Nuprl Lemma : geo-intersection-unicity
∀e:BasicGeometry. ∀a,b,c,d,p,q:Point.
((¬Colinear(a;b;c)) ⇒ c ≠ d ⇒ Colinear(a;b;p) ⇒ Colinear(a;b;q) ⇒ Colinear(c;d;p) ⇒ Colinear(c;d;q) ⇒ p ≡ q)
Proof
Definitions occuring in Statement :
basic-geometry: BasicGeometry,
geo-colinear: Colinear(a;b;c),
geo-eq: a ≡ b,
geo-sep: a ≠ b,
geo-point: Point,
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q
Definitions unfolded in proof :
or: P ∨ Q,
stable: Stable{P},
geo-eq: a ≡ b,
false: False,
not: ¬A,
uimplies: b supposing a,
guard: {T},
subtype_rel: A ⊆r B,
basic-geometry: BasicGeometry,
uall: ∀[x:A]. B[x],
prop: ℙ,
member: t ∈ T,
implies: P ⇒ Q,
all: ∀x:A. B[x],
geo-colinear: Colinear(a;b;c),
subtract: n - m,
cons: [a / b],
select: L[n],
true: True,
squash: ↓T,
less_than: a < b,
less_than': less_than'(a;b),
le: A ≤ B,
lelt: i ≤ j < k,
int_seg: {i..j-},
l_all: (∀x∈L.P[x]),
geo-colinear-set: geo-colinear-set(e; L),
so_apply: x[s1;s2;s3],
top: Top,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
append: as @ bs,
so_apply: x[s],
so_lambda: λ2x.t[x],
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
cand: A c∧ B,
exists: ∃x:A. B[x],
and: P ∧ Q
Lemmas referenced :
minimal-not-not-excluded-middle,
minimal-double-negation-hyp-elim,
or_wf,
false_wf,
stable__not,
geo-point_wf,
not_wf,
Error :basic-geo-primitives_wf,
Error :basic-geo-structure_wf,
basic-geometry_wf,
subtype_rel_transitivity,
basic-geometry-subtype,
geo-sep_wf,
geo-colinear_wf,
geo-eq_weakening,
geo-colinear_functionality,
geo-colinear-same,
lelt_wf,
length_of_nil_lemma,
length_of_cons_lemma,
list_ind_nil_lemma,
list_ind_cons_lemma,
geo-colinear-is-colinear-set,
exists_wf,
equal_wf,
l_member_wf,
cons_member,
nil_wf,
cons_wf,
geo-colinear-append,
geo-between_wf
Rules used in proof :
unionElimination,
voidElimination,
independent_functionElimination,
functionEquality,
because_Cache,
sqequalRule,
independent_isectElimination,
instantiate,
applyEquality,
hypothesis,
hypothesisEquality,
rename,
setElimination,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
impliesLevelFunctionality,
promote_hyp,
levelHypothesis,
impliesFunctionality,
addLevel,
baseClosed,
imageMemberEquality,
natural_numberEquality,
dependent_set_memberEquality,
voidEquality,
isect_memberEquality,
lambdaEquality,
inlFormation,
inrFormation,
productElimination,
independent_pairFormation,
dependent_pairFormation,
dependent_functionElimination,
productEquality
Latex:
\mforall{}e:BasicGeometry. \mforall{}a,b,c,d,p,q:Point.
((\mneg{}Colinear(a;b;c))
{}\mRightarrow{} c \mneq{} d
{}\mRightarrow{} Colinear(a;b;p)
{}\mRightarrow{} Colinear(a;b;q)
{}\mRightarrow{} Colinear(c;d;p)
{}\mRightarrow{} Colinear(c;d;q)
{}\mRightarrow{} p \mequiv{} q)
Date html generated:
2017_10_02-PM-06_20_33
Last ObjectModification:
2017_08_05-PM-04_15_10
Theory : euclidean!plane!geometry
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