Metamath Proof Explorer


Theorem scmatrngiso

Description: There is a ring isomorphism from a ring to the ring of scalar matrices over this ring with positive dimension. (Contributed by AV, 29-Dec-2019)

Ref Expression
Hypotheses scmatrhmval.k 𝐾 = ( Base ‘ 𝑅 )
scmatrhmval.a 𝐴 = ( 𝑁 Mat 𝑅 )
scmatrhmval.o 1 = ( 1r𝐴 )
scmatrhmval.t = ( ·𝑠𝐴 )
scmatrhmval.f 𝐹 = ( 𝑥𝐾 ↦ ( 𝑥 1 ) )
scmatrhmval.c 𝐶 = ( 𝑁 ScMat 𝑅 )
scmatghm.s 𝑆 = ( 𝐴s 𝐶 )
Assertion scmatrngiso ( ( 𝑁 ∈ Fin ∧ 𝑁 ≠ ∅ ∧ 𝑅 ∈ Ring ) → 𝐹 ∈ ( 𝑅 RingIso 𝑆 ) )

Proof

Step Hyp Ref Expression
1 scmatrhmval.k 𝐾 = ( Base ‘ 𝑅 )
2 scmatrhmval.a 𝐴 = ( 𝑁 Mat 𝑅 )
3 scmatrhmval.o 1 = ( 1r𝐴 )
4 scmatrhmval.t = ( ·𝑠𝐴 )
5 scmatrhmval.f 𝐹 = ( 𝑥𝐾 ↦ ( 𝑥 1 ) )
6 scmatrhmval.c 𝐶 = ( 𝑁 ScMat 𝑅 )
7 scmatghm.s 𝑆 = ( 𝐴s 𝐶 )
8 1 2 3 4 5 6 7 scmatrhm ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) → 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) )
9 8 3adant2 ( ( 𝑁 ∈ Fin ∧ 𝑁 ≠ ∅ ∧ 𝑅 ∈ Ring ) → 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) )
10 1 2 3 4 5 6 scmatf1o ( ( 𝑁 ∈ Fin ∧ 𝑁 ≠ ∅ ∧ 𝑅 ∈ Ring ) → 𝐹 : 𝐾1-1-onto𝐶 )
11 2 6 7 scmatstrbas ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) → ( Base ‘ 𝑆 ) = 𝐶 )
12 11 3adant2 ( ( 𝑁 ∈ Fin ∧ 𝑁 ≠ ∅ ∧ 𝑅 ∈ Ring ) → ( Base ‘ 𝑆 ) = 𝐶 )
13 12 f1oeq3d ( ( 𝑁 ∈ Fin ∧ 𝑁 ≠ ∅ ∧ 𝑅 ∈ Ring ) → ( 𝐹 : 𝐾1-1-onto→ ( Base ‘ 𝑆 ) ↔ 𝐹 : 𝐾1-1-onto𝐶 ) )
14 10 13 mpbird ( ( 𝑁 ∈ Fin ∧ 𝑁 ≠ ∅ ∧ 𝑅 ∈ Ring ) → 𝐹 : 𝐾1-1-onto→ ( Base ‘ 𝑆 ) )
15 eqid ( Base ‘ 𝑆 ) = ( Base ‘ 𝑆 )
16 1 15 isrim ( 𝐹 ∈ ( 𝑅 RingIso 𝑆 ) ↔ ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ 𝐹 : 𝐾1-1-onto→ ( Base ‘ 𝑆 ) ) )
17 9 14 16 sylanbrc ( ( 𝑁 ∈ Fin ∧ 𝑁 ≠ ∅ ∧ 𝑅 ∈ Ring ) → 𝐹 ∈ ( 𝑅 RingIso 𝑆 ) )