Nuprl Lemma : real-vec-norm-squared
∀[n:ℕ]. ∀[x:ℝ^n]. (||x||^2 = x ⋅ x)
Proof
Definitions occuring in Statement :
real-vec-norm: ||x||,
dot-product: x ⋅ y,
real-vec: ℝ^n,
rnexp: x^k1,
req: x = y,
nat: ℕ,
uall: ∀[x:A]. B[x],
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
real-vec: ℝ^n,
nat: ℕ,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
implies: P ⇒ Q,
prop: ℙ,
real-vec-norm: ||x||,
uimplies: b supposing a,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
req_witness,
rnexp_wf,
false_wf,
le_wf,
real-vec-norm_wf,
dot-product_wf,
real-vec_wf,
nat_wf,
rmul_wf,
rsqrt_squared,
dot-product-nonneg,
rleq_wf,
int-to-real_wf,
req_functionality,
rnexp2,
req_weakening
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
lemma_by_obid,
isectElimination,
thin,
dependent_set_memberEquality,
natural_numberEquality,
sqequalRule,
independent_pairFormation,
lambdaFormation,
hypothesis,
hypothesisEquality,
independent_functionElimination,
isect_memberEquality,
because_Cache,
independent_isectElimination,
productElimination
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[x:\mBbbR{}\^{}n]. (||x||\^{}2 = x \mcdot{} x)
Date html generated:
2016_05_18-AM-09_48_48
Last ObjectModification:
2015_12_27-PM-11_11_25
Theory : reals
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