Step
*
of Lemma
sum-reindex2
∀[n:ℕ]. ∀[a:ℕn ⟶ ℤ]. ∀[m:ℕ]. ∀[b:ℕm ⟶ ℤ]. ∀[f:{i:ℕn| ¬(a[i] = 0 ∈ ℤ)} ⟶ {j:ℕm| ¬(b[j] = 0 ∈ ℤ)} ].
(Σ(a[i] | i < n) = Σ(b[j] | j < m) ∈ ℤ) supposing
((∀i:{i:ℕn| ¬(a[i] = 0 ∈ ℤ)} . (a[i] = b[f i] ∈ ℤ)) and
Bij({i:ℕn| ¬(a[i] = 0 ∈ ℤ)} ;{j:ℕm| ¬(b[j] = 0 ∈ ℤ)} ;f))
BY
{ Auto }
1
1. n : ℕ
2. a : ℕn ⟶ ℤ
3. m : ℕ
4. b : ℕm ⟶ ℤ
5. f : {i:ℕn| ¬(a[i] = 0 ∈ ℤ)} ⟶ {j:ℕm| ¬(b[j] = 0 ∈ ℤ)}
6. Bij({i:ℕn| ¬(a[i] = 0 ∈ ℤ)} ;{j:ℕm| ¬(b[j] = 0 ∈ ℤ)} ;f)
7. ∀i:{i:ℕn| ¬(a[i] = 0 ∈ ℤ)} . (a[i] = b[f i] ∈ ℤ)
⊢ Σ(a[i] | i < n) = Σ(b[j] | j < m) ∈ ℤ
Latex:
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[a:\mBbbN{}n {}\mrightarrow{} \mBbbZ{}]. \mforall{}[m:\mBbbN{}]. \mforall{}[b:\mBbbN{}m {}\mrightarrow{} \mBbbZ{}]. \mforall{}[f:\{i:\mBbbN{}n| \mneg{}(a[i] = 0)\} {}\mrightarrow{} \{j:\mBbbN{}m| \mneg{}(b[j] = 0)\} ].
(\mSigma{}(a[i] | i < n) = \mSigma{}(b[j] | j < m)) supposing
((\mforall{}i:\{i:\mBbbN{}n| \mneg{}(a[i] = 0)\} . (a[i] = b[f i])) and
Bij(\{i:\mBbbN{}n| \mneg{}(a[i] = 0)\} ;\{j:\mBbbN{}m| \mneg{}(b[j] = 0)\} ;f))
By
Latex:
Auto
Home
Index