Description: An isomorphism of modules is a homomorphism. (Contributed by Stefan O'Rear, 21-Jan-2015)
Ref | Expression | ||
---|---|---|---|
Assertion | lmimlmhm | ⊢ ( 𝐹 ∈ ( 𝑅 LMIso 𝑆 ) → 𝐹 ∈ ( 𝑅 LMHom 𝑆 ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid | ⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 ) | |
2 | eqid | ⊢ ( Base ‘ 𝑆 ) = ( Base ‘ 𝑆 ) | |
3 | 1 2 | islmim | ⊢ ( 𝐹 ∈ ( 𝑅 LMIso 𝑆 ) ↔ ( 𝐹 ∈ ( 𝑅 LMHom 𝑆 ) ∧ 𝐹 : ( Base ‘ 𝑅 ) –1-1-onto→ ( Base ‘ 𝑆 ) ) ) |
4 | 3 | simplbi | ⊢ ( 𝐹 ∈ ( 𝑅 LMIso 𝑆 ) → 𝐹 ∈ ( 𝑅 LMHom 𝑆 ) ) |