Metamath Proof Explorer


Theorem rnglz

Description: The zero of a nonunital ring is a left-absorbing element. (Contributed by AV, 17-Apr-2020)

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

Proof

Step Hyp Ref Expression
1 rngcl.b 𝐵 = ( Base ‘ 𝑅 )
2 rngcl.t · = ( .r𝑅 )
3 rnglz.z 0 = ( 0g𝑅 )
4 rngabl ( 𝑅 ∈ Rng → 𝑅 ∈ Abel )
5 ablgrp ( 𝑅 ∈ Abel → 𝑅 ∈ Grp )
6 4 5 syl ( 𝑅 ∈ Rng → 𝑅 ∈ Grp )
7 1 3 grpidcl ( 𝑅 ∈ Grp → 0𝐵 )
8 eqid ( +g𝑅 ) = ( +g𝑅 )
9 1 8 3 grplid ( ( 𝑅 ∈ Grp ∧ 0𝐵 ) → ( 0 ( +g𝑅 ) 0 ) = 0 )
10 6 7 9 syl2anc2 ( 𝑅 ∈ Rng → ( 0 ( +g𝑅 ) 0 ) = 0 )
11 10 adantr ( ( 𝑅 ∈ Rng ∧ 𝑋𝐵 ) → ( 0 ( +g𝑅 ) 0 ) = 0 )
12 11 oveq1d ( ( 𝑅 ∈ Rng ∧ 𝑋𝐵 ) → ( ( 0 ( +g𝑅 ) 0 ) · 𝑋 ) = ( 0 · 𝑋 ) )
13 simpl ( ( 𝑅 ∈ Rng ∧ 𝑋𝐵 ) → 𝑅 ∈ Rng )
14 6 7 syl ( 𝑅 ∈ Rng → 0𝐵 )
15 14 14 jca ( 𝑅 ∈ Rng → ( 0𝐵0𝐵 ) )
16 15 anim1i ( ( 𝑅 ∈ Rng ∧ 𝑋𝐵 ) → ( ( 0𝐵0𝐵 ) ∧ 𝑋𝐵 ) )
17 df-3an ( ( 0𝐵0𝐵𝑋𝐵 ) ↔ ( ( 0𝐵0𝐵 ) ∧ 𝑋𝐵 ) )
18 16 17 sylibr ( ( 𝑅 ∈ Rng ∧ 𝑋𝐵 ) → ( 0𝐵0𝐵𝑋𝐵 ) )
19 1 8 2 rngdir ( ( 𝑅 ∈ Rng ∧ ( 0𝐵0𝐵𝑋𝐵 ) ) → ( ( 0 ( +g𝑅 ) 0 ) · 𝑋 ) = ( ( 0 · 𝑋 ) ( +g𝑅 ) ( 0 · 𝑋 ) ) )
20 13 18 19 syl2anc ( ( 𝑅 ∈ Rng ∧ 𝑋𝐵 ) → ( ( 0 ( +g𝑅 ) 0 ) · 𝑋 ) = ( ( 0 · 𝑋 ) ( +g𝑅 ) ( 0 · 𝑋 ) ) )
21 6 adantr ( ( 𝑅 ∈ Rng ∧ 𝑋𝐵 ) → 𝑅 ∈ Grp )
22 14 adantr ( ( 𝑅 ∈ Rng ∧ 𝑋𝐵 ) → 0𝐵 )
23 simpr ( ( 𝑅 ∈ Rng ∧ 𝑋𝐵 ) → 𝑋𝐵 )
24 1 2 rngcl ( ( 𝑅 ∈ Rng ∧ 0𝐵𝑋𝐵 ) → ( 0 · 𝑋 ) ∈ 𝐵 )
25 13 22 23 24 syl3anc ( ( 𝑅 ∈ Rng ∧ 𝑋𝐵 ) → ( 0 · 𝑋 ) ∈ 𝐵 )
26 1 8 3 grprid ( ( 𝑅 ∈ Grp ∧ ( 0 · 𝑋 ) ∈ 𝐵 ) → ( ( 0 · 𝑋 ) ( +g𝑅 ) 0 ) = ( 0 · 𝑋 ) )
27 26 eqcomd ( ( 𝑅 ∈ Grp ∧ ( 0 · 𝑋 ) ∈ 𝐵 ) → ( 0 · 𝑋 ) = ( ( 0 · 𝑋 ) ( +g𝑅 ) 0 ) )
28 21 25 27 syl2anc ( ( 𝑅 ∈ Rng ∧ 𝑋𝐵 ) → ( 0 · 𝑋 ) = ( ( 0 · 𝑋 ) ( +g𝑅 ) 0 ) )
29 12 20 28 3eqtr3d ( ( 𝑅 ∈ Rng ∧ 𝑋𝐵 ) → ( ( 0 · 𝑋 ) ( +g𝑅 ) ( 0 · 𝑋 ) ) = ( ( 0 · 𝑋 ) ( +g𝑅 ) 0 ) )
30 1 8 grplcan ( ( 𝑅 ∈ Grp ∧ ( ( 0 · 𝑋 ) ∈ 𝐵0𝐵 ∧ ( 0 · 𝑋 ) ∈ 𝐵 ) ) → ( ( ( 0 · 𝑋 ) ( +g𝑅 ) ( 0 · 𝑋 ) ) = ( ( 0 · 𝑋 ) ( +g𝑅 ) 0 ) ↔ ( 0 · 𝑋 ) = 0 ) )
31 21 25 22 25 30 syl13anc ( ( 𝑅 ∈ Rng ∧ 𝑋𝐵 ) → ( ( ( 0 · 𝑋 ) ( +g𝑅 ) ( 0 · 𝑋 ) ) = ( ( 0 · 𝑋 ) ( +g𝑅 ) 0 ) ↔ ( 0 · 𝑋 ) = 0 ) )
32 29 31 mpbid ( ( 𝑅 ∈ Rng ∧ 𝑋𝐵 ) → ( 0 · 𝑋 ) = 0 )