Description: An ideal is closed under left-multiplication by elements of the full ring. (Contributed by Stefan O'Rear, 3-Jan-2015) (Proof shortened by AV, 31-Mar-2025)
Ref | Expression | ||
---|---|---|---|
Hypotheses | lidlcl.u | |
|
lidlcl.b | |
||
lidlmcl.t | |
||
Assertion | lidlmcl | |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lidlcl.u | |
|
2 | lidlcl.b | |
|
3 | lidlmcl.t | |
|
4 | ringrng | |
|
5 | 4 | adantr | |
6 | simpr | |
|
7 | eqid | |
|
8 | 1 7 | lidl0cl | |
9 | 5 6 8 | 3jca | |
10 | 7 2 3 1 | rnglidlmcl | |
11 | 9 10 | sylan | |