Theorem esumeq2

Description: Equality theorem for extended sum. (Contributed by Thierry Arnoux, 24-Dec-2016)

	Ref	Expression
Assertion	esumeq2	$$⊢\forall k\in AB=C\to \underset{k\in A}{{\sum }^{*}}B=\underset{k\in A}{{\sum }^{*}}C$$

Proof

Step	Hyp	Ref	Expression
1		eqid	$$⊢A=A$$
2		mpteq12	$$⊢\left(A=A\wedge \forall k\in AB=C\right)\to \left(k\in A⟼B\right)=\left(k\in A⟼C\right)$$
3	1 2	mpan	$$⊢\forall k\in AB=C\to \left(k\in A⟼B\right)=\left(k\in A⟼C\right)$$
4	3	oveq2d	$$⊢\forall k\in AB=C\to \left({ℝ}_{𝑠}^{*}{\upharpoonright }_{𝑠}\left[0,\mathrm{+\infty }\right]\right)\mathrm{tsums}\left(k\in A⟼B\right)=\left({ℝ}_{𝑠}^{*}{\upharpoonright }_{𝑠}\left[0,\mathrm{+\infty }\right]\right)\mathrm{tsums}\left(k\in A⟼C\right)$$
5	4	unieqd	$$⊢\forall k\in AB=C\to \bigcup \left(\left({ℝ}_{𝑠}^{*}{\upharpoonright }_{𝑠}\left[0,\mathrm{+\infty }\right]\right)\mathrm{tsums}\left(k\in A⟼B\right)\right)=\bigcup \left(\left({ℝ}_{𝑠}^{*}{\upharpoonright }_{𝑠}\left[0,\mathrm{+\infty }\right]\right)\mathrm{tsums}\left(k\in A⟼C\right)\right)$$
6		df-esum	$$⊢\underset{k\in A}{{\sum }^{*}}B=\bigcup \left(\left({ℝ}_{𝑠}^{*}{\upharpoonright }_{𝑠}\left[0,\mathrm{+\infty }\right]\right)\mathrm{tsums}\left(k\in A⟼B\right)\right)$$
7		df-esum	$$⊢\underset{k\in A}{{\sum }^{*}}C=\bigcup \left(\left({ℝ}_{𝑠}^{*}{\upharpoonright }_{𝑠}\left[0,\mathrm{+\infty }\right]\right)\mathrm{tsums}\left(k\in A⟼C\right)\right)$$
8	5 6 7	3eqtr4g	$$⊢\forall k\in AB=C\to \underset{k\in A}{{\sum }^{*}}B=\underset{k\in A}{{\sum }^{*}}C$$