Theorem islmhm3

Description: Property of a module homomorphism, similar to ismhm . (Contributed by Stefan O'Rear, 7-Mar-2015)

	Ref	Expression
Hypotheses	islmhm.k	⊢ 𝐾 = ( Scalar ‘ 𝑆 )
	islmhm.l	⊢ 𝐿 = ( Scalar ‘ 𝑇 )
	islmhm.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
	islmhm.e	⊢ 𝐸 = ( Base ‘ 𝑆 )
	islmhm.m	⊢ · = ( ·𝑠 ‘ 𝑆 )
	islmhm.n	⊢ × = ( ·𝑠 ‘ 𝑇 )
Assertion	islmhm3	⊢ ( ( 𝑆 ∈ LMod ∧ 𝑇 ∈ LMod ) → ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ↔ ( 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) ∧ 𝐿 = 𝐾 ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐸 ( 𝐹 ‘ ( 𝑥 · 𝑦 ) ) = ( 𝑥 × ( 𝐹 ‘ 𝑦 ) ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		islmhm.k	⊢ 𝐾 = ( Scalar ‘ 𝑆 )
2		islmhm.l	⊢ 𝐿 = ( Scalar ‘ 𝑇 )
3		islmhm.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
4		islmhm.e	⊢ 𝐸 = ( Base ‘ 𝑆 )
5		islmhm.m	⊢ · = ( ·𝑠 ‘ 𝑆 )
6		islmhm.n	⊢ × = ( ·𝑠 ‘ 𝑇 )
7	1 2 3 4 5 6	islmhm	⊢ ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ↔ ( ( 𝑆 ∈ LMod ∧ 𝑇 ∈ LMod ) ∧ ( 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) ∧ 𝐿 = 𝐾 ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐸 ( 𝐹 ‘ ( 𝑥 · 𝑦 ) ) = ( 𝑥 × ( 𝐹 ‘ 𝑦 ) ) ) ) )
8	7	baib	⊢ ( ( 𝑆 ∈ LMod ∧ 𝑇 ∈ LMod ) → ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ↔ ( 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) ∧ 𝐿 = 𝐾 ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐸 ( 𝐹 ‘ ( 𝑥 · 𝑦 ) ) = ( 𝑥 × ( 𝐹 ‘ 𝑦 ) ) ) ) )