# Metamath Proof Explorer

## Theorem ringrz

Description: The zero of a unital ring is a right-absorbing element. (Contributed by FL, 31-Aug-2009)

Ref Expression
Hypotheses rngz.b 𝐵 = ( Base ‘ 𝑅 )
rngz.t · = ( .r𝑅 )
rngz.z 0 = ( 0g𝑅 )
Assertion ringrz ( ( 𝑅 ∈ Ring ∧ 𝑋𝐵 ) → ( 𝑋 · 0 ) = 0 )

### Proof

Step Hyp Ref Expression
1 rngz.b 𝐵 = ( Base ‘ 𝑅 )
2 rngz.t · = ( .r𝑅 )
3 rngz.z 0 = ( 0g𝑅 )
4 ringgrp ( 𝑅 ∈ Ring → 𝑅 ∈ Grp )
5 1 3 grpidcl ( 𝑅 ∈ Grp → 0𝐵 )
6 eqid ( +g𝑅 ) = ( +g𝑅 )
7 1 6 3 grplid ( ( 𝑅 ∈ Grp ∧ 0𝐵 ) → ( 0 ( +g𝑅 ) 0 ) = 0 )
8 4 5 7 syl2anc2 ( 𝑅 ∈ Ring → ( 0 ( +g𝑅 ) 0 ) = 0 )
9 8 adantr ( ( 𝑅 ∈ Ring ∧ 𝑋𝐵 ) → ( 0 ( +g𝑅 ) 0 ) = 0 )
10 9 oveq2d ( ( 𝑅 ∈ Ring ∧ 𝑋𝐵 ) → ( 𝑋 · ( 0 ( +g𝑅 ) 0 ) ) = ( 𝑋 · 0 ) )
11 simpr ( ( 𝑅 ∈ Ring ∧ 𝑋𝐵 ) → 𝑋𝐵 )
12 4 5 syl ( 𝑅 ∈ Ring → 0𝐵 )
13 12 adantr ( ( 𝑅 ∈ Ring ∧ 𝑋𝐵 ) → 0𝐵 )
14 11 13 13 3jca ( ( 𝑅 ∈ Ring ∧ 𝑋𝐵 ) → ( 𝑋𝐵0𝐵0𝐵 ) )
15 1 6 2 ringdi ( ( 𝑅 ∈ Ring ∧ ( 𝑋𝐵0𝐵0𝐵 ) ) → ( 𝑋 · ( 0 ( +g𝑅 ) 0 ) ) = ( ( 𝑋 · 0 ) ( +g𝑅 ) ( 𝑋 · 0 ) ) )
16 14 15 syldan ( ( 𝑅 ∈ Ring ∧ 𝑋𝐵 ) → ( 𝑋 · ( 0 ( +g𝑅 ) 0 ) ) = ( ( 𝑋 · 0 ) ( +g𝑅 ) ( 𝑋 · 0 ) ) )
17 4 adantr ( ( 𝑅 ∈ Ring ∧ 𝑋𝐵 ) → 𝑅 ∈ Grp )
18 1 2 ringcl ( ( 𝑅 ∈ Ring ∧ 𝑋𝐵0𝐵 ) → ( 𝑋 · 0 ) ∈ 𝐵 )
19 13 18 mpd3an3 ( ( 𝑅 ∈ Ring ∧ 𝑋𝐵 ) → ( 𝑋 · 0 ) ∈ 𝐵 )
20 1 6 3 grplid ( ( 𝑅 ∈ Grp ∧ ( 𝑋 · 0 ) ∈ 𝐵 ) → ( 0 ( +g𝑅 ) ( 𝑋 · 0 ) ) = ( 𝑋 · 0 ) )
21 20 eqcomd ( ( 𝑅 ∈ Grp ∧ ( 𝑋 · 0 ) ∈ 𝐵 ) → ( 𝑋 · 0 ) = ( 0 ( +g𝑅 ) ( 𝑋 · 0 ) ) )
22 17 19 21 syl2anc ( ( 𝑅 ∈ Ring ∧ 𝑋𝐵 ) → ( 𝑋 · 0 ) = ( 0 ( +g𝑅 ) ( 𝑋 · 0 ) ) )
23 10 16 22 3eqtr3d ( ( 𝑅 ∈ Ring ∧ 𝑋𝐵 ) → ( ( 𝑋 · 0 ) ( +g𝑅 ) ( 𝑋 · 0 ) ) = ( 0 ( +g𝑅 ) ( 𝑋 · 0 ) ) )
24 1 6 grprcan ( ( 𝑅 ∈ Grp ∧ ( ( 𝑋 · 0 ) ∈ 𝐵0𝐵 ∧ ( 𝑋 · 0 ) ∈ 𝐵 ) ) → ( ( ( 𝑋 · 0 ) ( +g𝑅 ) ( 𝑋 · 0 ) ) = ( 0 ( +g𝑅 ) ( 𝑋 · 0 ) ) ↔ ( 𝑋 · 0 ) = 0 ) )
25 17 19 13 19 24 syl13anc ( ( 𝑅 ∈ Ring ∧ 𝑋𝐵 ) → ( ( ( 𝑋 · 0 ) ( +g𝑅 ) ( 𝑋 · 0 ) ) = ( 0 ( +g𝑅 ) ( 𝑋 · 0 ) ) ↔ ( 𝑋 · 0 ) = 0 ) )
26 23 25 mpbid ( ( 𝑅 ∈ Ring ∧ 𝑋𝐵 ) → ( 𝑋 · 0 ) = 0 )