Theorem rnghmval2

Description: The non-unital ring homomorphisms between two non-unital rings. (Contributed by AV, 1-Mar-2020)

	Ref	Expression
Assertion	rnghmval2	⊢ ( ( 𝑅 ∈ Rng ∧ 𝑆 ∈ Rng ) → ( 𝑅 RngHom 𝑆 ) = ( ( 𝑅 GrpHom 𝑆 ) ∩ ( ( mulGrp ‘ 𝑅 ) MgmHom ( mulGrp ‘ 𝑆 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		eqid	⊢ ( mulGrp ‘ 𝑅 ) = ( mulGrp ‘ 𝑅 )
2		eqid	⊢ ( mulGrp ‘ 𝑆 ) = ( mulGrp ‘ 𝑆 )
3	1 2	isrnghmmul	⊢ ( ℎ ∈ ( 𝑅 RngHom 𝑆 ) ↔ ( ( 𝑅 ∈ Rng ∧ 𝑆 ∈ Rng ) ∧ ( ℎ ∈ ( 𝑅 GrpHom 𝑆 ) ∧ ℎ ∈ ( ( mulGrp ‘ 𝑅 ) MgmHom ( mulGrp ‘ 𝑆 ) ) ) ) )
4		elin	⊢ ( ℎ ∈ ( ( 𝑅 GrpHom 𝑆 ) ∩ ( ( mulGrp ‘ 𝑅 ) MgmHom ( mulGrp ‘ 𝑆 ) ) ) ↔ ( ℎ ∈ ( 𝑅 GrpHom 𝑆 ) ∧ ℎ ∈ ( ( mulGrp ‘ 𝑅 ) MgmHom ( mulGrp ‘ 𝑆 ) ) ) )
5		ibar	⊢ ( ( 𝑅 ∈ Rng ∧ 𝑆 ∈ Rng ) → ( ( ℎ ∈ ( 𝑅 GrpHom 𝑆 ) ∧ ℎ ∈ ( ( mulGrp ‘ 𝑅 ) MgmHom ( mulGrp ‘ 𝑆 ) ) ) ↔ ( ( 𝑅 ∈ Rng ∧ 𝑆 ∈ Rng ) ∧ ( ℎ ∈ ( 𝑅 GrpHom 𝑆 ) ∧ ℎ ∈ ( ( mulGrp ‘ 𝑅 ) MgmHom ( mulGrp ‘ 𝑆 ) ) ) ) ) )
6	4 5	bitr2id	⊢ ( ( 𝑅 ∈ Rng ∧ 𝑆 ∈ Rng ) → ( ( ( 𝑅 ∈ Rng ∧ 𝑆 ∈ Rng ) ∧ ( ℎ ∈ ( 𝑅 GrpHom 𝑆 ) ∧ ℎ ∈ ( ( mulGrp ‘ 𝑅 ) MgmHom ( mulGrp ‘ 𝑆 ) ) ) ) ↔ ℎ ∈ ( ( 𝑅 GrpHom 𝑆 ) ∩ ( ( mulGrp ‘ 𝑅 ) MgmHom ( mulGrp ‘ 𝑆 ) ) ) ) )
7	3 6	bitrid	⊢ ( ( 𝑅 ∈ Rng ∧ 𝑆 ∈ Rng ) → ( ℎ ∈ ( 𝑅 RngHom 𝑆 ) ↔ ℎ ∈ ( ( 𝑅 GrpHom 𝑆 ) ∩ ( ( mulGrp ‘ 𝑅 ) MgmHom ( mulGrp ‘ 𝑆 ) ) ) ) )
8	7	eqrdv	⊢ ( ( 𝑅 ∈ Rng ∧ 𝑆 ∈ Rng ) → ( 𝑅 RngHom 𝑆 ) = ( ( 𝑅 GrpHom 𝑆 ) ∩ ( ( mulGrp ‘ 𝑅 ) MgmHom ( mulGrp ‘ 𝑆 ) ) ) )