Metamath Proof Explorer


Theorem lmimcnv

Description: The converse of a bijective module homomorphism is a bijective module homomorphism. (Contributed by Stefan O'Rear, 25-Jan-2015) (Revised by Mario Carneiro, 6-May-2015)

Ref Expression
Assertion lmimcnv ( 𝐹 ∈ ( 𝑆 LMIso 𝑇 ) → 𝐹 ∈ ( 𝑇 LMIso 𝑆 ) )

Proof

Step Hyp Ref Expression
1 eqid ( Base ‘ 𝑆 ) = ( Base ‘ 𝑆 )
2 eqid ( Base ‘ 𝑇 ) = ( Base ‘ 𝑇 )
3 1 2 lmhmf ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) → 𝐹 : ( Base ‘ 𝑆 ) ⟶ ( Base ‘ 𝑇 ) )
4 frel ( 𝐹 : ( Base ‘ 𝑆 ) ⟶ ( Base ‘ 𝑇 ) → Rel 𝐹 )
5 3 4 syl ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) → Rel 𝐹 )
6 dfrel2 ( Rel 𝐹 𝐹 = 𝐹 )
7 5 6 sylib ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) → 𝐹 = 𝐹 )
8 id ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) → 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) )
9 7 8 eqeltrd ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) → 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) )
10 9 anim1ci ( ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ∧ 𝐹 ∈ ( 𝑇 LMHom 𝑆 ) ) → ( 𝐹 ∈ ( 𝑇 LMHom 𝑆 ) ∧ 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ) )
11 islmim2 ( 𝐹 ∈ ( 𝑆 LMIso 𝑇 ) ↔ ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ∧ 𝐹 ∈ ( 𝑇 LMHom 𝑆 ) ) )
12 islmim2 ( 𝐹 ∈ ( 𝑇 LMIso 𝑆 ) ↔ ( 𝐹 ∈ ( 𝑇 LMHom 𝑆 ) ∧ 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ) )
13 10 11 12 3imtr4i ( 𝐹 ∈ ( 𝑆 LMIso 𝑇 ) → 𝐹 ∈ ( 𝑇 LMIso 𝑆 ) )