Step |
Hyp |
Ref |
Expression |
1 |
|
irredn0.i |
⊢ 𝐼 = ( Irred ‘ 𝑅 ) |
2 |
|
irredn0.z |
⊢ 0 = ( 0g ‘ 𝑅 ) |
3 |
|
eqid |
⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 ) |
4 |
3 2
|
ring0cl |
⊢ ( 𝑅 ∈ Ring → 0 ∈ ( Base ‘ 𝑅 ) ) |
5 |
4
|
anim1i |
⊢ ( ( 𝑅 ∈ Ring ∧ ¬ 0 ∈ ( Unit ‘ 𝑅 ) ) → ( 0 ∈ ( Base ‘ 𝑅 ) ∧ ¬ 0 ∈ ( Unit ‘ 𝑅 ) ) ) |
6 |
|
eldif |
⊢ ( 0 ∈ ( ( Base ‘ 𝑅 ) ∖ ( Unit ‘ 𝑅 ) ) ↔ ( 0 ∈ ( Base ‘ 𝑅 ) ∧ ¬ 0 ∈ ( Unit ‘ 𝑅 ) ) ) |
7 |
5 6
|
sylibr |
⊢ ( ( 𝑅 ∈ Ring ∧ ¬ 0 ∈ ( Unit ‘ 𝑅 ) ) → 0 ∈ ( ( Base ‘ 𝑅 ) ∖ ( Unit ‘ 𝑅 ) ) ) |
8 |
|
eqid |
⊢ ( .r ‘ 𝑅 ) = ( .r ‘ 𝑅 ) |
9 |
3 8 2
|
ringlz |
⊢ ( ( 𝑅 ∈ Ring ∧ 0 ∈ ( Base ‘ 𝑅 ) ) → ( 0 ( .r ‘ 𝑅 ) 0 ) = 0 ) |
10 |
4 9
|
mpdan |
⊢ ( 𝑅 ∈ Ring → ( 0 ( .r ‘ 𝑅 ) 0 ) = 0 ) |
11 |
10
|
adantr |
⊢ ( ( 𝑅 ∈ Ring ∧ ¬ 0 ∈ ( Unit ‘ 𝑅 ) ) → ( 0 ( .r ‘ 𝑅 ) 0 ) = 0 ) |
12 |
|
oveq1 |
⊢ ( 𝑥 = 0 → ( 𝑥 ( .r ‘ 𝑅 ) 𝑦 ) = ( 0 ( .r ‘ 𝑅 ) 𝑦 ) ) |
13 |
12
|
eqeq1d |
⊢ ( 𝑥 = 0 → ( ( 𝑥 ( .r ‘ 𝑅 ) 𝑦 ) = 0 ↔ ( 0 ( .r ‘ 𝑅 ) 𝑦 ) = 0 ) ) |
14 |
|
oveq2 |
⊢ ( 𝑦 = 0 → ( 0 ( .r ‘ 𝑅 ) 𝑦 ) = ( 0 ( .r ‘ 𝑅 ) 0 ) ) |
15 |
14
|
eqeq1d |
⊢ ( 𝑦 = 0 → ( ( 0 ( .r ‘ 𝑅 ) 𝑦 ) = 0 ↔ ( 0 ( .r ‘ 𝑅 ) 0 ) = 0 ) ) |
16 |
13 15
|
rspc2ev |
⊢ ( ( 0 ∈ ( ( Base ‘ 𝑅 ) ∖ ( Unit ‘ 𝑅 ) ) ∧ 0 ∈ ( ( Base ‘ 𝑅 ) ∖ ( Unit ‘ 𝑅 ) ) ∧ ( 0 ( .r ‘ 𝑅 ) 0 ) = 0 ) → ∃ 𝑥 ∈ ( ( Base ‘ 𝑅 ) ∖ ( Unit ‘ 𝑅 ) ) ∃ 𝑦 ∈ ( ( Base ‘ 𝑅 ) ∖ ( Unit ‘ 𝑅 ) ) ( 𝑥 ( .r ‘ 𝑅 ) 𝑦 ) = 0 ) |
17 |
7 7 11 16
|
syl3anc |
