Step |
Hyp |
Ref |
Expression |
1 |
|
1idl.1 |
⊢ 𝐺 = ( 1st ‘ 𝑅 ) |
2 |
|
1idl.2 |
⊢ 𝐻 = ( 2nd ‘ 𝑅 ) |
3 |
|
1idl.3 |
⊢ 𝑋 = ran 𝐺 |
4 |
|
1idl.4 |
⊢ 𝑈 = ( GId ‘ 𝐻 ) |
5 |
1 3
|
idlss |
⊢ ( ( 𝑅 ∈ RingOps ∧ 𝐼 ∈ ( Idl ‘ 𝑅 ) ) → 𝐼 ⊆ 𝑋 ) |
6 |
5
|
adantr |
⊢ ( ( ( 𝑅 ∈ RingOps ∧ 𝐼 ∈ ( Idl ‘ 𝑅 ) ) ∧ 𝑈 ∈ 𝐼 ) → 𝐼 ⊆ 𝑋 ) |
7 |
1
|
rneqi |
⊢ ran 𝐺 = ran ( 1st ‘ 𝑅 ) |
8 |
3 7
|
eqtri |
⊢ 𝑋 = ran ( 1st ‘ 𝑅 ) |
9 |
2 8 4
|
rngolidm |
⊢ ( ( 𝑅 ∈ RingOps ∧ 𝑥 ∈ 𝑋 ) → ( 𝑈 𝐻 𝑥 ) = 𝑥 ) |
10 |
9
|
ad2ant2rl |
⊢ ( ( ( 𝑅 ∈ RingOps ∧ 𝐼 ∈ ( Idl ‘ 𝑅 ) ) ∧ ( 𝑈 ∈ 𝐼 ∧ 𝑥 ∈ 𝑋 ) ) → ( 𝑈 𝐻 𝑥 ) = 𝑥 ) |
11 |
1 2 3
|
idlrmulcl |
⊢ ( ( ( 𝑅 ∈ RingOps ∧ 𝐼 ∈ ( Idl ‘ 𝑅 ) ) ∧ ( 𝑈 ∈ 𝐼 ∧ 𝑥 ∈ 𝑋 ) ) → ( 𝑈 𝐻 𝑥 ) ∈ 𝐼 ) |
12 |
10 11
|
eqeltrrd |
⊢ ( ( ( 𝑅 ∈ RingOps ∧ 𝐼 ∈ ( Idl ‘ 𝑅 ) ) ∧ ( 𝑈 ∈ 𝐼 ∧ 𝑥 ∈ 𝑋 ) ) → 𝑥 ∈ 𝐼 ) |
13 |
12
|
expr |
⊢ ( ( ( 𝑅 ∈ RingOps ∧ 𝐼 ∈ ( Idl ‘ 𝑅 ) ) ∧ 𝑈 ∈ 𝐼 ) → ( 𝑥 ∈ 𝑋 → 𝑥 ∈ 𝐼 ) ) |
14 |
13
|
ssrdv |
⊢ ( ( ( 𝑅 ∈ RingOps ∧ 𝐼 ∈ ( Idl ‘ 𝑅 ) ) ∧ 𝑈 ∈ 𝐼 ) → 𝑋 ⊆ 𝐼 ) |
15 |
6 14
|
eqssd |
⊢ ( ( ( 𝑅 ∈ RingOps ∧ 𝐼 ∈ ( Idl ‘ 𝑅 ) ) ∧ 𝑈 ∈ 𝐼 ) → 𝐼 = 𝑋 ) |
16 |
15
|
ex |
⊢ ( ( 𝑅 ∈ RingOps ∧ 𝐼 ∈ ( Idl ‘ 𝑅 ) ) → ( 𝑈 ∈ 𝐼 → 𝐼 = 𝑋 ) ) |
17 |
8 2 4
|
rngo1cl |
⊢ ( 𝑅 ∈ RingOps → 𝑈 ∈ 𝑋 ) |
18 |
17
|
adantr |
⊢ ( ( 𝑅 ∈ RingOps ∧ 𝐼 ∈ ( Idl ‘ 𝑅 ) ) → 𝑈 ∈ 𝑋 ) |
19 |
|
eleq2 |
⊢ ( 𝐼 = 𝑋 → ( 𝑈 ∈ 𝐼 ↔ 𝑈 ∈ 𝑋 ) ) |
20 |
18 19
|
syl5ibrcom |
⊢ ( ( 𝑅 ∈ RingOps ∧ 𝐼 ∈ ( Idl ‘ 𝑅 ) ) → ( 𝐼 = 𝑋 → 𝑈 ∈ 𝐼 ) ) |
21 |
16 20
|
impbid |
⊢ ( ( 𝑅 ∈ RingOps ∧ 𝐼 ∈ ( Idl ‘ 𝑅 ) ) → ( 𝑈 ∈ 𝐼 ↔ 𝐼 = 𝑋 ) ) |