Nuprl Lemma : real-vec-norm-is-0
∀[n:ℕ]. ∀[x:ℝ^n]. uiff(||x|| = r0;req-vec(n;x;λi.r0))
Proof
Definitions occuring in Statement :
real-vec-norm: ||x||,
req-vec: req-vec(n;x;y),
real-vec: ℝ^n,
req: x = y,
int-to-real: r(n),
nat: ℕ,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
lambda: λx.A[x],
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
req-vec: req-vec(n;x;y),
all: ∀x:A. B[x],
real-vec: ℝ^n,
implies: P ⇒ Q,
prop: ℙ,
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
nat: ℕ,
iff: P ⇐⇒ Q,
rev_uimplies: rev_uimplies(P;Q),
not: ¬A,
false: False,
less_than': less_than'(a;b),
cand: A c∧ B,
so_apply: x[s],
so_lambda: λ2x.t[x],
dot-product: x⋅y
Lemmas referenced :
req_witness,
int-to-real_wf,
req_wf,
real-vec-norm_wf,
req-vec_wf,
int_seg_wf,
real-vec_wf,
istype-nat,
rabs-is-zero,
rleq_antisymmetry,
rabs_wf,
zero-rleq-rabs,
component-rleq-real-vec-norm,
rleq_functionality,
req_weakening,
req_inversion,
dot-product_functionality,
req_functionality,
le_wf,
false_wf,
rnexp_wf,
dot-product_wf,
rleq_weakening_equal,
real-vec-norm-eq-iff,
rsum-constant,
rmul_wf,
subtract_wf,
rsum_wf,
rmul-zero-both,
rmul-int,
rmul_functionality,
uiff_transitivity,
rnexp2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
independent_pairFormation,
sqequalRule,
sqequalHypSubstitution,
lambdaEquality_alt,
dependent_functionElimination,
thin,
hypothesisEquality,
extract_by_obid,
isectElimination,
applyEquality,
closedConclusion,
natural_numberEquality,
hypothesis,
independent_functionElimination,
functionIsTypeImplies,
inhabitedIsType,
universeIsType,
setElimination,
rename,
productElimination,
independent_pairEquality,
isect_memberEquality_alt,
because_Cache,
isectIsTypeImplies,
lambdaFormation_alt,
independent_isectElimination,
lambdaFormation,
dependent_set_memberEquality,
lambdaEquality,
addEquality,
multiplyEquality
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[x:\mBbbR{}\^{}n]. uiff(||x|| = r0;req-vec(n;x;\mlambda{}i.r0))
Date html generated:
2019_10_30-AM-08_07_34
Last ObjectModification:
2019_06_26-PM-00_48_18
Theory : reals
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