Nuprl Lemma : rsum_product
∀[a,b,c,d:ℤ]. ∀[x:{a..b + 1-} ⟶ ℝ]. ∀[y:{c..d + 1-} ⟶ ℝ].
((Σ{x[i] | a≤i≤b} * Σ{y[j] | c≤j≤d}) = Σ{Σ{x[i] * y[j] | c≤j≤d} | a≤i≤b})
Proof
Definitions occuring in Statement :
rsum: Σ{x[k] | n≤k≤m},
req: x = y,
rmul: a * b,
real: ℝ,
int_seg: {i..j-},
uall: ∀[x:A]. B[x],
so_apply: x[s],
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
so_apply: x[s],
implies: P ⇒ Q,
uimplies: b supposing a,
uiff: uiff(P;Q),
and: P ∧ Q,
rev_uimplies: rev_uimplies(P;Q),
pointwise-req: x[k] = y[k] for k ∈ [n,m],
all: ∀x:A. B[x],
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
prop: ℙ,
less_than: a < b,
squash: ↓T
Latex:
\mforall{}[a,b,c,d:\mBbbZ{}]. \mforall{}[x:\{a..b + 1\msupminus{}\} {}\mrightarrow{} \mBbbR{}]. \mforall{}[y:\{c..d + 1\msupminus{}\} {}\mrightarrow{} \mBbbR{}].
((\mSigma{}\{x[i] | a\mleq{}i\mleq{}b\} * \mSigma{}\{y[j] | c\mleq{}j\mleq{}d\}) = \mSigma{}\{\mSigma{}\{x[i] * y[j] | c\mleq{}j\mleq{}d\} | a\mleq{}i\mleq{}b\})
Date html generated:
2020_05_20-AM-11_11_47
Last ObjectModification:
2020_01_02-PM-02_11_55
Theory : reals
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