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Metamath Proof Explorer

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 ) ) ) )