Nuprl Lemma : real-vec-norm-positive-iff
∀n:ℕ. ∀x:ℝ^n. (r0 < ||x|| ⇐⇒ ∃i:ℕn. r0 ≠ x i)
Proof
Definitions occuring in Statement :
real-vec-norm: ||x||,
real-vec: ℝ^n,
rneq: x ≠ y,
rless: x < y,
int-to-real: r(n),
int_seg: {i..j-},
nat: ℕ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
apply: f a,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
real-vec-norm: ||x||,
member: t ∈ T,
uall: ∀[x:A]. B[x],
dot-product: x⋅y,
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
nat: ℕ,
so_lambda: λ2x.t[x],
real-vec: ℝ^n,
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
less_than: a < b,
squash: ↓T,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
prop: ℙ,
so_apply: x[s],
rev_implies: P ⇐ Q,
rless: x < y,
sq_exists: ∃x:A [B[x]],
nat_plus: ℕ+,
rneq: x ≠ y,
cand: A c∧ B,
subtype_rel: A ⊆r B
Lemmas referenced :
real-vec_wf,
istype-nat,
rless_wf,
int-to-real_wf,
rsum_wf,
subtract_wf,
rmul_wf,
int_seg_properties,
nat_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
decidable__lt,
intformless_wf,
itermAdd_wf,
itermSubtract_wf,
int_formula_prop_less_lemma,
int_term_value_add_lemma,
int_term_value_subtract_lemma,
istype-le,
istype-less_than,
int_seg_wf,
rneq_wf,
rsum-positive-implies,
nat_plus_properties,
subtract-add-cancel,
rabs_wf,
square-nonneg,
rless_functionality,
req_weakening,
rabs-of-nonneg,
rmul-is-positive,
rsum-of-nonneg-positive-iff,
rsqrt-positive-iff,
dot-product-nonneg,
dot-product_wf,
rleq_wf,
rsqrt_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
universeIsType,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
independent_pairFormation,
natural_numberEquality,
setElimination,
rename,
sqequalRule,
lambdaEquality_alt,
applyEquality,
dependent_set_memberEquality_alt,
productElimination,
imageElimination,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
productIsType,
because_Cache,
addEquality,
inlFormation_alt,
inrFormation_alt
Latex:
\mforall{}n:\mBbbN{}. \mforall{}x:\mBbbR{}\^{}n. (r0 < ||x|| \mLeftarrow{}{}\mRightarrow{} \mexists{}i:\mBbbN{}n. r0 \mneq{} x i)
Date html generated:
2019_10_30-AM-08_06_42
Last ObjectModification:
2019_04_02-AM-09_54_10
Theory : reals
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