Nuprl Lemma : real-vec-norm-diff-squared
∀[n:ℕ]. ∀[x,y:ℝ^n]. (||x - y||^2 = ((||x||^2 + ||y||^2) + (r(-2) * x⋅y)))
Proof
Definitions occuring in Statement :
real-vec-norm: ||x||,
dot-product: x⋅y,
real-vec-sub: X - Y,
real-vec: ℝ^n,
rnexp: x^k1,
req: x = y,
rmul: a * b,
radd: a + b,
int-to-real: r(n),
nat: ℕ,
uall: ∀[x:A]. B[x],
minus: -n,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
implies: P ⇒ Q,
prop: ℙ,
uimplies: b supposing a,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
rsub: x - y
Lemmas referenced :
req_witness,
rnexp_wf,
false_wf,
le_wf,
real-vec-norm_wf,
real-vec-sub_wf,
radd_wf,
rmul_wf,
int-to-real_wf,
dot-product_wf,
real-vec_wf,
nat_wf,
rsub_wf,
req_functionality,
req_transitivity,
real-vec-norm-squared,
dot-product-linearity1-sub,
rsub_functionality,
radd_functionality,
req_weakening,
dot-product-comm,
req_wf,
rminus_wf,
uiff_transitivity,
rminus-radd,
req_inversion,
radd-assoc,
radd_comm,
rmul_functionality,
rminus-as-rmul,
rmul-distrib2,
radd-int,
rminus-rminus
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
dependent_set_memberEquality,
natural_numberEquality,
sqequalRule,
independent_pairFormation,
lambdaFormation,
hypothesis,
hypothesisEquality,
because_Cache,
minusEquality,
independent_functionElimination,
isect_memberEquality,
independent_isectElimination,
productElimination,
addEquality
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[x,y:\mBbbR{}\^{}n]. (||x - y||\^{}2 = ((||x||\^{}2 + ||y||\^{}2) + (r(-2) * x\mcdot{}y)))
Date html generated:
2016_10_26-AM-10_21_23
Last ObjectModification:
2016_09_30-PM-00_25_05
Theory : reals
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