ∀l,m:LINE. ((l = m ∈ (l,m:Line//l || m)) ⇒ l || m)
Proof
Definitions occuring in Statement :
euclidean-parallel-plane: EuclideanParPlane,
geo-Aparallel: l || m,
geoline: LINE,
geo-line: Line,
quotient: x,y:A//B[x; y],
all: ∀x:A. B[x],
implies: P ⇒ Q,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
geo-Aparallel: l || m,
not: ¬A,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
euclidean-parallel-plane: EuclideanParPlane,
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
quotient: x,y:A//B[x; y],
false: False,
and: P ∧ Q,
cand: A c∧ B
Lemmas referenced :
geo-intersect_wf,
equal_wf,
quotient_wf,
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-Aparallel_wf,
geoline-subtype1,
geo-Aparallel-equiv,
geoline-subtype2,
geoline_wf,
quotient_subtype_quotient,
geo-Aparallel-equiv2,
equal_functionality_wrt_subtype_rel2,
member_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
because_Cache,
hypothesis,
hypothesisEquality,
dependent_functionElimination,
applyEquality,
instantiate,
independent_isectElimination,
sqequalRule,
lambdaEquality,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
pertypeElimination,
productElimination,
voidElimination,
productEquality
Latex:
\mforall{}g:EuclideanParPlane. \mforall{}l,m:LINE. ((l = m) {}\mRightarrow{} l || m)
Date html generated:
2018_05_22-PM-01_11_38
Last ObjectModification:
2018_05_14-AM-11_56_37
Theory : euclidean!plane!geometry
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