Theorem mat1rhmcl

Description: The value of the ring homomorphism F is a matrix with dimension 1. (Contributed by AV, 22-Dec-2019)

	Ref	Expression
Hypotheses	mat1rhmval.k	⊢ 𝐾 = ( Base ‘ 𝑅 )
	mat1rhmval.a	⊢ 𝐴 = ( { 𝐸 } Mat 𝑅 )
	mat1rhmval.b	⊢ 𝐵 = ( Base ‘ 𝐴 )
	mat1rhmval.o	⊢ 𝑂 = ⟨ 𝐸 , 𝐸 ⟩
	mat1rhmval.f	⊢ 𝐹 = ( 𝑥 ∈ 𝐾 ↦ { ⟨ 𝑂 , 𝑥 ⟩ } )
Assertion	mat1rhmcl	⊢ ( ( 𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝐾 ) → ( 𝐹 ‘ 𝑋 ) ∈ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		mat1rhmval.k	⊢ 𝐾 = ( Base ‘ 𝑅 )
2		mat1rhmval.a	⊢ 𝐴 = ( { 𝐸 } Mat 𝑅 )
3		mat1rhmval.b	⊢ 𝐵 = ( Base ‘ 𝐴 )
4		mat1rhmval.o	⊢ 𝑂 = ⟨ 𝐸 , 𝐸 ⟩
5		mat1rhmval.f	⊢ 𝐹 = ( 𝑥 ∈ 𝐾 ↦ { ⟨ 𝑂 , 𝑥 ⟩ } )
6	2 1 4	mat1dimbas	⊢ ( ( 𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝐾 ) → { ⟨ 𝑂 , 𝑋 ⟩ } ∈ ( Base ‘ 𝐴 ) )
7	1 2 3 4 5	mat1rhmval	⊢ ( ( 𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝐾 ) → ( 𝐹 ‘ 𝑋 ) = { ⟨ 𝑂 , 𝑋 ⟩ } )
8	3	a1i	⊢ ( ( 𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝐾 ) → 𝐵 = ( Base ‘ 𝐴 ) )
9	6 7 8	3eltr4d	⊢ ( ( 𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝐾 ) → ( 𝐹 ‘ 𝑋 ) ∈ 𝐵 )