Nuprl Lemma : geo-incident-line
∀[e:EuclideanPlane]. ∀[p:Point]. ∀[l:Line]. uiff(p I l;Colinear(p;fst(l);fst(snd(l))))
Proof
Definitions occuring in Statement :
geo-incident: p I L,
geo-line: Line,
euclidean-plane: EuclideanPlane,
geo-colinear: Colinear(a;b;c),
geo-point: Point,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
pi1: fst(t),
pi2: snd(t)
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
geoline: LINE,
member: t ∈ T,
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
uiff: uiff(P;Q),
and: P ∧ Q,
geo-colinear: Colinear(a;b;c),
not: ¬A,
implies: P ⇒ Q,
false: False,
geo-line: Line,
pi1: fst(t),
pi2: snd(t),
prop: ℙ,
geo-incident: p I L,
quotient: x,y:A//B[x; y],
top: Top,
sq_stable: SqStable(P),
squash: ↓T,
oriented-plane: OrientedPlane,
exists: ∃x:A. B[x],
cand: A c∧ B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
or: P ∨ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
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),
less_than: a < b,
true: True,
select: L[n],
cons: [a / b],
subtract: n - m
Lemmas referenced :
subtype_quotient,
geo-line_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-line-eq_wf,
geo-line-eq-equiv,
not_wf,
geo-between_wf,
geo-incident_wf,
equal_wf,
geoline_wf,
geo-colinear_wf,
geo-point_wf,
trivial-equal,
member_wf,
sq_stable__colinear,
pi1_wf_top,
geo-line-eq-to-col,
oriented-colinear-append,
cons_wf,
nil_wf,
cons_member,
l_member_wf,
geo-sep_wf,
exists_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
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
cut,
sqequalRule,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
dependent_functionElimination,
hypothesisEquality,
applyEquality,
hypothesis,
instantiate,
independent_isectElimination,
lambdaEquality,
because_Cache,
independent_pairFormation,
productEquality,
productElimination,
lambdaFormation,
rename,
voidElimination,
independent_pairEquality,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
pertypeElimination,
voidEquality,
imageMemberEquality,
baseClosed,
imageElimination,
dependent_pairFormation,
inrFormation,
inlFormation,
dependent_set_memberEquality,
natural_numberEquality
Latex:
\mforall{}[e:EuclideanPlane]. \mforall{}[p:Point]. \mforall{}[l:Line]. uiff(p I l;Colinear(p;fst(l);fst(snd(l))))
Date html generated:
2018_05_22-PM-01_03_42
Last ObjectModification:
2018_05_10-PM-05_17_54
Theory : euclidean!plane!geometry
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