Theorem drgextvsca

Description: The scalar multiplication operation of a division ring extension. (Contributed by Thierry Arnoux, 17-Jul-2023)

	Ref	Expression
Hypotheses	drgext.b	|- B = ( ( subringAlg ` E ) ` U )
	drgext.1	|- ( ph -> E e. DivRing )
	drgext.2	|- ( ph -> U e. ( SubRing ` E ) )
Assertion	drgextvsca	|- ( ph -> ( .r ` E ) = ( .s ` B ) )

Proof

Step	Hyp	Ref	Expression
1		drgext.b	|- B = ( ( subringAlg ` E ) ` U )
2		drgext.1	|- ( ph -> E e. DivRing )
3		drgext.2	|- ( ph -> U e. ( SubRing ` E ) )
4	1	a1i	|- ( ph -> B = ( ( subringAlg ` E ) ` U ) )
5		eqid	|- ( Base ` E ) = ( Base ` E )
6	5	subrgss	|- ( U e. ( SubRing ` E ) -> U C_ ( Base ` E ) )
7	3 6	syl	|- ( ph -> U C_ ( Base ` E ) )
8	4 7	sravsca	|- ( ph -> ( .r ` E ) = ( .s ` B ) )