Nuprl Lemma : Euclid-Prop17
∀g:EuclideanPlane. ∀a,b,c,i,j,k:Point. (a # bc ⇒ acb + abc ≅ ijk ⇒ (∀a1,a2,a3:Point. (a1-a2-a3 ⇒ ijk < a1a2a3)))
Proof
Definitions occuring in Statement :
hp-angle-sum: abc + xyz ≅ def,
geo-lt-angle: abc < xyz,
euclidean-plane: EuclideanPlane,
geo-lsep: a # bc,
geo-strict-between: a-b-c,
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,
basic-geometry: BasicGeometry,
euclidean-plane: EuclideanPlane,
guard: {T},
and: P ∧ Q,
exists: ∃x:A. B[x],
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
prop: ℙ,
cand: A c∧ B,
basic-geometry-: BasicGeometry-,
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
Lemmas referenced :
geo-proper-extend-exists,
geo-O_wf,
geo-X_wf,
lsep-implies-sep,
geo-sep-O-X,
Euclid-prop16,
geo-lt-angle-symm2,
geo-strict-between_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
hp-angle-sum_wf,
geo-lsep_wf,
geo-point_wf,
Euclid-prop13,
colinear-lsep-cycle,
lsep-all-sym,
geo-sep-sym,
geo-strict-between-sep1,
geo-colinear-is-colinear-set,
geo-strict-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-strict-between-sep3,
hp-angle-sum-subst4,
straight-angles-congruent,
geo-cong-angle-refl,
lsep-symmetry2,
hp-angle-sum-lt2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
sqequalRule,
hypothesisEquality,
setElimination,
rename,
hypothesis,
because_Cache,
independent_functionElimination,
productElimination,
universeIsType,
isectElimination,
applyEquality,
instantiate,
independent_isectElimination,
inhabitedIsType,
isect_memberEquality_alt,
voidElimination,
dependent_set_memberEquality_alt,
natural_numberEquality,
independent_pairFormation,
unionElimination,
approximateComputation,
dependent_pairFormation_alt,
lambdaEquality_alt,
productIsType
Latex:
\mforall{}g:EuclideanPlane. \mforall{}a,b,c,i,j,k:Point.
(a \# bc {}\mRightarrow{} acb + abc \mcong{} ijk {}\mRightarrow{} (\mforall{}a1,a2,a3:Point. (a1-a2-a3 {}\mRightarrow{} ijk < a1a2a3)))
Date html generated:
2019_10_16-PM-02_32_50
Last ObjectModification:
2019_09_12-AM-11_55_22
Theory : euclidean!plane!geometry
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