⊢ ( ( 𝑅 ∈ Ring ∧ ¬ 0 ∈ ( Unit ‘ 𝑅 ) ) → ∃ 𝑥 ∈ ( ( Base ‘ 𝑅 ) ∖ ( Unit ‘ 𝑅 ) ) ∃ 𝑦 ∈ ( ( Base ‘ 𝑅 ) ∖ ( Unit ‘ 𝑅 ) ) ( 𝑥 ( .r ‘ 𝑅 ) 𝑦 ) = 0 ) |
18 |
17
|
ex |
⊢ ( 𝑅 ∈ Ring → ( ¬ 0 ∈ ( Unit ‘ 𝑅 ) → ∃ 𝑥 ∈ ( ( Base ‘ 𝑅 ) ∖ ( Unit ‘ 𝑅 ) ) ∃ 𝑦 ∈ ( ( Base ‘ 𝑅 ) ∖ ( Unit ‘ 𝑅 ) ) ( 𝑥 ( .r ‘ 𝑅 ) 𝑦 ) = 0 ) ) |
19 |
18
|
orrd |
⊢ ( 𝑅 ∈ Ring → ( 0 ∈ ( Unit ‘ 𝑅 ) ∨ ∃ 𝑥 ∈ ( ( Base ‘ 𝑅 ) ∖ ( Unit ‘ 𝑅 ) ) ∃ 𝑦 ∈ ( ( Base ‘ 𝑅 ) ∖ ( Unit ‘ 𝑅 ) ) ( 𝑥 ( .r ‘ 𝑅 ) 𝑦 ) = 0 ) ) |
20 |
|
eqid |
⊢ ( Unit ‘ 𝑅 ) = ( Unit ‘ 𝑅 ) |
21 |
|
eqid |
⊢ ( ( Base ‘ 𝑅 ) ∖ ( Unit ‘ 𝑅 ) ) = ( ( Base ‘ 𝑅 ) ∖ ( Unit ‘ 𝑅 ) ) |
22 |
3 20 1 21 8
|
isnirred |
⊢ ( 0 ∈ ( Base ‘ 𝑅 ) → ( ¬ 0 ∈ 𝐼 ↔ ( 0 ∈ ( Unit ‘ 𝑅 ) ∨ ∃ 𝑥 ∈ ( ( Base ‘ 𝑅 ) ∖ ( Unit ‘ 𝑅 ) ) ∃ 𝑦 ∈ ( ( Base ‘ 𝑅 ) ∖ ( Unit ‘ 𝑅 ) ) ( 𝑥 ( .r ‘ 𝑅 ) 𝑦 ) = 0 ) ) ) |
23 |
4 22
|
syl |
⊢ ( 𝑅 ∈ Ring → ( ¬ 0 ∈ 𝐼 ↔ ( 0 ∈ ( Unit ‘ 𝑅 ) ∨ ∃ 𝑥 ∈ ( ( Base ‘ 𝑅 ) ∖ ( Unit ‘ 𝑅 ) ) ∃ 𝑦 ∈ ( ( Base ‘ 𝑅 ) ∖ ( Unit ‘ 𝑅 ) ) ( 𝑥 ( .r ‘ 𝑅 ) 𝑦 ) = 0 ) ) ) |
24 |
19 23
|
mpbird |
⊢ ( 𝑅 ∈ Ring → ¬ 0 ∈ 𝐼 ) |
25 |
24
|
adantr |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐼 ) → ¬ 0 ∈ 𝐼 ) |
26 |
|
simpr |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐼 ) → 𝑋 ∈ 𝐼 ) |
27 |
|
eleq1 |
⊢ ( 𝑋 = 0 → ( 𝑋 ∈ 𝐼 ↔ 0 ∈ 𝐼 ) ) |
28 |
26 27
|
syl5ibcom |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐼 ) → ( 𝑋 = 0 → 0 ∈ 𝐼 ) ) |
29 |
28
|
necon3bd |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐼 ) → ( ¬ 0 ∈ 𝐼 → 𝑋 ≠ 0 ) ) |
30 |
25 29
|
mpd |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐼 ) → 𝑋 ≠ 0 ) |