Metamath Proof Explorer


Theorem mat1f

Description: There is a function from a ring to the ring of matrices with dimension 1 over this ring. (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 = ( 𝑥𝐾 ↦ { ⟨ 𝑂 , 𝑥 ⟩ } )
Assertion mat1f ( ( 𝑅 ∈ Ring ∧ 𝐸𝑉 ) → 𝐹 : 𝐾𝐵 )

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 = ( 𝑥𝐾 ↦ { ⟨ 𝑂 , 𝑥 ⟩ } )
6 1 2 3 4 5 mat1f1o ( ( 𝑅 ∈ Ring ∧ 𝐸𝑉 ) → 𝐹 : 𝐾1-1-onto𝐵 )
7 f1of ( 𝐹 : 𝐾1-1-onto𝐵𝐹 : 𝐾𝐵 )
8 6 7 syl ( ( 𝑅 ∈ Ring ∧ 𝐸𝑉 ) → 𝐹 : 𝐾𝐵 )