Nuprl Lemma : p_inf-l_eu-sep-exists
∀eu:EuclideanParPlane. ∀p:l,m:Line//l || m. ∀m:Line. ((m = p ∈ (l,m:Line//l || m)) ⇒ (∃L:Line. (¬m || L)))
Proof
Definitions occuring in Statement :
euclidean-parallel-plane: EuclideanParPlane,
geo-Aparallel: l || m,
geo-line: Line,
quotient: x,y:A//B[x; y],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
so_lambda: λ2x y.t[x; y],
euclidean-parallel-plane: EuclideanParPlane,
so_apply: x[s1;s2],
geo-line: Line,
exists: ∃x:A. B[x],
geo-equilateral: EQΔ(a;b;c),
and: P ∧ Q,
not: ¬A,
geo-Aparallel: l || m,
false: False,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
pi1: fst(t),
pi2: snd(t),
geo-lsep: a # bc,
or: P ∨ Q,
cand: A c∧ B,
geo-colinear-set: geo-colinear-set(e; L),
l_all: (∀x∈L.P[x]),
select: L[n],
cons: [a / b],
subtract: n - m,
so_lambda: λ2x.t[x],
so_apply: x[s],
geo-strict-between: a-b-c
Lemmas referenced :
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,
subtype_quotient,
Euclid-Prop1,
lsep-implies-sep,
geo-sep_wf,
geo-point_wf,
not_wf,
geo-intersect-lines,
left-symmetry,
euclidean-plane-subtype-basic,
basic-geometry_wf,
geo-sep-sym,
geo-colinear-is-colinear-set,
geo-strict-between-implies-colinear,
geo-colinear-same,
geo-colinear_wf,
geo-left_wf,
exists_wf,
geo-proper-extend-exists,
left-between-implies-right2,
euclidean-plane-subtype-oriented,
oriented-plane_wf,
left-between-implies-right1
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
dependent_functionElimination,
hypothesisEquality,
applyEquality,
hypothesis,
instantiate,
independent_isectElimination,
sqequalRule,
lambdaEquality,
setElimination,
rename,
because_Cache,
productElimination,
independent_functionElimination,
dependent_pairFormation,
dependent_pairEquality,
productEquality,
voidElimination,
unionElimination,
independent_pairFormation
Latex:
\mforall{}eu:EuclideanParPlane. \mforall{}p:l,m:Line//l || m. \mforall{}m:Line. ((m = p) {}\mRightarrow{} (\mexists{}L:Line. (\mneg{}m || L)))
Date html generated:
2018_05_22-PM-01_15_37
Last ObjectModification:
2018_05_12-AM-09_04_57
Theory : euclidean!plane!geometry
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