Theorem sraaddg

Description: Additive operation of a subring algebra. (Contributed by Stefan O'Rear, 27-Nov-2014) (Revised by Mario Carneiro, 4-Oct-2015) (Revised by Thierry Arnoux, 16-Jun-2019)

	Ref	Expression
Hypotheses	srapart.a	$$⊢\phi \to A=\mathrm{subringAlg}\left(W\right)\left(S\right)$$
	srapart.s	$$⊢\phi \to S\subseteq {\mathrm{Base}}_W$$
Assertion	sraaddg	$$⊢\phi \to {+}_W={+}_A$$

Proof

Step	Hyp	Ref	Expression
1		srapart.a	$$⊢\phi \to A=\mathrm{subringAlg}\left(W\right)\left(S\right)$$
2		srapart.s	$$⊢\phi \to S\subseteq {\mathrm{Base}}_W$$
3		df-plusg	$$⊢{+}_{𝑔}=\mathrm{Slot}2$$
4		2nn	$$⊢2\in \mathbb{N}$$
5		2lt5	$$⊢2<5$$
6	5	orci	$$⊢\left(2<5\vee 8<2\right)$$
7	1 2 3 4 6	sralem	$$⊢\phi \to {+}_W={+}_A$$