Nuprl Lemma : Prop22-inequality-to-triangle-construction
∀e:EuclideanPlane. ∀a,b,c:Point.
(|ac| < |ab| + |bc|
⇒ |ab| < |ac| + |bc|
⇒ |bc| < |ac| + |ab|
⇒ (∃c1,c2:Point. ((ac ≅ ac1 ∧ bc2 > bc1) ∧ bc ≅ bc2 ∧ ac1 > ac2)))
Proof
Definitions occuring in Statement :
geo-lt: p < q,
geo-add-length: p + q,
geo-length: |s|,
geo-mk-seg: ab,
euclidean-plane: EuclideanPlane,
geo-gt: cd > ab,
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,
euclidean-plane: EuclideanPlane,
exists: ∃x:A. B[x],
uall: ∀[x:A]. B[x],
basic-geometry: BasicGeometry,
prop: ℙ,
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
squash: ↓T,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
cand: A c∧ B,
uiff: uiff(P;Q),
basic-geometry-: BasicGeometry-,
geo-strict-between: a-b-c,
or: P ∨ Q,
l_member: (x ∈ l),
nat: ℕ,
decidable: Dec(P),
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
false: False,
select: L[n],
cons: [a / b],
subtract: n - m,
less_than: a < b,
less_than': less_than'(a;b),
ge: i ≥ j ,
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
geo-colinear-set: geo-colinear-set(e; L),
l_all: (∀x∈L.P[x]),
int_seg: {i..j-},
lelt: i ≤ j < k,
geo-gt: cd > ab,
geo-eq: a ≡ b
Lemmas referenced :
geo-proper-extend-exists,
geo-O_wf,
geo-X_wf,
geo-sep-sym,
geo-sep-O-X,
geo-lt_wf,
geo-length_wf,
geo-mk-seg_wf,
geo-add-length_wf,
geo-point_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
squash_wf,
true_wf,
geo-length-type_wf,
basic-geometry_wf,
geo-add-length-comm,
subtype_rel_self,
iff_weakening_equal,
geo-strict-between-sep3,
geo-triangle-inequality-lt-sep,
extend-using-SC,
geo-congruent-iff-length,
geo-between-symmetry,
geo-strict-between-implies-between,
geo-add-length-between,
geo-lt-out-to-between,
geo-out-iff-between1,
geo-strict-between-sep1,
extended-out-preserves-between,
geo-between-outer-trans,
geo-between-exchange4,
geo-between-implies-out2,
geo-strict-between-sym,
geo-length-flip,
geo-length_wf1,
geo-add-length_wf1,
geo-zero-point-sep-iff-sep,
geo-lt-sep,
geo-length-equality,
geo-lt-add1-iff,
geo-between-inner-trans,
geo-out_inversion,
geo-between-out,
geo-strict-between-sep2,
euclidean-plane-axioms,
geo-out_transitivity,
geo-lt-add1-iff2,
geo-add-length-assoc,
equal_wf,
istype-universe,
geo-sep-or,
geo-sep_wf,
colinear-implies-midpoint,
geo-colinear-append,
cons_wf,
nil_wf,
decidable__le,
full-omega-unsat,
intformnot_wf,
intformle_wf,
itermConstant_wf,
istype-int,
int_formula_prop_not_lemma,
istype-void,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
istype-le,
length_of_cons_lemma,
length_of_nil_lemma,
istype-less_than,
length_wf,
select_wf,
nat_properties,
intformand_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_term_value_var_lemma,
l_member_wf,
geo-colinear-is-colinear-set,
geo-strict-between-implies-colinear,
list_ind_cons_lemma,
list_ind_nil_lemma,
geo-between-implies-colinear,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
midpoint-sep,
geo-between-same-side-or,
geo-between-exchange3,
geo-congruent_wf,
geo-gt_wf,
geo-congruent-refl,
geo-between_wf,
geo-construction-unicity,
congruence-preserves-between_symmetric-points2,
geo-between-same,
geo-construction-unicity-from-first,
geo-eq_inversion,
geo-congruent-preserves-gt,
geo-between-sep,
geo-between-outer-trans2,
geo-congruent-sep
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
sqequalHypSubstitution,
productElimination,
thin,
introduction,
extract_by_obid,
dependent_functionElimination,
because_Cache,
setElimination,
rename,
hypothesis,
independent_functionElimination,
hypothesisEquality,
universeIsType,
isectElimination,
sqequalRule,
inhabitedIsType,
applyEquality,
instantiate,
independent_isectElimination,
lambdaEquality_alt,
imageElimination,
equalityTransitivity,
equalitySymmetry,
natural_numberEquality,
imageMemberEquality,
baseClosed,
universeEquality,
independent_pairFormation,
dependent_set_memberEquality_alt,
productIsType,
equalityIstype,
applyLambdaEquality,
hyp_replacement,
unionElimination,
dependent_pairFormation_alt,
approximateComputation,
isect_memberEquality_alt,
voidElimination,
int_eqEquality
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a,b,c:Point.
(|ac| < |ab| + |bc|
{}\mRightarrow{} |ab| < |ac| + |bc|
{}\mRightarrow{} |bc| < |ac| + |ab|
{}\mRightarrow{} (\mexists{}c1,c2:Point. ((ac \mcong{} ac1 \mwedge{} bc2 > bc1) \mwedge{} bc \mcong{} bc2 \mwedge{} ac1 > ac2)))
Date html generated:
2019_10_16-PM-02_23_02
Last ObjectModification:
2019_02_26-PM-02_33_53
Theory : euclidean!plane!geometry
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