Nuprl Lemma : geo-triangle-same-line
∀e:HeytingGeometry. ∀x,a:Point. ∀b:{b:Point| a ≠ b} . ∀c:Point. ∀d:{d:Point| c ≠ d} .
(line(a;b)=line(c;d) ⇒ {x # ab ⇐⇒ x # cd})
Proof
Definitions occuring in Statement :
geo-triangle: a # bc,
heyting-geometry: HeytingGeometry,
geo-same-line: line(a;b)=line(c;d),
geo-sep: a ≠ b,
geo-point: Point,
guard: {T},
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
set: {x:A| B[x]}
Definitions unfolded in proof :
member: t ∈ T,
all: ∀x:A. B[x],
so_apply: x[s],
so_lambda: λ2x.t[x],
uimplies: b supposing a,
guard: {T},
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
prop: ℙ,
squash: ↓T,
and: P ∧ Q,
geo-same-line: line(a;b)=line(c;d),
sq_stable: SqStable(P),
heyting-geometry: Error :heyting-geometry,
implies: P ⇒ Q,
or: P ∨ Q,
so_apply: x[s1;s2;s3],
so_lambda: so_lambda(x,y,z.t[x; y; z]),
append: as @ bs,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
exists: ∃x:A. B[x],
subtract: n - m,
cons: [a / b],
select: L[n],
true: True,
less_than: a < b,
not: ¬A,
false: False,
less_than': less_than'(a;b),
le: A ≤ B,
lelt: i ≤ j < k,
int_seg: {i..j-},
top: Top,
l_all: (∀x∈L.P[x]),
geo-colinear-set: geo-colinear-set(e; L),
cand: A c∧ B
Lemmas referenced :
Error :heyting-geometry_wf,
geo-sep_wf,
Error :basic-geo-primitives_wf,
Error :basic-geo-structure_wf,
basic-geometry_wf,
subtype_rel_transitivity,
basic-geometry-subtype,
geo-point_wf,
set_wf,
heyting-geometry-subtype,
geo-same-line_wf,
Error :geo-triangle_wf,
Error :sq_stable__geo-triangle,
geo-sep-or,
list_ind_nil_lemma,
list_ind_cons_lemma,
exists_wf,
equal_wf,
l_member_wf,
cons_member,
nil_wf,
cons_wf,
geo-colinear-append,
lelt_wf,
false_wf,
length_of_nil_lemma,
length_of_cons_lemma,
geo-colinear-is-colinear-set,
geo-sep-sym,
geo-triangle-colinear,
geo-triangle-symmetry,
geo-same-line-symmetry
Rules used in proof :
hypothesis,
extract_by_obid,
introduction,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
because_Cache,
lambdaEquality,
independent_isectElimination,
instantiate,
applyEquality,
isectElimination,
imageElimination,
baseClosed,
imageMemberEquality,
sqequalRule,
productElimination,
independent_functionElimination,
hypothesisEquality,
dependent_functionElimination,
sqequalHypSubstitution,
rename,
thin,
setElimination,
unionElimination,
dependent_set_memberEquality,
productEquality,
inrFormation,
inlFormation,
dependent_pairFormation,
independent_pairFormation,
natural_numberEquality,
voidEquality,
voidElimination,
isect_memberEquality
Latex:
\mforall{}e:HeytingGeometry. \mforall{}x,a:Point. \mforall{}b:\{b:Point| a \mneq{} b\} . \mforall{}c:Point. \mforall{}d:\{d:Point| c \mneq{} d\} .
(line(a;b)=line(c;d) {}\mRightarrow{} \{x \# ab \mLeftarrow{}{}\mRightarrow{} x \# cd\})
Date html generated:
2017_10_02-PM-07_03_11
Last ObjectModification:
2017_08_08-PM-00_39_42
Theory : euclidean!plane!geometry
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