Step |
Hyp |
Ref |
Expression |
1 |
|
scmatval.k |
⊢ 𝐾 = ( Base ‘ 𝑅 ) |
2 |
|
scmatval.a |
⊢ 𝐴 = ( 𝑁 Mat 𝑅 ) |
3 |
|
scmatval.b |
⊢ 𝐵 = ( Base ‘ 𝐴 ) |
4 |
|
scmatval.1 |
⊢ 1 = ( 1r ‘ 𝐴 ) |
5 |
|
scmatval.t |
⊢ · = ( ·𝑠 ‘ 𝐴 ) |
6 |
|
scmatval.s |
⊢ 𝑆 = ( 𝑁 ScMat 𝑅 ) |
7 |
1 2 3 4 5 6
|
scmatval |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ) → 𝑆 = { 𝑚 ∈ 𝐵 ∣ ∃ 𝑐 ∈ 𝐾 𝑚 = ( 𝑐 · 1 ) } ) |
8 |
7
|
eleq2d |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ) → ( 𝑀 ∈ 𝑆 ↔ 𝑀 ∈ { 𝑚 ∈ 𝐵 ∣ ∃ 𝑐 ∈ 𝐾 𝑚 = ( 𝑐 · 1 ) } ) ) |
9 |
|
eqeq1 |
⊢ ( 𝑚 = 𝑀 → ( 𝑚 = ( 𝑐 · 1 ) ↔ 𝑀 = ( 𝑐 · 1 ) ) ) |
10 |
9
|
rexbidv |
⊢ ( 𝑚 = 𝑀 → ( ∃ 𝑐 ∈ 𝐾 𝑚 = ( 𝑐 · 1 ) ↔ ∃ 𝑐 ∈ 𝐾 𝑀 = ( 𝑐 · 1 ) ) ) |
11 |
10
|
elrab |
⊢ ( 𝑀 ∈ { 𝑚 ∈ 𝐵 ∣ ∃ 𝑐 ∈ 𝐾 𝑚 = ( 𝑐 · 1 ) } ↔ ( 𝑀 ∈ 𝐵 ∧ ∃ 𝑐 ∈ 𝐾 𝑀 = ( 𝑐 · 1 ) ) ) |
12 |
8 11
|
bitrdi |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ) → ( 𝑀 ∈ 𝑆 ↔ ( 𝑀 ∈ 𝐵 ∧ ∃ 𝑐 ∈ 𝐾 𝑀 = ( 𝑐 · 1 ) ) ) ) |