Step |
Hyp |
Ref |
Expression |
1 |
|
islnr3.b |
⊢ 𝐵 = ( Base ‘ 𝑅 ) |
2 |
|
islnr3.u |
⊢ 𝑈 = ( LIdeal ‘ 𝑅 ) |
3 |
|
eqid |
⊢ ( RSpan ‘ 𝑅 ) = ( RSpan ‘ 𝑅 ) |
4 |
1 2 3
|
islnr2 |
⊢ ( 𝑅 ∈ LNoeR ↔ ( 𝑅 ∈ Ring ∧ ∀ 𝑥 ∈ 𝑈 ∃ 𝑦 ∈ ( 𝒫 𝐵 ∩ Fin ) 𝑥 = ( ( RSpan ‘ 𝑅 ) ‘ 𝑦 ) ) ) |
5 |
|
eqid |
⊢ ( mrCls ‘ 𝑈 ) = ( mrCls ‘ 𝑈 ) |
6 |
2 3 5
|
mrcrsp |
⊢ ( 𝑅 ∈ Ring → ( RSpan ‘ 𝑅 ) = ( mrCls ‘ 𝑈 ) ) |
7 |
6
|
fveq1d |
⊢ ( 𝑅 ∈ Ring → ( ( RSpan ‘ 𝑅 ) ‘ 𝑦 ) = ( ( mrCls ‘ 𝑈 ) ‘ 𝑦 ) ) |
8 |
7
|
eqeq2d |
⊢ ( 𝑅 ∈ Ring → ( 𝑥 = ( ( RSpan ‘ 𝑅 ) ‘ 𝑦 ) ↔ 𝑥 = ( ( mrCls ‘ 𝑈 ) ‘ 𝑦 ) ) ) |
9 |
8
|
rexbidv |
⊢ ( 𝑅 ∈ Ring → ( ∃ 𝑦 ∈ ( 𝒫 𝐵 ∩ Fin ) 𝑥 = ( ( RSpan ‘ 𝑅 ) ‘ 𝑦 ) ↔ ∃ 𝑦 ∈ ( 𝒫 𝐵 ∩ Fin ) 𝑥 = ( ( mrCls ‘ 𝑈 ) ‘ 𝑦 ) ) ) |
10 |
9
|
ralbidv |
⊢ ( 𝑅 ∈ Ring → ( ∀ 𝑥 ∈ 𝑈 ∃ 𝑦 ∈ ( 𝒫 𝐵 ∩ Fin ) 𝑥 = ( ( RSpan ‘ 𝑅 ) ‘ 𝑦 ) ↔ ∀ 𝑥 ∈ 𝑈 ∃ 𝑦 ∈ ( 𝒫 𝐵 ∩ Fin ) 𝑥 = ( ( mrCls ‘ 𝑈 ) ‘ 𝑦 ) ) ) |
11 |
1 2
|
lidlacs |
⊢ ( 𝑅 ∈ Ring → 𝑈 ∈ ( ACS ‘ 𝐵 ) ) |
12 |
11
|
biantrurd |
⊢ ( 𝑅 ∈ Ring → ( ∀ 𝑥 ∈ 𝑈 ∃ 𝑦 ∈ ( 𝒫 𝐵 ∩ Fin ) 𝑥 = ( ( mrCls ‘ 𝑈 ) ‘ 𝑦 ) ↔ ( 𝑈 ∈ ( ACS ‘ 𝐵 ) ∧ ∀ 𝑥 ∈ 𝑈 ∃ 𝑦 ∈ ( 𝒫 𝐵 ∩ Fin ) 𝑥 = ( ( mrCls ‘ 𝑈 ) ‘ 𝑦 ) ) ) ) |
13 |
10 12
|
bitrd |
⊢ ( 𝑅 ∈ Ring → ( ∀ 𝑥 ∈ 𝑈 ∃ 𝑦 ∈ ( 𝒫 𝐵 ∩ Fin ) 𝑥 = ( ( RSpan ‘ 𝑅 ) ‘ 𝑦 ) ↔ ( 𝑈 ∈ ( ACS ‘ 𝐵 ) ∧ ∀ 𝑥 ∈ 𝑈 ∃ 𝑦 ∈ ( 𝒫 𝐵 ∩ Fin ) 𝑥 = ( ( mrCls ‘ 𝑈 ) ‘ 𝑦 ) ) ) ) |
14 |
5
|
isnacs |
⊢ ( 𝑈 ∈ ( NoeACS ‘ 𝐵 ) ↔ ( 𝑈 ∈ ( ACS ‘ 𝐵 ) ∧ ∀ 𝑥 ∈ 𝑈 ∃ 𝑦 ∈ ( 𝒫 𝐵 ∩ Fin ) 𝑥 = ( ( mrCls ‘ 𝑈 ) ‘ 𝑦 ) ) ) |
15 |
13 14
|
bitr4di |
⊢ ( 𝑅 ∈ Ring → ( ∀ 𝑥 ∈ 𝑈 ∃ 𝑦 ∈ ( 𝒫 𝐵 ∩ Fin ) 𝑥 = ( ( RSpan ‘ 𝑅 ) ‘ 𝑦 ) ↔ 𝑈 ∈ ( NoeACS ‘ 𝐵 ) ) ) |
16 |
15
|
pm5.32i |
⊢ ( ( 𝑅 ∈ Ring ∧ ∀ 𝑥 ∈ 𝑈 ∃ 𝑦 ∈ ( 𝒫 𝐵 ∩ Fin ) 𝑥 = ( ( RSpan ‘ 𝑅 ) ‘ 𝑦 ) ) ↔ ( 𝑅 ∈ Ring ∧ 𝑈 ∈ ( NoeACS ‘ 𝐵 ) ) ) |
17 |
4 16
|
bitri |
⊢ ( 𝑅 ∈ LNoeR ↔ ( 𝑅 ∈ Ring ∧ 𝑈 ∈ ( NoeACS ‘ 𝐵 ) ) ) |