Nuprl Lemma : common-P_point-parallel-P_lines
∀e:EuclideanParPlane. ∀P:P_point(e). ∀L:P_line(e).
((¬P_point-line-sep(e;P;L))
⇒ (fst(snd(P)) || fst(snd(snd(P))) ∧ (∀l,m,n:Line. (l \/ m ⇒ (l \/ n ∨ m \/ n))))
⇒ fst(L) || fst(snd(snd(P))))
Proof
Definitions occuring in Statement :
P_point-line-sep: P_point-line-sep(e;P;L),
P_line: P_line(eu),
P_point: P_point(eu),
euclidean-parallel-plane: EuclideanParPlane,
geo-Aparallel: l || m,
geo-intersect: L \/ M,
geo-line: Line,
pi1: fst(t),
pi2: snd(t),
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
or: P ∨ Q,
and: P ∧ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
P_point: P_point(eu),
pi2: snd(t),
pi1: fst(t),
geo-Aparallel: l || m,
not: ¬A,
P_line: P_line(eu),
P_point-line-sep: P_point-line-sep(e;P;L),
false: False,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
euclidean-parallel-plane: EuclideanParPlane,
subtype_rel: A ⊆r B,
top: Top,
guard: {T},
uimplies: b supposing a,
so_lambda: λ2x.t[x],
or: P ∨ Q,
so_apply: x[s],
cand: A c∧ B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
exists: ∃x:A. B[x],
geo-line-sep: (l # m),
geo-line: Line,
geo-plsep: p # l,
geo-incident: p I L
Lemmas referenced :
geo-intersect_wf,
pi1_wf_top,
geoline_wf,
geoline-subtype1,
geo-Aparallel_wf,
all_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,
or_wf,
not_wf,
P_point-line-sep_wf,
P_line_wf,
P_point_wf,
set_wf,
geo-point_wf,
geo-incident_wf,
geo-intersect-symmetry,
geo-intersect-lines-iff,
geo-colinear_wf,
geo-lsep_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
sqequalRule,
independent_functionElimination,
voidElimination,
cut,
introduction,
extract_by_obid,
isectElimination,
setElimination,
rename,
hypothesisEquality,
hypothesis,
because_Cache,
independent_pairEquality,
applyEquality,
isect_memberEquality,
voidEquality,
productEquality,
dependent_functionElimination,
instantiate,
independent_isectElimination,
lambdaEquality,
functionEquality,
unionElimination,
independent_pairFormation,
inlFormation,
inrFormation,
dependent_pairFormation,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}e:EuclideanParPlane. \mforall{}P:P\_point(e). \mforall{}L:P\_line(e).
((\mneg{}P\_point-line-sep(e;P;L))
{}\mRightarrow{} (fst(snd(P)) || fst(snd(snd(P))) \mwedge{} (\mforall{}l,m,n:Line. (l \mbackslash{}/ m {}\mRightarrow{} (l \mbackslash{}/ n \mvee{} m \mbackslash{}/ n))))
{}\mRightarrow{} fst(L) || fst(snd(snd(P))))
Date html generated:
2019_10_16-PM-03_04_11
Last ObjectModification:
2018_08_14-PM-02_07_47
Theory : euclidean!plane!geometry
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