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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 ) )