Step |
Hyp |
Ref |
Expression |
1 |
|
igenval.1 |
⊢ 𝐺 = ( 1st ‘ 𝑅 ) |
2 |
|
igenval.2 |
⊢ 𝑋 = ran 𝐺 |
3 |
1 2
|
rngoidl |
⊢ ( 𝑅 ∈ RingOps → 𝑋 ∈ ( Idl ‘ 𝑅 ) ) |
4 |
|
sseq2 |
⊢ ( 𝑗 = 𝑋 → ( 𝑆 ⊆ 𝑗 ↔ 𝑆 ⊆ 𝑋 ) ) |
5 |
4
|
rspcev |
⊢ ( ( 𝑋 ∈ ( Idl ‘ 𝑅 ) ∧ 𝑆 ⊆ 𝑋 ) → ∃ 𝑗 ∈ ( Idl ‘ 𝑅 ) 𝑆 ⊆ 𝑗 ) |
6 |
3 5
|
sylan |
⊢ ( ( 𝑅 ∈ RingOps ∧ 𝑆 ⊆ 𝑋 ) → ∃ 𝑗 ∈ ( Idl ‘ 𝑅 ) 𝑆 ⊆ 𝑗 ) |
7 |
|
rabn0 |
⊢ ( { 𝑗 ∈ ( Idl ‘ 𝑅 ) ∣ 𝑆 ⊆ 𝑗 } ≠ ∅ ↔ ∃ 𝑗 ∈ ( Idl ‘ 𝑅 ) 𝑆 ⊆ 𝑗 ) |
8 |
6 7
|
sylibr |
⊢ ( ( 𝑅 ∈ RingOps ∧ 𝑆 ⊆ 𝑋 ) → { 𝑗 ∈ ( Idl ‘ 𝑅 ) ∣ 𝑆 ⊆ 𝑗 } ≠ ∅ ) |
9 |
|
intex |
⊢ ( { 𝑗 ∈ ( Idl ‘ 𝑅 ) ∣ 𝑆 ⊆ 𝑗 } ≠ ∅ ↔ ∩ { 𝑗 ∈ ( Idl ‘ 𝑅 ) ∣ 𝑆 ⊆ 𝑗 } ∈ V ) |
10 |
8 9
|
sylib |
⊢ ( ( 𝑅 ∈ RingOps ∧ 𝑆 ⊆ 𝑋 ) → ∩ { 𝑗 ∈ ( Idl ‘ 𝑅 ) ∣ 𝑆 ⊆ 𝑗 } ∈ V ) |
11 |
1
|
fvexi |
⊢ 𝐺 ∈ V |
12 |
11
|
rnex |
⊢ ran 𝐺 ∈ V |
13 |
2 12
|
eqeltri |
⊢ 𝑋 ∈ V |
14 |
13
|
elpw2 |
⊢ ( 𝑆 ∈ 𝒫 𝑋 ↔ 𝑆 ⊆ 𝑋 ) |
15 |
|
simpl |
⊢ ( ( 𝑟 = 𝑅 ∧ 𝑠 = 𝑆 ) → 𝑟 = 𝑅 ) |
16 |
15
|
fveq2d |
⊢ ( ( 𝑟 = 𝑅 ∧ 𝑠 = 𝑆 ) → ( Idl ‘ 𝑟 ) = ( Idl ‘ 𝑅 ) ) |
17 |
|
sseq1 |
⊢ ( 𝑠 = 𝑆 → ( 𝑠 ⊆ 𝑗 ↔ 𝑆 ⊆ 𝑗 ) ) |
18 |
17
|
adantl |
⊢ ( ( 𝑟 = 𝑅 ∧ 𝑠 = 𝑆 ) → ( 𝑠 ⊆ 𝑗 ↔ 𝑆 ⊆ 𝑗 ) ) |
19 |
16 18
|
rabeqbidv |
⊢ ( ( 𝑟 = 𝑅 ∧ 𝑠 = 𝑆 ) → { 𝑗 ∈ ( Idl ‘ 𝑟 ) ∣ 𝑠 ⊆ 𝑗 } = { 𝑗 ∈ ( Idl ‘ 𝑅 ) ∣ 𝑆 ⊆ 𝑗 } ) |
20 |
19
|
inteqd |
⊢ ( ( 𝑟 = 𝑅 ∧ 𝑠 = 𝑆 ) → ∩ { 𝑗 ∈ ( Idl ‘ 𝑟 ) ∣ 𝑠 ⊆ 𝑗 } = ∩ { 𝑗 ∈ ( Idl ‘ 𝑅 ) ∣ 𝑆 ⊆ 𝑗 } ) |
21 |
|
fveq2 |
⊢ ( 𝑟 = 𝑅 → ( 1st ‘ 𝑟 ) = ( 1st ‘ 𝑅 ) ) |
22 |
21 1
|
eqtr4di |
⊢ ( 𝑟 = 𝑅 → ( 1st ‘ 𝑟 ) = 𝐺 ) |
23 |
22
|
rneqd |
⊢ ( 𝑟 = 𝑅 → ran ( 1st ‘ 𝑟 ) = ran 𝐺 ) |
24 |
23 2
|
eqtr4di |
⊢ ( 𝑟 = 𝑅 → ran ( 1st ‘ 𝑟 ) = 𝑋 ) |
25 |
24
|
pweqd |
⊢ ( 𝑟 = 𝑅 → 𝒫 ran ( 1st ‘ 𝑟 ) = 𝒫 𝑋 ) |
26 |
|
df-igen |
⊢ IdlGen = ( 𝑟 ∈ RingOps , 𝑠 ∈ 𝒫 ran ( 1st ‘ 𝑟 ) ↦ ∩ { 𝑗 ∈ ( Idl ‘ 𝑟 ) ∣ 𝑠 ⊆ 𝑗 } ) |
27 |
20 25 26
|
ovmpox |
⊢ ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ 𝒫 𝑋 ∧ ∩ { 𝑗 ∈ ( Idl ‘ 𝑅 ) ∣ 𝑆 ⊆ 𝑗 } ∈ V ) → ( 𝑅 IdlGen 𝑆 ) = ∩ { 𝑗 ∈ ( Idl ‘ 𝑅 ) ∣ 𝑆 ⊆ 𝑗 } ) |
28 |
14 27
|
syl3an2br |
⊢ ( ( 𝑅 ∈ RingOps ∧ 𝑆 ⊆ 𝑋 ∧ ∩ { 𝑗 ∈ ( Idl ‘ 𝑅 ) ∣ 𝑆 ⊆ 𝑗 } ∈ V ) → ( 𝑅 IdlGen 𝑆 ) = ∩ { 𝑗 ∈ ( Idl ‘ 𝑅 ) ∣ 𝑆 ⊆ 𝑗 } ) |
29 |
10 28
|
mpd3an3 |
⊢ ( ( 𝑅 ∈ RingOps ∧ 𝑆 ⊆ 𝑋 ) → ( 𝑅 IdlGen 𝑆 ) = ∩ { 𝑗 ∈ ( Idl ‘ 𝑅 ) ∣ 𝑆 ⊆ 𝑗 } ) |