Theorem rngoisohom

Description: A ring isomorphism is a ring homomorphism. (Contributed by Jeff Madsen, 16-Jun-2011)

	Ref	Expression
Assertion	rngoisohom	Could not format assertion : No typesetting found for |- ( ( R e. RingOps /\ S e. RingOps /\ F e. ( R RingOpsIso S ) ) -> F e. ( R RingOpsHom S ) ) with typecode |-

Proof

Step	Hyp	Ref	Expression
1		eqid	$$⊢1^{st}\left(R\right)=1^{st}\left(R\right)$$
2		eqid	$$⊢\mathrm{ran}{1}^{st}\left(R\right)=\mathrm{ran}{1}^{st}\left(R\right)$$
3		eqid	$$⊢1^{st}\left(S\right)=1^{st}\left(S\right)$$
4		eqid	$$⊢\mathrm{ran}{1}^{st}\left(S\right)=\mathrm{ran}{1}^{st}\left(S\right)$$
5	1 2 3 4	isrngoiso	Could not format ( ( R e. RingOps /\ S e. RingOps ) -> ( F e. ( R RingOpsIso S ) <-> ( F e. ( R RingOpsHom S ) /\ F : ran ( 1st ` R ) -1-1-onto-> ran ( 1st ` S ) ) ) ) : No typesetting found for |- ( ( R e. RingOps /\ S e. RingOps ) -> ( F e. ( R RingOpsIso S ) <-> ( F e. ( R RingOpsHom S ) /\ F : ran ( 1st ` R ) -1-1-onto-> ran ( 1st ` S ) ) ) ) with typecode |-
6	5	simprbda	Could not format ( ( ( R e. RingOps /\ S e. RingOps ) /\ F e. ( R RingOpsIso S ) ) -> F e. ( R RingOpsHom S ) ) : No typesetting found for |- ( ( ( R e. RingOps /\ S e. RingOps ) /\ F e. ( R RingOpsIso S ) ) -> F e. ( R RingOpsHom S ) ) with typecode |-
7	6	3impa	Could not format ( ( R e. RingOps /\ S e. RingOps /\ F e. ( R RingOpsIso S ) ) -> F e. ( R RingOpsHom S ) ) : No typesetting found for |- ( ( R e. RingOps /\ S e. RingOps /\ F e. ( R RingOpsIso S ) ) -> F e. ( R RingOpsHom S ) ) with typecode |-