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) (Revised by AV, 29-Oct-2024)

	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		baseid	⊢ Base = Slot ( Base ‘ ndx )
4		scandxnbasendx	⊢ ( Scalar ‘ ndx ) ≠ ( Base ‘ ndx )
5		vscandxnbasendx	⊢ ( ·𝑠 ‘ ndx ) ≠ ( Base ‘ ndx )
6		ipndxnbasendx	⊢ ( ·𝑖 ‘ ndx ) ≠ ( Base ‘ ndx )
7	1 2 3 4 5 6	sralem	⊢ ( 𝜑 → ( Base ‘ 𝑊 ) = ( Base ‘ 𝐴 ) )