Theorem drgextsubrg

Description: The scalar field is a subring 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 ) )
	drgext.f	|- F = ( E |`s U )
	drgext.3	|- ( ph -> F e. DivRing )
Assertion	drgextsubrg	|- ( ph -> U e. ( SubRing ` 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		drgext.f	|- F = ( E |`s U )
5		drgext.3	|- ( ph -> F e. DivRing )
6	1	a1i	|- ( ph -> B = ( ( subringAlg ` E ) ` U ) )
7		eqid	|- ( Base ` E ) = ( Base ` E )
8	7	subrgss	|- ( U e. ( SubRing ` E ) -> U C_ ( Base ` E ) )
9	3 8	syl	|- ( ph -> U C_ ( Base ` E ) )
10	6 3 9	srasubrg	|- ( ph -> U e. ( SubRing ` B ) )