Theorem 0ghm

Description: The constant zero linear function between two groups. (Contributed by Stefan O'Rear, 5-Sep-2015)

	Ref	Expression
Hypotheses	0ghm.z	⊢ 0 = ( 0g ‘ 𝑁 )
	0ghm.b	⊢ 𝐵 = ( Base ‘ 𝑀 )
Assertion	0ghm	⊢ ( ( 𝑀 ∈ Grp ∧ 𝑁 ∈ Grp ) → ( 𝐵 × { 0 } ) ∈ ( 𝑀 GrpHom 𝑁 ) )

Proof

Step	Hyp	Ref	Expression
1		0ghm.z	⊢ 0 = ( 0g ‘ 𝑁 )
2		0ghm.b	⊢ 𝐵 = ( Base ‘ 𝑀 )
3		grpmnd	⊢ ( 𝑀 ∈ Grp → 𝑀 ∈ Mnd )
4		grpmnd	⊢ ( 𝑁 ∈ Grp → 𝑁 ∈ Mnd )
5	1 2	0mhm	⊢ ( ( 𝑀 ∈ Mnd ∧ 𝑁 ∈ Mnd ) → ( 𝐵 × { 0 } ) ∈ ( 𝑀 MndHom 𝑁 ) )
6	3 4 5	syl2an	⊢ ( ( 𝑀 ∈ Grp ∧ 𝑁 ∈ Grp ) → ( 𝐵 × { 0 } ) ∈ ( 𝑀 MndHom 𝑁 ) )
7		ghmmhmb	⊢ ( ( 𝑀 ∈ Grp ∧ 𝑁 ∈ Grp ) → ( 𝑀 GrpHom 𝑁 ) = ( 𝑀 MndHom 𝑁 ) )
8	6 7	eleqtrrd	⊢ ( ( 𝑀 ∈ Grp ∧ 𝑁 ∈ Grp ) → ( 𝐵 × { 0 } ) ∈ ( 𝑀 GrpHom 𝑁 ) )