Step |
Hyp |
Ref |
Expression |
1 |
|
scmatrhmval.k |
⊢ 𝐾 = ( Base ‘ 𝑅 ) |
2 |
|
scmatrhmval.a |
⊢ 𝐴 = ( 𝑁 Mat 𝑅 ) |
3 |
|
scmatrhmval.o |
⊢ 1 = ( 1r ‘ 𝐴 ) |
4 |
|
scmatrhmval.t |
⊢ ∗ = ( ·𝑠 ‘ 𝐴 ) |
5 |
|
scmatrhmval.f |
⊢ 𝐹 = ( 𝑥 ∈ 𝐾 ↦ ( 𝑥 ∗ 1 ) ) |
6 |
|
scmatrhmval.c |
⊢ 𝐶 = ( 𝑁 ScMat 𝑅 ) |
7 |
|
scmatghm.s |
⊢ 𝑆 = ( 𝐴 ↾s 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 : 𝐾 –1-1-onto→ 𝐶 ) |
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 : 𝐾 –1-1-onto→ ( Base ‘ 𝑆 ) ↔ 𝐹 : 𝐾 –1-1-onto→ 𝐶 ) ) |
14 |
10 13
|
mpbird |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑁 ≠ ∅ ∧ 𝑅 ∈ Ring ) → 𝐹 : 𝐾 –1-1-onto→ ( Base ‘ 𝑆 ) ) |
15 |
|
eqid |
⊢ ( Base ‘ 𝑆 ) = ( Base ‘ 𝑆 ) |
16 |
1 15
|
isrim |
⊢ ( 𝐹 ∈ ( 𝑅 RingIso 𝑆 ) ↔ ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ 𝐹 : 𝐾 –1-1-onto→ ( Base ‘ 𝑆 ) ) ) |
17 |
9 14 16
|
sylanbrc |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑁 ≠ ∅ ∧ 𝑅 ∈ Ring ) → 𝐹 ∈ ( 𝑅 RingIso 𝑆 ) ) |