Metamath Proof Explorer


Theorem lidlmcl

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 U=LIdealR
lidlcl.b B=BaseR
lidlmcl.t ·˙=R
Assertion lidlmcl RRingIUXBYIX·˙YI

Proof

Step Hyp Ref Expression
1 lidlcl.u U=LIdealR
2 lidlcl.b B=BaseR
3 lidlmcl.t ·˙=R
4 ringrng RRingRRng
5 4 adantr RRingIURRng
6 simpr RRingIUIU
7 eqid 0R=0R
8 1 7 lidl0cl RRingIU0RI
9 5 6 8 3jca RRingIURRngIU0RI
10 7 2 3 1 rnglidlmcl RRngIU0RIXBYIX·˙YI
11 9 10 sylan RRingIUXBYIX·˙YI