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