Theorem mat1rngiso

Description: There is a ring isomorphism 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 = ( 𝑥 ∈ 𝐾 ↦ { ⟨ 𝑂 , 𝑥 ⟩ } )
Assertion	mat1rngiso	⊢ ( ( 𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉 ) → 𝐹 ∈ ( 𝑅 RingIso 𝐴 ) )

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	1 2 3 4 5	mat1rhm	⊢ ( ( 𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉 ) → 𝐹 ∈ ( 𝑅 RingHom 𝐴 ) )
7	1 2 3 4 5	mat1f1o	⊢ ( ( 𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉 ) → 𝐹 : 𝐾 –1-1-onto→ 𝐵 )
8		snfi	⊢ { 𝐸 } ∈ Fin
9		simpl	⊢ ( ( 𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉 ) → 𝑅 ∈ Ring )
10	2	matring	⊢ ( ( { 𝐸 } ∈ Fin ∧ 𝑅 ∈ Ring ) → 𝐴 ∈ Ring )
11	8 9 10	sylancr	⊢ ( ( 𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉 ) → 𝐴 ∈ Ring )
12	1 3	isrim	⊢ ( ( 𝑅 ∈ Ring ∧ 𝐴 ∈ Ring ) → ( 𝐹 ∈ ( 𝑅 RingIso 𝐴 ) ↔ ( 𝐹 ∈ ( 𝑅 RingHom 𝐴 ) ∧ 𝐹 : 𝐾 –1-1-onto→ 𝐵 ) ) )
13	11 12	syldan	⊢ ( ( 𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉 ) → ( 𝐹 ∈ ( 𝑅 RingIso 𝐴 ) ↔ ( 𝐹 ∈ ( 𝑅 RingHom 𝐴 ) ∧ 𝐹 : 𝐾 –1-1-onto→ 𝐵 ) ) )
14	6 7 13	mpbir2and	⊢ ( ( 𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉 ) → 𝐹 ∈ ( 𝑅 RingIso 𝐴 ) )