Nuprl Lemma : Euclid-Prop19
∀e:EuclideanPlane. ∀a,b,c:Point. (a # bc ⇒ bca < abc ⇒ |ab| < |ac|)
Proof
Definitions occuring in Statement :
geo-lt-angle: abc < xyz,
geo-lt: p < q,
geo-length: |s|,
geo-mk-seg: ab,
euclidean-plane: EuclideanPlane,
geo-lsep: a # bc,
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,
guard: {T},
and: P ∧ Q,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
prop: ℙ,
sq_exists: ∃x:A [B[x]],
euclidean-plane: EuclideanPlane,
sq_stable: SqStable(P),
squash: ↓T,
exists: ∃x:A. B[x],
basic-geometry: BasicGeometry,
geo-midpoint: a=m=b,
cand: A c∧ 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,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
select: L[n],
cons: [a / b],
subtract: n - m,
basic-geometry-: BasicGeometry-,
uiff: uiff(P;Q),
geo-cong-angle: abc ≅a xyz,
geo-lsep: a # bc,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
Euclid-midpoint,
lsep-implies-sep,
geo-sep_wf,
sq_stable__midpoint,
geo-midpoint_wf,
midpoint-sep,
colinear-lsep,
lsep-all-sym,
geo-sep-sym,
geo-colinear-is-colinear-set,
geo-between-implies-colinear,
length_of_cons_lemma,
istype-void,
length_of_nil_lemma,
decidable__le,
full-omega-unsat,
intformnot_wf,
intformle_wf,
itermConstant_wf,
istype-int,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
istype-le,
istype-less_than,
geo-lt-angle_wf,
geo-lsep_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-point_wf,
colinear-implies-midpoint,
geo-strict-between-sep1,
geo-strict-between-implies-colinear,
geo-congruent-iff-length,
geo-length-flip,
geo-midpoint-diagonals-congruent,
geo-midpoint-symmetry,
geo-congruent_wf,
geo-proper-extend-exists,
geo-between-sep,
geo-between-trivial,
geo-congruent-right-comm,
geo-between_wf,
geo-out_weakening,
geo-congruent-sep,
geo-eq_weakening,
geo-between-out,
geo-congruent-symmetry,
geo-between-symmetry,
euclidean-plane-axioms,
geo-out_inversion,
left-implies-sep,
geo-cong-angle-symm2,
lsep-symmetry,
out-preserves-angle-cong_1,
out-cong-angle,
geo-cong-angle-preserves-lt-angle2,
geo-cong-angle-preserves-lt-angle,
Euclid-Prop9-with-between,
colinear-lsep-cycle,
Euclid-Prop19-lemma2,
geo-lt-angle-symm2,
geo-cong-angle_wf,
geo-strict-between_wf,
geo-lt_wf,
geo-length_wf,
geo-mk-seg_wf,
Euclid-Prop19-lemma1,
squash_wf,
true_wf,
geo-length-type_wf,
basic-geometry_wf,
subtype_rel_self,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
because_Cache,
independent_functionElimination,
hypothesis,
productElimination,
dependent_set_memberEquality_alt,
universeIsType,
isectElimination,
applyEquality,
sqequalRule,
setElimination,
rename,
imageMemberEquality,
baseClosed,
imageElimination,
dependent_pairFormation_alt,
independent_isectElimination,
isect_memberEquality_alt,
voidElimination,
natural_numberEquality,
independent_pairFormation,
unionElimination,
approximateComputation,
lambdaEquality_alt,
productIsType,
instantiate,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
universeEquality
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a,b,c:Point. (a \# bc {}\mRightarrow{} bca < abc {}\mRightarrow{} |ab| < |ac|)
Date html generated:
2019_10_16-PM-02_19_09
Last ObjectModification:
2019_09_12-AM-11_43_02
Theory : euclidean!plane!geometry
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