Nuprl Lemma : geo-non-parallel_wf
∀[g:EuclideanParPlane]. ∀[l:Line]. ∀[L:l,m:Line//l || m]. (geo-non-parallel(g;l;L) ∈ ℙ)
Proof
Definitions occuring in Statement :
geo-non-parallel: geo-non-parallel(g;l;L),
euclidean-parallel-plane: EuclideanParPlane,
geo-Aparallel: l || m,
geo-line: Line,
quotient: x,y:A//B[x; y],
uall: ∀[x:A]. B[x],
prop: ℙ,
member: t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
geo-non-parallel: geo-non-parallel(g;l;L),
so_lambda: λ2x.t[x],
implies: P ⇒ Q,
prop: ℙ,
so_lambda: λ2x y.t[x; y],
subtype_rel: A ⊆r B,
so_apply: x[s1;s2],
uimplies: b supposing a,
all: ∀x:A. B[x],
and: P ∧ Q,
so_apply: x[s],
exists: ∃x:A. B[x],
guard: {T}
Lemmas referenced :
squash_wf,
all_wf,
equal_wf,
quotient_wf,
geo-Aparallel_wf,
euclidean-planes-subtype,
geoline-subtype1,
geo-Aparallel-equiv,
subtype_quotient,
geo-line_wf,
exists_wf,
geo-point_wf,
geo-incident_wf,
euclidean-plane-structure-subtype,
subtype_rel_transitivity,
euclidean-parallel-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
lambdaEquality,
functionEquality,
hypothesisEquality,
applyEquality,
hypothesis,
independent_isectElimination,
dependent_functionElimination,
productEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
instantiate,
isect_memberEquality
Latex:
\mforall{}[g:EuclideanParPlane]. \mforall{}[l:Line]. \mforall{}[L:l,m:Line//l || m]. (geo-non-parallel(g;l;L) \mmember{} \mBbbP{})
Date html generated:
2018_05_22-PM-01_11_59
Last ObjectModification:
2018_05_11-PM-01_16_28
Theory : euclidean!plane!geometry
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