Nuprl Lemma : real-vec-norm-mul
∀[n:ℕ]. ∀[x:ℝ^n]. ∀[a:ℝ]. (||a*x|| = (|a| * ||x||))
Proof
Definitions occuring in Statement :
real-vec-norm: ||x||,
real-vec-mul: a*X,
real-vec: ℝ^n,
rabs: |x|,
req: x = y,
rmul: a * b,
real: ℝ,
nat: ℕ,
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
nat_plus: ℕ+,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
and: P ∧ Q,
prop: ℙ,
implies: P ⇒ Q,
uimplies: b supposing a,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
nat: ℕ,
le: A ≤ B,
false: False,
not: ¬A,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
itermConstant: "const",
req_int_terms: t1 ≡ t2,
top: Top
Lemmas referenced :
rnexp-req-iff,
less_than_wf,
real-vec-norm_wf,
real-vec-mul_wf,
rmul_wf,
rabs_wf,
real-vec-norm-nonneg,
rmul-nonneg-case1,
zero-rleq-rabs,
req_witness,
real_wf,
real-vec_wf,
nat_wf,
rnexp_wf,
false_wf,
le_wf,
dot-product_wf,
req_functionality,
real-vec-norm-squared,
req_weakening,
req_wf,
uiff_transitivity,
req_transitivity,
dot-product-linearity2,
rmul_functionality,
rmul-assoc,
req_inversion,
rnexp2,
square-nonneg,
real_term_polynomial,
itermSubtract_wf,
itermMultiply_wf,
itermVar_wf,
int-to-real_wf,
real_term_value_const_lemma,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
req-iff-rsub-is-0,
rabs-rmul,
rabs-of-nonneg
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
dependent_set_memberEquality,
natural_numberEquality,
sqequalRule,
independent_pairFormation,
imageMemberEquality,
hypothesisEquality,
baseClosed,
hypothesis,
isectElimination,
independent_functionElimination,
independent_isectElimination,
because_Cache,
productElimination,
isect_memberEquality,
lambdaFormation,
computeAll,
lambdaEquality,
int_eqEquality,
intEquality,
voidElimination,
voidEquality
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[x:\mBbbR{}\^{}n]. \mforall{}[a:\mBbbR{}]. (||a*x|| = (|a| * ||x||))
Date html generated:
2017_10_03-AM-10_49_56
Last ObjectModification:
2017_07_28-AM-08_20_26
Theory : reals
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