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Metamath Proof Explorer

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 )