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 |
4
|
fveq1d |
⊢ ( 𝜑 → ( 𝑁 ‘ 𝐴 ) = ( ( RingSpan ‘ 𝑅 ) ‘ 𝐴 ) ) |
7 |
|
elex |
⊢ ( 𝑅 ∈ Ring → 𝑅 ∈ V ) |
8 |
|
fveq2 |
⊢ ( 𝑎 = 𝑅 → ( Base ‘ 𝑎 ) = ( Base ‘ 𝑅 ) ) |
9 |
8
|
pweqd |
⊢ ( 𝑎 = 𝑅 → 𝒫 ( Base ‘ 𝑎 ) = 𝒫 ( Base ‘ 𝑅 ) ) |
10 |
|
fveq2 |
⊢ ( 𝑎 = 𝑅 → ( SubRing ‘ 𝑎 ) = ( SubRing ‘ 𝑅 ) ) |
11 |
|
rabeq |
⊢ ( ( SubRing ‘ 𝑎 ) = ( SubRing ‘ 𝑅 ) → { 𝑡 ∈ ( SubRing ‘ 𝑎 ) ∣ 𝑏 ⊆ 𝑡 } = { 𝑡 ∈ ( SubRing ‘ 𝑅 ) ∣ 𝑏 ⊆ 𝑡 } ) |
12 |
10 11
|
syl |
⊢ ( 𝑎 = 𝑅 → { 𝑡 ∈ ( SubRing ‘ 𝑎 ) ∣ 𝑏 ⊆ 𝑡 } = { 𝑡 ∈ ( SubRing ‘ 𝑅 ) ∣ 𝑏 ⊆ 𝑡 } ) |
13 |
12
|
inteqd |
⊢ ( 𝑎 = 𝑅 → ∩ { 𝑡 ∈ ( SubRing ‘ 𝑎 ) ∣ 𝑏 ⊆ 𝑡 } = ∩ { 𝑡 ∈ ( SubRing ‘ 𝑅 ) ∣ 𝑏 ⊆ 𝑡 } ) |
14 |
9 13
|
mpteq12dv |
⊢ ( 𝑎 = 𝑅 → ( 𝑏 ∈ 𝒫 ( Base ‘ 𝑎 ) ↦ ∩ { 𝑡 ∈ ( SubRing ‘ 𝑎 ) ∣ 𝑏 ⊆ 𝑡 } ) = ( 𝑏 ∈ 𝒫 ( Base ‘ 𝑅 ) ↦ ∩ { 𝑡 ∈ ( SubRing ‘ 𝑅 ) ∣ 𝑏 ⊆ 𝑡 } ) ) |
15 |
|
df-rgspn |
⊢ RingSpan = ( 𝑎 ∈ V ↦ ( 𝑏 ∈ 𝒫 ( Base ‘ 𝑎 ) ↦ ∩ { 𝑡 ∈ ( SubRing ‘ 𝑎 ) ∣ 𝑏 ⊆ 𝑡 } ) ) |
16 |
|
fvex |
⊢ ( Base ‘ 𝑅 ) ∈ V |
17 |
16
|
pwex |
⊢ 𝒫 ( Base ‘ 𝑅 ) ∈ V |
18 |
17
|
mptex |
⊢ ( 𝑏 ∈ 𝒫 ( Base ‘ 𝑅 ) ↦ ∩ { 𝑡 ∈ ( SubRing ‘ 𝑅 ) ∣ 𝑏 ⊆ 𝑡 } ) ∈ V |
19 |
14 15 18
|
fvmpt |
⊢ ( 𝑅 ∈ V → ( RingSpan ‘ 𝑅 ) = ( 𝑏 ∈ 𝒫 ( Base ‘ 𝑅 ) ↦ ∩ { 𝑡 ∈ ( SubRing ‘ 𝑅 ) ∣ 𝑏 ⊆ 𝑡 } ) ) |
20 |
1 7 19
|
3syl |
⊢ ( 𝜑 → ( RingSpan ‘ 𝑅 ) = ( 𝑏 ∈ 𝒫 ( Base ‘ 𝑅 ) ↦ ∩ { 𝑡 ∈ ( SubRing ‘ 𝑅 ) ∣ 𝑏 ⊆ 𝑡 } ) ) |
21 |
20
|
fveq1d |
⊢ ( 𝜑 → ( ( RingSpan ‘ 𝑅 ) ‘ 𝐴 ) = ( ( 𝑏 ∈ 𝒫 ( Base ‘ 𝑅 ) ↦ ∩ { 𝑡 ∈ ( SubRing ‘ 𝑅 ) ∣ 𝑏 ⊆ 𝑡 } ) ‘ 𝐴 ) ) |
22 |
|
eqid |
