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Metamath Proof Explorer

Theorem lmimco

Description: The composition of two isomorphisms of modules is an isomorphism of modules. (Contributed by AV, 10-Mar-2019)

Ref Expression
Assertion lmimco ( ( 𝐹 ∈ ( 𝑆 LMIso 𝑇 ) ∧ 𝐺 ∈ ( 𝑅 LMIso 𝑆 ) ) → ( 𝐹 ∘ 𝐺 ) ∈ ( 𝑅 LMIso 𝑇 ) )

Proof

Step Hyp Ref Expression
1 eqid ⊢ ( Base ‘ 𝑆 ) = ( Base ‘ 𝑆 )
2 eqid ⊢ ( Base ‘ 𝑇 ) = ( Base ‘ 𝑇 )
3 1 2 islmim ⊢ ( 𝐹 ∈ ( 𝑆 LMIso 𝑇 ) ↔ ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ∧ 𝐹 : ( Base ‘ 𝑆 ) –1-1-onto→ ( Base ‘ 𝑇 ) ) )
4 eqid ⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 )
5 4 1 islmim ⊢ ( 𝐺 ∈ ( 𝑅 LMIso 𝑆 ) ↔ ( 𝐺 ∈ ( 𝑅 LMHom 𝑆 ) ∧ 𝐺 : ( Base ‘ 𝑅 ) –1-1-onto→ ( Base ‘ 𝑆 ) ) )
6 lmhmco ⊢ ( ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ∧ 𝐺 ∈ ( 𝑅 LMHom 𝑆 ) ) → ( 𝐹 ∘ 𝐺 ) ∈ ( 𝑅 LMHom 𝑇 ) )
7 6 ad2ant2r ⊢ ( ( ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ∧ 𝐹 : ( Base ‘ 𝑆 ) –1-1-onto→ ( Base ‘ 𝑇 ) ) ∧ ( 𝐺 ∈ ( 𝑅 LMHom 𝑆 ) ∧ 𝐺 : ( Base ‘ 𝑅 ) –1-1-onto→ ( Base ‘ 𝑆 ) ) ) → ( 𝐹 ∘ 𝐺 ) ∈ ( 𝑅 LMHom 𝑇 ) )
8 f1oco ⊢ ( ( 𝐹 : ( Base ‘ 𝑆 ) –1-1-onto→ ( Base ‘ 𝑇 ) ∧ 𝐺 : ( Base ‘ 𝑅 ) –1-1-onto→ ( Base ‘ 𝑆 ) ) → ( 𝐹 ∘ 𝐺 ) : ( Base ‘ 𝑅 ) –1-1-onto→ ( Base ‘ 𝑇 ) )
9 8 ad2ant2l ⊢ ( ( ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ∧ 𝐹 : ( Base ‘ 𝑆 ) –1-1-onto→ ( Base ‘ 𝑇 ) ) ∧ ( 𝐺 ∈ ( 𝑅 LMHom 𝑆 ) ∧ 𝐺 : ( Base ‘ 𝑅 ) –1-1-onto→ ( Base ‘ 𝑆 ) ) ) → ( 𝐹 ∘ 𝐺 ) : ( Base ‘ 𝑅 ) –1-1-onto→ ( Base ‘ 𝑇 ) )
10 4 2 islmim ⊢ ( ( 𝐹 ∘ 𝐺 ) ∈ ( 𝑅 LMIso 𝑇 ) ↔ ( ( 𝐹 ∘ 𝐺 ) ∈ ( 𝑅 LMHom 𝑇 ) ∧ ( 𝐹 ∘ 𝐺 ) : ( Base ‘ 𝑅 ) –1-1-onto→ ( Base ‘ 𝑇 ) ) )
11 7 9 10 sylanbrc ⊢ ( ( ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ∧ 𝐹 : ( Base ‘ 𝑆 ) –1-1-onto→ ( Base ‘ 𝑇 ) ) ∧ ( 𝐺 ∈ ( 𝑅 LMHom 𝑆 ) ∧ 𝐺 : ( Base ‘ 𝑅 ) –1-1-onto→ ( Base ‘ 𝑆 ) ) ) → ( 𝐹 ∘ 𝐺 ) ∈ ( 𝑅 LMIso 𝑇 ) )
12 3 5 11 syl2anb ⊢ ( ( 𝐹 ∈ ( 𝑆 LMIso 𝑇 ) ∧ 𝐺 ∈ ( 𝑅 LMIso 𝑆 ) ) → ( 𝐹 ∘ 𝐺 ) ∈ ( 𝑅 LMIso 𝑇 ) )