Nuprl Lemma : real-vec-norm-positive-iff2
∀n:ℕ. ∀x:ℝ^n. (∃i:ℕn. x i ≠ r0 ⇐⇒ r0 < ||x||)
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],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
exists: ∃x:A. B[x],
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
real-vec: ℝ^n,
nat: ℕ,
so_lambda: λ2x.t[x],
so_apply: x[s],
rev_implies: P ⇐ Q
Lemmas referenced :
rneq-symmetry,
rneq_wf,
int-to-real_wf,
exists_wf,
int_seg_wf,
real-vec_wf,
nat_wf,
real-vec-norm-positive-iff,
rless_wf,
real-vec-norm_wf,
all_wf,
iff_wf
Rules used in proof :
cut,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
independent_pairFormation,
sqequalHypSubstitution,
productElimination,
thin,
dependent_pairFormation,
hypothesisEquality,
introduction,
extract_by_obid,
dependent_functionElimination,
because_Cache,
independent_functionElimination,
hypothesis,
isectElimination,
natural_numberEquality,
applyEquality,
setElimination,
rename,
sqequalRule,
lambdaEquality,
addLevel,
allFunctionality,
impliesFunctionality
Latex:
\mforall{}n:\mBbbN{}. \mforall{}x:\mBbbR{}\^{}n. (\mexists{}i:\mBbbN{}n. x i \mneq{} r0 \mLeftarrow{}{}\mRightarrow{} r0 < ||x||)
Date html generated:
2017_10_03-AM-10_49_22
Last ObjectModification:
2017_06_18-PM-05_34_35
Theory : reals
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