Theorem subrngin

Description: The intersection of two subrings is a subring. (Contributed by AV, 15-Feb-2025)

	Ref	Expression
Assertion	subrngin	Could not format assertion : No typesetting found for |- ( ( A e. ( SubRng ` R ) /\ B e. ( SubRng ` R ) ) -> ( A i^i B ) e. ( SubRng ` R ) ) with typecode |-

Proof

Step	Hyp	Ref	Expression
1		intprg	Could not format ( ( A e. ( SubRng ` R ) /\ B e. ( SubRng ` R ) ) -> |^| { A , B } = ( A i^i B ) ) : No typesetting found for |- ( ( A e. ( SubRng ` R ) /\ B e. ( SubRng ` R ) ) -> |^| { A , B } = ( A i^i B ) ) with typecode |-
2		prssi	Could not format ( ( A e. ( SubRng ` R ) /\ B e. ( SubRng ` R ) ) -> { A , B } C_ ( SubRng ` R ) ) : No typesetting found for |- ( ( A e. ( SubRng ` R ) /\ B e. ( SubRng ` R ) ) -> { A , B } C_ ( SubRng ` R ) ) with typecode |-
3		prnzg	Could not format ( A e. ( SubRng ` R ) -> { A , B } =/= (/) ) : No typesetting found for |- ( A e. ( SubRng ` R ) -> { A , B } =/= (/) ) with typecode |-
4	3	adantr	Could not format ( ( A e. ( SubRng ` R ) /\ B e. ( SubRng ` R ) ) -> { A , B } =/= (/) ) : No typesetting found for |- ( ( A e. ( SubRng ` R ) /\ B e. ( SubRng ` R ) ) -> { A , B } =/= (/) ) with typecode |-
5		subrngint	Could not format ( ( { A , B } C_ ( SubRng ` R ) /\ { A , B } =/= (/) ) -> |^| { A , B } e. ( SubRng ` R ) ) : No typesetting found for |- ( ( { A , B } C_ ( SubRng ` R ) /\ { A , B } =/= (/) ) -> |^| { A , B } e. ( SubRng ` R ) ) with typecode |-
6	2 4 5	syl2anc	Could not format ( ( A e. ( SubRng ` R ) /\ B e. ( SubRng ` R ) ) -> |^| { A , B } e. ( SubRng ` R ) ) : No typesetting found for |- ( ( A e. ( SubRng ` R ) /\ B e. ( SubRng ` R ) ) -> |^| { A , B } e. ( SubRng ` R ) ) with typecode |-
7	1 6	eqeltrrd	Could not format ( ( A e. ( SubRng ` R ) /\ B e. ( SubRng ` R ) ) -> ( A i^i B ) e. ( SubRng ` R ) ) : No typesetting found for |- ( ( A e. ( SubRng ` R ) /\ B e. ( SubRng ` R ) ) -> ( A i^i B ) e. ( SubRng ` R ) ) with typecode |-