Nuprl Lemma : real-ball-0
∀[r:{r:ℝ| r0 ≤ r} ]. B(0;r) ≡ Top
Proof
Definitions occuring in Statement :
real-ball: B(n;r),
rleq: x ≤ y,
int-to-real: r(n),
real: ℝ,
ext-eq: A ≡ B,
uall: ∀[x:A]. B[x],
top: Top,
set: {x:A| B[x]} ,
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,
top: Top,
nat: ℕ,
le: A ≤ B,
less_than': less_than'(a;b),
not: ¬A,
implies: P ⇒ Q,
false: False,
real-ball: B(n;r),
real-vec: ℝ^n,
int_seg: {i..j-},
lelt: i ≤ j < k,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
all: ∀x:A. B[x],
prop: ℙ,
real-vec-norm: ||x||,
dot-product: x⋅y,
subtract: n - m,
so_lambda: λ2x.t[x],
so_apply: x[s],
less_than: a < b,
squash: ↓T,
true: True,
sq_stable: SqStable(P),
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
istype-void,
real-ball_wf,
istype-le,
int_seg_properties,
full-omega-unsat,
intformand_wf,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
istype-int,
int_formula_prop_and_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
int_seg_wf,
rleq_wf,
real-vec-norm_wf,
istype-top,
real_wf,
int-to-real_wf,
rsum-empty,
rsqrt_wf,
rleq_weakening_equal,
sq_stable__rleq,
rleq_functionality,
rsqrt0,
req_weakening
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
independent_pairFormation,
lambdaEquality_alt,
isect_memberEquality_alt,
voidElimination,
extract_by_obid,
hypothesis,
universeIsType,
sqequalHypSubstitution,
isectElimination,
thin,
dependent_set_memberEquality_alt,
natural_numberEquality,
sqequalRule,
lambdaFormation_alt,
hypothesisEquality,
setElimination,
rename,
functionExtensionality,
productElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
int_eqEquality,
dependent_functionElimination,
because_Cache,
independent_pairEquality,
axiomEquality,
applyEquality,
setIsType,
minusEquality,
imageMemberEquality,
baseClosed,
imageElimination
Latex:
\mforall{}[r:\{r:\mBbbR{}| r0 \mleq{} r\} ]. B(0;r) \mequiv{} Top
Date html generated:
2019_10_30-AM-10_14_49
Last ObjectModification:
2019_06_28-PM-01_52_09
Theory : real!vectors
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