This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem lidl1

Description: Every ring contains a unit ideal. (Contributed by Stefan O'Rear, 3-Jan-2015) (Proof shortened by AV, 18-Apr-2025)

Ref Expression
Hypotheses rnglidl0.u ⊢ U = LIdeal ⁡ R
rnglidl1.b ⊢ B = Base R
Assertion lidl1 ⊢ R ∈ Ring → B ∈ U

Proof

Step Hyp Ref Expression
1 rnglidl0.u ⊢ U = LIdeal ⁡ R
2 rnglidl1.b ⊢ B = Base R
3 ringrng ⊢ R ∈ Ring → R ∈ Rng
4 1 2 rnglidl1 ⊢ R ∈ Rng → B ∈ U
5 3 4 syl ⊢ R ∈ Ring → B ∈ U