Theorem cbvesumv

Description: Change bound variable in an extended sum. (Contributed by Thierry Arnoux, 19-Jun-2017)

	Ref	Expression
Hypothesis	cbvesum.1	$$⊢j=k\to B=C$$
Assertion	cbvesumv	$$⊢\underset{j\in A}{{\sum }^{*}}B=\underset{k\in A}{{\sum }^{*}}C$$

Proof

Step	Hyp	Ref	Expression
1		cbvesum.1	$$⊢j=k\to B=C$$
2	1	cbvmptv	$$⊢\left(j\in A⟼B\right)=\left(k\in A⟼C\right)$$
3	2	oveq2i	$$⊢\left({ℝ}_{𝑠}^{*}{\upharpoonright }_{𝑠}\left[0,\mathrm{+\infty }\right]\right)\mathrm{tsums}\left(j\in A⟼B\right)=\left({ℝ}_{𝑠}^{*}{\upharpoonright }_{𝑠}\left[0,\mathrm{+\infty }\right]\right)\mathrm{tsums}\left(k\in A⟼C\right)$$
4	3	unieqi	$$⊢\bigcup \left(\left({ℝ}_{𝑠}^{*}{\upharpoonright }_{𝑠}\left[0,\mathrm{+\infty }\right]\right)\mathrm{tsums}\left(j\in A⟼B\right)\right)=\bigcup \left(\left({ℝ}_{𝑠}^{*}{\upharpoonright }_{𝑠}\left[0,\mathrm{+\infty }\right]\right)\mathrm{tsums}\left(k\in A⟼C\right)\right)$$
5		df-esum	$$⊢\underset{j\in A}{{\sum }^{*}}B=\bigcup \left(\left({ℝ}_{𝑠}^{*}{\upharpoonright }_{𝑠}\left[0,\mathrm{+\infty }\right]\right)\mathrm{tsums}\left(j\in A⟼B\right)\right)$$
6		df-esum	$$⊢\underset{k\in A}{{\sum }^{*}}C=\bigcup \left(\left({ℝ}_{𝑠}^{*}{\upharpoonright }_{𝑠}\left[0,\mathrm{+\infty }\right]\right)\mathrm{tsums}\left(k\in A⟼C\right)\right)$$
7	4 5 6	3eqtr4i	$$⊢\underset{j\in A}{{\sum }^{*}}B=\underset{k\in A}{{\sum }^{*}}C$$