Nuprl Lemma : perp-aux2
∀e:HeytingGeometry. ∀a,b,c,p,q,r:Point. (a # bc ⇒ a-p-b ⇒ c-q-b ⇒ a-r-c ⇒ p # qr)
Proof
Definitions occuring in Statement :
geo-triangle: a # bc,
heyting-geometry: HeytingGeometry,
geo-strict-between: a-b-c,
geo-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
exists: ∃x:A. B[x],
and: P ∧ Q,
heyting-geometry: HeytingGeometry,
euclidean-plane: EuclideanPlane,
basic-geometry: BasicGeometry,
geo-colinear-set: geo-colinear-set(e; L),
l_all: (∀x∈L.P[x]),
top: Top,
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,
cand: A c∧ B,
basic-geometry-: BasicGeometry-
Lemmas referenced :
geo-strict-between_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
heyting-geometry-subtype,
subtype_rel_transitivity,
heyting-geometry_wf,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-triangle_wf,
geo-point_wf,
perp-aux1,
geo-triangle-colinear,
geo-strict-between-sep3,
geo-colinear-is-colinear-set,
geo-strict-between-implies-colinear,
subtype_rel_self,
basic-geo-axioms_wf,
geo-left-axioms_wf,
length_of_cons_lemma,
length_of_nil_lemma,
false_wf,
lelt_wf,
geo-triangle-symmetry,
geo-strict-between-sep2,
geo-sep-sym,
geo-strict-between-sep1
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
applyEquality,
hypothesis,
instantiate,
independent_isectElimination,
sqequalRule,
because_Cache,
dependent_functionElimination,
independent_functionElimination,
productElimination,
setEquality,
productEquality,
cumulativity,
isect_memberEquality,
voidElimination,
voidEquality,
dependent_set_memberEquality,
natural_numberEquality,
independent_pairFormation,
imageMemberEquality,
baseClosed
Latex:
\mforall{}e:HeytingGeometry. \mforall{}a,b,c,p,q,r:Point. (a \# bc {}\mRightarrow{} a-p-b {}\mRightarrow{} c-q-b {}\mRightarrow{} a-r-c {}\mRightarrow{} p \# qr)
Date html generated:
2017_10_02-PM-07_07_45
Last ObjectModification:
2017_08_10-PM-03_38_38
Theory : euclidean!plane!geometry
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