Theorem cbvesum

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

	Ref	Expression
Hypotheses	cbvesum.1	$$⊢j=k\to B=C$$
	cbvesum.2	$$⊢\underset{\_}{Ⅎ}kA$$
	cbvesum.3	$$⊢\underset{\_}{Ⅎ}jA$$
	cbvesum.4	$$⊢\underset{\_}{Ⅎ}kB$$
	cbvesum.5	$$⊢\underset{\_}{Ⅎ}jC$$
Assertion	cbvesum	$$⊢\underset{j\in A}{{\sum }^{*}}B=\underset{k\in A}{{\sum }^{*}}C$$

Proof

Step	Hyp	Ref	Expression
1		cbvesum.1	$$⊢j=k\to B=C$$
2		cbvesum.2	$$⊢\underset{\_}{Ⅎ}kA$$
3		cbvesum.3	$$⊢\underset{\_}{Ⅎ}jA$$
4		cbvesum.4	$$⊢\underset{\_}{Ⅎ}kB$$
5		cbvesum.5	$$⊢\underset{\_}{Ⅎ}jC$$
6	3 2 4 5 1	cbvmptf	$$⊢\left(j\in A⟼B\right)=\left(k\in A⟼C\right)$$
7	6	oveq2i	$$⊢\left({ℝ}_{𝑠}^{*}{\upharpoonright }_{𝑠}\left[0,\mathrm{+\infty }\right]\right)\mathrm{tsums}\left(j\in A⟼B\right)=\left({ℝ}_{𝑠}^{*}{\upharpoonright }_{𝑠}\left[0,\mathrm{+\infty }\right]\right)\mathrm{tsums}\left(k\in A⟼C\right)$$
8	7	unieqi	$$⊢\bigcup \left(\left({ℝ}_{𝑠}^{*}{\upharpoonright }_{𝑠}\left[0,\mathrm{+\infty }\right]\right)\mathrm{tsums}\left(j\in A⟼B\right)\right)=\bigcup \left(\left({ℝ}_{𝑠}^{*}{\upharpoonright }_{𝑠}\left[0,\mathrm{+\infty }\right]\right)\mathrm{tsums}\left(k\in A⟼C\right)\right)$$
9		df-esum	$$⊢\underset{j\in A}{{\sum }^{*}}B=\bigcup \left(\left({ℝ}_{𝑠}^{*}{\upharpoonright }_{𝑠}\left[0,\mathrm{+\infty }\right]\right)\mathrm{tsums}\left(j\in A⟼B\right)\right)$$
10		df-esum	$$⊢\underset{k\in A}{{\sum }^{*}}C=\bigcup \left(\left({ℝ}_{𝑠}^{*}{\upharpoonright }_{𝑠}\left[0,\mathrm{+\infty }\right]\right)\mathrm{tsums}\left(k\in A⟼C\right)\right)$$
11	8 9 10	3eqtr4i	$$⊢\underset{j\in A}{{\sum }^{*}}B=\underset{k\in A}{{\sum }^{*}}C$$