⊢ ( 𝑏 ∈ 𝒫 ( Base ‘ 𝑅 ) ↦ ∩ { 𝑡 ∈ ( SubRing ‘ 𝑅 ) ∣ 𝑏 ⊆ 𝑡 } ) = ( 𝑏 ∈ 𝒫 ( Base ‘ 𝑅 ) ↦ ∩ { 𝑡 ∈ ( SubRing ‘ 𝑅 ) ∣ 𝑏 ⊆ 𝑡 } ) |
23 |
|
sseq1 |
⊢ ( 𝑏 = 𝐴 → ( 𝑏 ⊆ 𝑡 ↔ 𝐴 ⊆ 𝑡 ) ) |
24 |
23
|
rabbidv |
⊢ ( 𝑏 = 𝐴 → { 𝑡 ∈ ( SubRing ‘ 𝑅 ) ∣ 𝑏 ⊆ 𝑡 } = { 𝑡 ∈ ( SubRing ‘ 𝑅 ) ∣ 𝐴 ⊆ 𝑡 } ) |
25 |
24
|
inteqd |
⊢ ( 𝑏 = 𝐴 → ∩ { 𝑡 ∈ ( SubRing ‘ 𝑅 ) ∣ 𝑏 ⊆ 𝑡 } = ∩ { 𝑡 ∈ ( SubRing ‘ 𝑅 ) ∣ 𝐴 ⊆ 𝑡 } ) |
26 |
3 2
|
sseqtrd |
⊢ ( 𝜑 → 𝐴 ⊆ ( Base ‘ 𝑅 ) ) |
27 |
16
|
elpw2 |
⊢ ( 𝐴 ∈ 𝒫 ( Base ‘ 𝑅 ) ↔ 𝐴 ⊆ ( Base ‘ 𝑅 ) ) |
28 |
26 27
|
sylibr |
⊢ ( 𝜑 → 𝐴 ∈ 𝒫 ( Base ‘ 𝑅 ) ) |
29 |
|
eqid |
⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 ) |
30 |
29
|
subrgid |
⊢ ( 𝑅 ∈ Ring → ( Base ‘ 𝑅 ) ∈ ( SubRing ‘ 𝑅 ) ) |
31 |
1 30
|
syl |
⊢ ( 𝜑 → ( Base ‘ 𝑅 ) ∈ ( SubRing ‘ 𝑅 ) ) |
32 |
2 31
|
eqeltrd |
⊢ ( 𝜑 → 𝐵 ∈ ( SubRing ‘ 𝑅 ) ) |
33 |
|
sseq2 |
⊢ ( 𝑡 = 𝐵 → ( 𝐴 ⊆ 𝑡 ↔ 𝐴 ⊆ 𝐵 ) ) |
34 |
33
|
rspcev |
⊢ ( ( 𝐵 ∈ ( SubRing ‘ 𝑅 ) ∧ 𝐴 ⊆ 𝐵 ) → ∃ 𝑡 ∈ ( SubRing ‘ 𝑅 ) 𝐴 ⊆ 𝑡 ) |
35 |
32 3 34
|
syl2anc |
⊢ ( 𝜑 → ∃ 𝑡 ∈ ( SubRing ‘ 𝑅 ) 𝐴 ⊆ 𝑡 ) |
36 |
|
intexrab |
⊢ ( ∃ 𝑡 ∈ ( SubRing ‘ 𝑅 ) 𝐴 ⊆ 𝑡 ↔ ∩ { 𝑡 ∈ ( SubRing ‘ 𝑅 ) ∣ 𝐴 ⊆ 𝑡 } ∈ V ) |
37 |
35 36
|
sylib |
⊢ ( 𝜑 → ∩ { 𝑡 ∈ ( SubRing ‘ 𝑅 ) ∣ 𝐴 ⊆ 𝑡 } ∈ V ) |
38 |
22 25 28 37
|
fvmptd3 |
⊢ ( 𝜑 → ( ( 𝑏 ∈ 𝒫 ( Base ‘ 𝑅 ) ↦ ∩ { 𝑡 ∈ ( SubRing ‘ 𝑅 ) ∣ 𝑏 ⊆ 𝑡 } ) ‘ 𝐴 ) = ∩ { 𝑡 ∈ ( SubRing ‘ 𝑅 ) ∣ 𝐴 ⊆ 𝑡 } ) |
39 |
21 38
|
eqtrd |
⊢ ( 𝜑 → ( ( RingSpan ‘ 𝑅 ) ‘ 𝐴 ) = ∩ { 𝑡 ∈ ( SubRing ‘ 𝑅 ) ∣ 𝐴 ⊆ 𝑡 } ) |
40 |
5 6 39
|
3eqtrd |
⊢ ( 𝜑 → 𝑈 = ∩ { 𝑡 ∈ ( SubRing ‘ 𝑅 ) ∣ 𝐴 ⊆ 𝑡 } ) |