Metamath Proof Explorer


Theorem mat1rhmval

Description: The value of the ring homomorphism F . (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 mat1rhmval ( ( 𝑅 ∈ Ring ∧ 𝐸𝑉𝑋𝐾 ) → ( 𝐹𝑋 ) = { ⟨ 𝑂 , 𝑋 ⟩ } )

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 opeq2 ( 𝑥 = 𝑋 → ⟨ 𝑂 , 𝑥 ⟩ = ⟨ 𝑂 , 𝑋 ⟩ )
7 6 sneqd ( 𝑥 = 𝑋 → { ⟨ 𝑂 , 𝑥 ⟩ } = { ⟨ 𝑂 , 𝑋 ⟩ } )
8 simp3 ( ( 𝑅 ∈ Ring ∧ 𝐸𝑉𝑋𝐾 ) → 𝑋𝐾 )
9 snex { ⟨ 𝑂 , 𝑋 ⟩ } ∈ V
10 9 a1i ( ( 𝑅 ∈ Ring ∧ 𝐸𝑉𝑋𝐾 ) → { ⟨ 𝑂 , 𝑋 ⟩ } ∈ V )
11 5 7 8 10 fvmptd3 ( ( 𝑅 ∈ Ring ∧ 𝐸𝑉𝑋𝐾 ) → ( 𝐹𝑋 ) = { ⟨ 𝑂 , 𝑋 ⟩ } )