Nuprl Lemma : real-ball-1
∀[r:{r:ℝ| r0 ≤ r} ]. B(1;r) ≡ {x:ℝ^1| x 0 ∈ [-(r), r]}
Proof
Definitions occuring in Statement :
real-ball: B(n;r),
real-vec: ℝ^n,
rccint: [l, u],
i-member: r ∈ I,
rleq: x ≤ y,
rminus: -(x),
int-to-real: r(n),
real: ℝ,
ext-eq: A ≡ B,
uall: ∀[x:A]. B[x],
set: {x:A| B[x]} ,
apply: f a,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
ext-eq: A ≡ B,
and: P ∧ Q,
subtype_rel: A ⊆r B,
real-ball: B(n;r),
real-vec: ℝ^n,
int_seg: {i..j-},
lelt: i ≤ j < k,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
prop: ℙ,
false: False,
nat: ℕ,
le: A ≤ B,
less_than': less_than'(a;b),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
i-member_wf,
rccint_wf,
rminus_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,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
istype-le,
istype-less_than,
real-ball_wf,
rleq_wf,
real-vec-norm_wf,
real-vec_wf,
real_wf,
int-to-real_wf,
rabs_wf,
member_rccint_lemma,
iff_transitivity,
iff_weakening_uiff,
rleq_functionality,
real-vec-norm-dim1,
req_weakening,
rabs-rleq-iff
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
independent_pairFormation,
lambdaEquality_alt,
sqequalHypSubstitution,
setElimination,
thin,
rename,
dependent_set_memberEquality_alt,
hypothesisEquality,
universeIsType,
extract_by_obid,
isectElimination,
because_Cache,
hypothesis,
applyEquality,
natural_numberEquality,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
isect_memberEquality_alt,
voidElimination,
sqequalRule,
productIsType,
lambdaFormation_alt,
setIsType,
productElimination,
independent_pairEquality,
axiomEquality,
productEquality
Latex:
\mforall{}[r:\{r:\mBbbR{}| r0 \mleq{} r\} ]. B(1;r) \mequiv{} \{x:\mBbbR{}\^{}1| x 0 \mmember{} [-(r), r]\}
Date html generated:
2019_10_30-AM-10_14_51
Last ObjectModification:
2019_06_28-PM-01_52_10
Theory : real!vectors
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