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Metamath Proof Explorer

Theorem lmhmlem

Description: Non-quantified consequences of a left module homomorphism. (Contributed by Stefan O'Rear, 1-Jan-2015)

Ref Expression
Hypotheses lmhmlem.k ⊢ 𝐾 = ( Scalar ‘ 𝑆 )
lmhmlem.l ⊢ 𝐿 = ( Scalar ‘ 𝑇 )
Assertion lmhmlem ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) → ( ( 𝑆 ∈ LMod ∧ 𝑇 ∈ LMod ) ∧ ( 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) ∧ 𝐿 = 𝐾 ) ) )

Proof

Step Hyp Ref Expression
1 lmhmlem.k ⊢ 𝐾 = ( Scalar ‘ 𝑆 )
2 lmhmlem.l ⊢ 𝐿 = ( Scalar ‘ 𝑇 )
3 eqid ⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
4 eqid ⊢ ( Base ‘ 𝑆 ) = ( Base ‘ 𝑆 )
5 eqid ⊢ ( ·𝑠 ‘ 𝑆 ) = ( ·𝑠 ‘ 𝑆 )
6 eqid ⊢ ( ·𝑠 ‘ 𝑇 ) = ( ·𝑠 ‘ 𝑇 )
7 1 2 3 4 5 6 islmhm ⊢ ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ↔ ( ( 𝑆 ∈ LMod ∧ 𝑇 ∈ LMod ) ∧ ( 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) ∧ 𝐿 = 𝐾 ∧ ∀ 𝑎 ∈ ( Base ‘ 𝐾 ) ∀ 𝑏 ∈ ( Base ‘ 𝑆 ) ( 𝐹 ‘ ( 𝑎 ( ·𝑠 ‘ 𝑆 ) 𝑏 ) ) = ( 𝑎 ( ·𝑠 ‘ 𝑇 ) ( 𝐹 ‘ 𝑏 ) ) ) ) )
8 3simpa ⊢ ( ( 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) ∧ 𝐿 = 𝐾 ∧ ∀ 𝑎 ∈ ( Base ‘ 𝐾 ) ∀ 𝑏 ∈ ( Base ‘ 𝑆 ) ( 𝐹 ‘ ( 𝑎 ( ·𝑠 ‘ 𝑆 ) 𝑏 ) ) = ( 𝑎 ( ·𝑠 ‘ 𝑇 ) ( 𝐹 ‘ 𝑏 ) ) ) → ( 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) ∧ 𝐿 = 𝐾 ) )
9 8 anim2i ⊢ ( ( ( 𝑆 ∈ LMod ∧ 𝑇 ∈ LMod ) ∧ ( 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) ∧ 𝐿 = 𝐾 ∧ ∀ 𝑎 ∈ ( Base ‘ 𝐾 ) ∀ 𝑏 ∈ ( Base ‘ 𝑆 ) ( 𝐹 ‘ ( 𝑎 ( ·𝑠 ‘ 𝑆 ) 𝑏 ) ) = ( 𝑎 ( ·𝑠 ‘ 𝑇 ) ( 𝐹 ‘ 𝑏 ) ) ) ) → ( ( 𝑆 ∈ LMod ∧ 𝑇 ∈ LMod ) ∧ ( 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) ∧ 𝐿 = 𝐾 ) ) )
10 7 9 sylbi ⊢ ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) → ( ( 𝑆 ∈ LMod ∧ 𝑇 ∈ LMod ) ∧ ( 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) ∧ 𝐿 = 𝐾 ) ) )