Theorem srabase

Description: Base set 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	srabase	⊢ ( 𝜑 → ( Base ‘ 𝑊 ) = ( Base ‘ 𝐴 ) )

Proof

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