Nuprl Lemma : geo-intersect-lines-iff
∀e:EuclideanPlane. ∀p,l:Line. (p \/ l ⇐⇒ geo-line-sep(e;p;l) ∧ (∃x:Point. (x I p ∧ x I l)))
Proof
Definitions occuring in Statement :
geo-intersect: L \/ M,
geo-incident: p I L,
geo-line-sep: geo-line-sep(g;l;m),
geo-line: Line,
euclidean-plane: EuclideanPlane,
geo-point: Point,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
rev_implies: P ⇐ Q,
guard: {T},
uimplies: b supposing a,
so_lambda: λ2x.t[x],
so_apply: x[s],
exists: ∃x:A. B[x],
geo-line-sep: geo-line-sep(g;l;m),
cand: A c∧ B,
geo-line: Line,
pi1: fst(t),
pi2: snd(t),
geo-incident: p I L,
oriented-plane: OrientedPlane,
uiff: uiff(P;Q),
or: P ∨ Q,
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
top: Top,
so_apply: x[s1;s2;s3],
geo-colinear-set: geo-colinear-set(e; L),
l_all: (∀x∈L.P[x]),
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
less_than: a < b,
squash: ↓T,
true: True,
select: L[n],
cons: [a / b],
subtract: n - m,
basic-geometry: BasicGeometry,
geo-lsep: a # bc,
geo-midpoint: a=m=b,
geo-strict-between: a-b-c
Lemmas referenced :
geo-intersect_wf,
geoline-subtype1,
geo-line-sep_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
exists_wf,
geo-point_wf,
geo-incident_wf,
geo-line_wf,
geo-intersect-iff2,
geo-colinear_wf,
geo-lsep_wf,
lsep-all-sym2,
colinear-lsep-general,
geo-incident-line,
geo-sep_wf,
oriented-colinear-append,
cons_wf,
nil_wf,
cons_member,
l_member_wf,
equal_wf,
geo-colinear-is-colinear-set,
list_ind_cons_lemma,
list_ind_nil_lemma,
length_of_cons_lemma,
length_of_nil_lemma,
false_wf,
lelt_wf,
geo-strict-between-incident,
geo-strict-between_wf,
geoline_wf,
uall_wf,
pi1_wf_top,
pi2_wf,
subtype_rel_product,
top_wf,
geo-intersect-lines,
lsep-colinear-sep,
all_wf,
symmetric-point-construction,
geo-sep-sym,
geo-between-implies-colinear,
geo-left_wf,
geo-congruent-symmetry,
geo-congruent-sep,
strict-between-left-right,
geo-colinear-symmetry
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
independent_pairFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
applyEquality,
hypothesis,
sqequalRule,
because_Cache,
productElimination,
productEquality,
instantiate,
independent_isectElimination,
lambdaEquality,
dependent_functionElimination,
independent_functionElimination,
dependent_pairFormation,
equalityTransitivity,
equalitySymmetry,
dependent_pairEquality,
inrFormation,
inlFormation,
isect_memberEquality,
voidElimination,
voidEquality,
dependent_set_memberEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
functionEquality,
addLevel,
existsFunctionality,
andLevelFunctionality,
independent_pairEquality,
levelHypothesis,
promote_hyp,
rename,
existsLevelFunctionality,
unionElimination
Latex:
\mforall{}e:EuclideanPlane. \mforall{}p,l:Line. (p \mbackslash{}/ l \mLeftarrow{}{}\mRightarrow{} geo-line-sep(e;p;l) \mwedge{} (\mexists{}x:Point. (x I p \mwedge{} x I l)))
Date html generated:
2018_05_22-PM-01_06_24
Last ObjectModification:
2018_05_12-AM-10_59_56
Theory : euclidean!plane!geometry
Home
Index