Nuprl Lemma : isosc-colinear-mid-exists
∀e:HeytingGeometry. ∀a,b,c,m,a',b',m':Point.
(c # ab ⇒ ac ≅ bc ⇒ (out(c aa') ∧ out(c bb')) ⇒ a=m=b ⇒ a'c ≅ b'c ⇒ (∃m':Point. (out(c m'm) ∧ a'=m'=b')))
Proof
Definitions occuring in Statement :
geo-triangle: a # bc,
heyting-geometry: HeytingGeometry,
geo-out: out(p ab),
geo-midpoint: a=m=b,
geo-congruent: ab ≅ cd,
geo-point: Point,
all: ∀x:A. B[x],
exists: ∃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,
member: t ∈ T,
cand: A c∧ B,
guard: {T},
exists: ∃x:A. B[x],
geo-triangle: a # bc,
subtype_rel: A ⊆r B,
heyting-geometry: HeytingGeometry,
euclidean-plane: EuclideanPlane,
oriented-plane: OrientedPlane,
uall: ∀[x:A]. B[x],
prop: ℙ,
uimplies: b supposing a,
geo-midpoint: a=m=b,
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,
geo-lsep: a # bc,
or: P ∨ Q,
geo-out: out(p ab),
basic-geometry: BasicGeometry,
stable: Stable{P},
geo-eq: a ≡ b,
iff: P ⇐⇒ Q,
uiff: uiff(P;Q),
rev_implies: P ⇐ Q
Lemmas referenced :
geo-out-triangle,
geo-triangle-symmetry,
isosceles-mid-exists,
lsep-colinear-sep,
subtype_rel_self,
euclidean-plane-structure_wf,
basic-geo-axioms_wf,
geo-left-axioms_wf,
lsep-all-sym,
geo-colinear-is-colinear-set,
geo-between-implies-colinear,
length_of_cons_lemma,
length_of_nil_lemma,
false_wf,
lelt_wf,
not_wf,
geo-between_wf,
geo-out_wf,
geo-midpoint_wf,
geo-congruent_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
heyting-geometry-subtype,
subtype_rel_transitivity,
heyting-geometry_wf,
euclidean-plane_wf,
geo-primitives_wf,
geo-triangle_wf,
geo-point_wf,
stable__not,
or_wf,
geo-sep_wf,
minimal-double-negation-hyp-elim,
geo-midpoint_functionality,
geo-eq_weakening,
geo-out_functionality,
geo-congruent_functionality,
geo-triangle_functionality,
minimal-not-not-excluded-middle,
geo-congruent-iff-length,
geo-length-flip,
geo-out-cong-cong,
isosc-bisectors-between-ns,
geo-between-symmetry,
geo-add-length-between,
equal_wf,
geo-length-type_wf,
and_wf,
geo-add-length_wf,
geo-length_wf,
geo-mk-seg_wf,
squash_wf,
true_wf,
basic-geometry_wf,
iff_weakening_equal,
geo-add-length-implies-eq-zero,
geo-add-length-comm,
geo-add-length-assoc,
geo-add-length-is-zero,
geo-zero-length-iff,
geo-congruent-symmetry,
geo-congruent-sep,
isosc-bisectors-between_1-ns,
geo-out-unicity,
at-most-one-midpoint,
geo-between-trivial,
geo-between_functionality,
geo-sep_functionality
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,
because_Cache,
dependent_pairFormation,
applyEquality,
sqequalRule,
instantiate,
isectElimination,
setEquality,
productEquality,
cumulativity,
independent_isectElimination,
isect_memberEquality,
voidElimination,
voidEquality,
dependent_set_memberEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
functionEquality,
unionElimination,
equalityTransitivity,
equalitySymmetry,
hyp_replacement,
applyLambdaEquality,
setElimination,
rename,
lambdaEquality,
imageElimination,
universeEquality,
addLevel,
impliesFunctionality,
levelHypothesis,
promote_hyp,
impliesLevelFunctionality
Latex:
\mforall{}e:HeytingGeometry. \mforall{}a,b,c,m,a',b',m':Point.
(c \# ab
{}\mRightarrow{} ac \00D0 bc
{}\mRightarrow{} (out(c aa') \mwedge{} out(c bb'))
{}\mRightarrow{} a=m=b
{}\mRightarrow{} a'c \00D0 b'c
{}\mRightarrow{} (\mexists{}m':Point. (out(c m'm) \mwedge{} a'=m'=b')))
Date html generated:
2017_10_02-PM-07_07_10
Last ObjectModification:
2017_08_10-PM-05_27_03
Theory : euclidean!plane!geometry
Home
Index