Nuprl Lemma : geo-same-line-transitivity
∀[e:BasicGeometry]. ∀[a:Point]. ∀[b:{b:Point| a ≠ b} ]. ∀[c:Point]. ∀[d:{d:Point| c ≠ d} ]. ∀[x:Point]. ∀[y:{y:Point|
x ≠ y} ].
(line(a;b)=line(c;d) ⇒ line(c;d)=line(x;y) ⇒ line(a;b)=line(x;y))
Proof
Definitions occuring in Statement :
geo-same-line: line(a;b)=line(c;d),
basic-geometry: BasicGeometry,
geo-sep: a ≠ b,
geo-point: Point,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]}
Definitions unfolded in proof :
so_apply: x[s],
so_lambda: λ2x.t[x],
uimplies: b supposing a,
guard: {T},
subtype_rel: A ⊆r B,
false: False,
not: ¬A,
geo-colinear: Colinear(a;b;c),
and: P ∧ Q,
geo-same-line: line(a;b)=line(c;d),
prop: ℙ,
squash: ↓T,
sq_stable: SqStable(P),
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
basic-geometry: BasicGeometry,
subtract: n - m,
cons: [a / b],
select: L[n],
true: True,
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,
or: P ∨ Q,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
exists: ∃x:A. B[x],
cand: A c∧ B
Lemmas referenced :
geo-sep_wf,
geo-point_wf,
set_wf,
Error :basic-geo-primitives_wf,
Error :basic-geo-structure_wf,
basic-geometry_wf,
subtype_rel_transitivity,
basic-geometry-subtype,
geo-between_wf,
not_wf,
geo-same-line_wf,
squash_wf,
sq_stable__colinear,
geo-colinear_wf,
sq_stable__and,
lelt_wf,
false_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
Rules used in proof :
voidElimination,
isect_memberEquality,
independent_isectElimination,
instantiate,
applyEquality,
productEquality,
because_Cache,
independent_pairEquality,
productElimination,
dependent_functionElimination,
lambdaEquality,
isectElimination,
extract_by_obid,
imageElimination,
hypothesis,
baseClosed,
hypothesisEquality,
imageMemberEquality,
sqequalRule,
independent_functionElimination,
sqequalHypSubstitution,
rename,
thin,
setElimination,
lambdaFormation,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
natural_numberEquality,
dependent_set_memberEquality,
voidEquality,
inlFormation,
inrFormation,
independent_pairFormation,
dependent_pairFormation
Latex:
\mforall{}[e:BasicGeometry]. \mforall{}[a:Point]. \mforall{}[b:\{b:Point| a \mneq{} b\} ]. \mforall{}[c:Point]. \mforall{}[d:\{d:Point| c \mneq{} d\} ].
\mforall{}[x:Point]. \mforall{}[y:\{y:Point| x \mneq{} y\} ].
(line(a;b)=line(c;d) {}\mRightarrow{} line(c;d)=line(x;y) {}\mRightarrow{} line(a;b)=line(x;y))
Date html generated:
2017_10_02-PM-06_22_52
Last ObjectModification:
2017_08_05-PM-04_17_04
Theory : euclidean!plane!geometry
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