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Metamath Proof Explorer

Theorem rimgim

Description: An isomorphism of rings is an isomorphism of their additive groups. (Contributed by AV, 24-Dec-2019)

Ref Expression
Assertion rimgim ( 𝐹 ∈ ( 𝑅 RingIso 𝑆 ) → 𝐹 ∈ ( 𝑅 GrpIso 𝑆 ) )

Proof

Step Hyp Ref Expression
1 rimrhm ⊢ ( 𝐹 ∈ ( 𝑅 RingIso 𝑆 ) → 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) )
2 rhmghm ⊢ ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) → 𝐹 ∈ ( 𝑅 GrpHom 𝑆 ) )
3 1 2 syl ⊢ ( 𝐹 ∈ ( 𝑅 RingIso 𝑆 ) → 𝐹 ∈ ( 𝑅 GrpHom 𝑆 ) )
4 eqid ⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 )
5 eqid ⊢ ( Base ‘ 𝑆 ) = ( Base ‘ 𝑆 )
6 4 5 rimf1o ⊢ ( 𝐹 ∈ ( 𝑅 RingIso 𝑆 ) → 𝐹 : ( Base ‘ 𝑅 ) –1-1-onto→ ( Base ‘ 𝑆 ) )
7 4 5 isgim ⊢ ( 𝐹 ∈ ( 𝑅 GrpIso 𝑆 ) ↔ ( 𝐹 ∈ ( 𝑅 GrpHom 𝑆 ) ∧ 𝐹 : ( Base ‘ 𝑅 ) –1-1-onto→ ( Base ‘ 𝑆 ) ) )
8 3 6 7 sylanbrc ⊢ ( 𝐹 ∈ ( 𝑅 RingIso 𝑆 ) → 𝐹 ∈ ( 𝑅 GrpIso 𝑆 ) )