Nuprl Lemma : Euclid-Prop18
∀e:EuclideanPlane. ∀a,b,c:Point. (a # bc ⇒ |ab| < |ac| ⇒ bca < abc)
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,
uall: ∀[x:A]. B[x],
basic-geometry: BasicGeometry,
euclidean-plane: EuclideanPlane,
prop: ℙ,
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
exists: ∃x:A. B[x],
and: P ∧ Q,
cand: A c∧ 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,
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,
geo-strict-between: a-b-c,
uiff: uiff(P;Q),
geo-lt-angle: abc < xyz,
geo-out: out(p ab)
Lemmas referenced :
geo-lt_wf,
geo-length_wf,
geo-mk-seg_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,
geo-lt-implies-gt-strong-1,
Euclid-prop16,
colinear-lsep-cycle,
lsep-all-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-between-symmetry,
geo-sep-sym,
geo-congruent-sep,
lsep-implies-sep,
Euclid-Prop5_1-not-tri,
geo-congruent-iff-length,
Euclid-Prop18-lemma,
geo-between-out,
geo-cong-angle-refl,
geo-out_weakening,
geo-eq_weakening,
out-preserves-angle-cong_1,
geo-lt-angle-symm2,
geo-cong-angle-preserves-lt-angle,
geo-lt-angle-symm,
geo-lt-angle-trans
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
universeIsType,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
sqequalRule,
hypothesisEquality,
setElimination,
rename,
hypothesis,
because_Cache,
applyEquality,
instantiate,
independent_isectElimination,
inhabitedIsType,
productElimination,
dependent_functionElimination,
independent_functionElimination,
isect_memberEquality_alt,
voidElimination,
dependent_set_memberEquality_alt,
natural_numberEquality,
independent_pairFormation,
unionElimination,
approximateComputation,
dependent_pairFormation_alt,
lambdaEquality_alt,
productIsType,
equalitySymmetry
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a,b,c:Point. (a \# bc {}\mRightarrow{} |ab| < |ac| {}\mRightarrow{} bca < abc)
Date html generated:
2019_10_16-PM-02_15_33
Last ObjectModification:
2019_09_12-AM-11_39_29
Theory : euclidean!plane!geometry
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