Nuprl Lemma : perp-aux-separations
∀e:HeytingGeometry. ∀a,b,c,x,c1,c',p:Point.
((((c # ba ∧ ab ⊥x cx) ∧ a ≠ x)
∧ ((c=a=c1 ∧ c=x=c') ∧ c'a ≅ ca)
∧ c' # c1a
∧ ((a # cc' ∧ c1=p=c') ∧ ab ⊥a pa)
∧ p # ab)
⇒ ((c' ≠ c1 ∧ c ≠ x ∧ (c' ≠ x ∧ c1 ≠ p) ∧ c' ≠ p ∧ c ≠ c1) ∧ c # c'c1))
Proof
Definitions occuring in Statement :
geo-triangle: a # bc,
heyting-geometry: HeytingGeometry,
geo-perp-in: ab ⊥x cd,
geo-midpoint: a=m=b,
geo-congruent: ab ≅ cd,
geo-sep: a ≠ b,
geo-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
cand: A c∧ B,
member: t ∈ T,
guard: {T},
subtype_rel: A ⊆r B,
heyting-geometry: HeytingGeometry,
euclidean-plane: EuclideanPlane,
basic-geometry: BasicGeometry,
uall: ∀[x:A]. B[x],
prop: ℙ,
geo-midpoint: a=m=b,
uimplies: b supposing a,
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
Lemmas referenced :
geo-triangle-property,
midpoint-sep,
subtype_rel_self,
euclidean-plane-structure_wf,
basic-geo-axioms_wf,
euclidean-plane-structure-subtype,
geo-left-axioms_wf,
geo-sep-sym,
geo-between-sep,
geo-triangle-colinear,
geo-triangle-symmetry,
geo-colinear-is-colinear-set,
geo-between-implies-colinear,
length_of_cons_lemma,
length_of_nil_lemma,
false_wf,
lelt_wf,
geo-triangle_wf,
euclidean-plane-subtype,
heyting-geometry-subtype,
subtype_rel_transitivity,
heyting-geometry_wf,
euclidean-plane_wf,
geo-primitives_wf,
geo-perp-in_wf,
geo-sep_wf,
geo-midpoint_wf,
geo-congruent_wf,
geo-point_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
cut,
introduction,
extract_by_obid,
dependent_functionElimination,
hypothesisEquality,
independent_functionElimination,
hypothesis,
independent_pairFormation,
applyEquality,
sqequalRule,
instantiate,
isectElimination,
setEquality,
productEquality,
cumulativity,
because_Cache,
independent_isectElimination,
isect_memberEquality,
voidElimination,
voidEquality,
dependent_set_memberEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed
Latex:
\mforall{}e:HeytingGeometry. \mforall{}a,b,c,x,c1,c',p:Point.
((((c \# ba \mwedge{} ab \mbot{}x cx) \mwedge{} a \mneq{} x)
\mwedge{} ((c=a=c1 \mwedge{} c=x=c') \mwedge{} c'a \00D0 ca)
\mwedge{} c' \# c1a
\mwedge{} ((a \# cc' \mwedge{} c1=p=c') \mwedge{} ab \mbot{}a pa)
\mwedge{} p \# ab)
{}\mRightarrow{} ((c' \mneq{} c1 \mwedge{} c \mneq{} x \mwedge{} (c' \mneq{} x \mwedge{} c1 \mneq{} p) \mwedge{} c' \mneq{} p \mwedge{} c \mneq{} c1) \mwedge{} c \# c'c1))
Date html generated:
2017_10_02-PM-07_07_54
Last ObjectModification:
2017_08_16-AM-11_04_52
Theory : euclidean!plane!geometry
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