Step |
Hyp |
Ref |
Expression |
1 |
|
rgspnval.r |
⊢ ( 𝜑 → 𝑅 ∈ Ring ) |
2 |
|
rgspnval.b |
⊢ ( 𝜑 → 𝐵 = ( Base ‘ 𝑅 ) ) |
3 |
|
rgspnval.ss |
⊢ ( 𝜑 → 𝐴 ⊆ 𝐵 ) |
4 |
|
rgspnval.n |
⊢ ( 𝜑 → 𝑁 = ( RingSpan ‘ 𝑅 ) ) |
5 |
|
rgspnval.sp |
⊢ ( 𝜑 → 𝑈 = ( 𝑁 ‘ 𝐴 ) ) |
6 |
|
rgspnmin.sr |
⊢ ( 𝜑 → 𝑆 ∈ ( SubRing ‘ 𝑅 ) ) |
7 |
|
rgspnmin.ss |
⊢ ( 𝜑 → 𝐴 ⊆ 𝑆 ) |
8 |
1 2 3 4 5
|
rgspnval |
⊢ ( 𝜑 → 𝑈 = ∩ { 𝑡 ∈ ( SubRing ‘ 𝑅 ) ∣ 𝐴 ⊆ 𝑡 } ) |
9 |
|
sseq2 |
⊢ ( 𝑡 = 𝑆 → ( 𝐴 ⊆ 𝑡 ↔ 𝐴 ⊆ 𝑆 ) ) |
10 |
9
|
elrab |
⊢ ( 𝑆 ∈ { 𝑡 ∈ ( SubRing ‘ 𝑅 ) ∣ 𝐴 ⊆ 𝑡 } ↔ ( 𝑆 ∈ ( SubRing ‘ 𝑅 ) ∧ 𝐴 ⊆ 𝑆 ) ) |
11 |
6 7 10
|
sylanbrc |
⊢ ( 𝜑 → 𝑆 ∈ { 𝑡 ∈ ( SubRing ‘ 𝑅 ) ∣ 𝐴 ⊆ 𝑡 } ) |
12 |
|
intss1 |
⊢ ( 𝑆 ∈ { 𝑡 ∈ ( SubRing ‘ 𝑅 ) ∣ 𝐴 ⊆ 𝑡 } → ∩ { 𝑡 ∈ ( SubRing ‘ 𝑅 ) ∣ 𝐴 ⊆ 𝑡 } ⊆ 𝑆 ) |
13 |
11 12
|
syl |
⊢ ( 𝜑 → ∩ { 𝑡 ∈ ( SubRing ‘ 𝑅 ) ∣ 𝐴 ⊆ 𝑡 } ⊆ 𝑆 ) |
14 |
8 13
|
eqsstrd |
⊢ ( 𝜑 → 𝑈 ⊆ 𝑆 ) |