Theorem sramulr

Description: Multiplicative 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	⊢ ( 𝜑 → 𝐴 = ( ( subringAlg ‘ 𝑊 ) ‘ 𝑆 ) )
	srapart.s	⊢ ( 𝜑 → 𝑆 ⊆ ( Base ‘ 𝑊 ) )
Assertion	sramulr	⊢ ( 𝜑 → ( .r ‘ 𝑊 ) = ( .r ‘ 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		srapart.a	⊢ ( 𝜑 → 𝐴 = ( ( subringAlg ‘ 𝑊 ) ‘ 𝑆 ) )
2		srapart.s	⊢ ( 𝜑 → 𝑆 ⊆ ( Base ‘ 𝑊 ) )
3		df-mulr	⊢ .r = Slot 3
4		3nn	⊢ 3 ∈ ℕ
5		3lt5	⊢ 3 < 5
6	5	orci	⊢ ( 3 < 5 ∨ 8 < 3 )
7	1 2 3 4 6	sralem	⊢ ( 𝜑 → ( .r ‘ 𝑊 ) = ( .r ‘ 𝐴 ) )