Step |
Hyp |
Ref |
Expression |
1 |
|
rrgnz.t |
⊢ 𝐸 = ( RLReg ‘ 𝑅 ) |
2 |
|
rrgnz.z |
⊢ 0 = ( 0g ‘ 𝑅 ) |
3 |
|
eqid |
⊢ ( 1r ‘ 𝑅 ) = ( 1r ‘ 𝑅 ) |
4 |
3 2
|
nzrnz |
⊢ ( 𝑅 ∈ NzRing → ( 1r ‘ 𝑅 ) ≠ 0 ) |
5 |
4
|
neneqd |
⊢ ( 𝑅 ∈ NzRing → ¬ ( 1r ‘ 𝑅 ) = 0 ) |
6 |
|
nzrring |
⊢ ( 𝑅 ∈ NzRing → 𝑅 ∈ Ring ) |
7 |
6
|
adantr |
⊢ ( ( 𝑅 ∈ NzRing ∧ 0 ∈ 𝐸 ) → 𝑅 ∈ Ring ) |
8 |
|
simpr |
⊢ ( ( 𝑅 ∈ NzRing ∧ 0 ∈ 𝐸 ) → 0 ∈ 𝐸 ) |
9 |
|
eqid |
⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 ) |
10 |
9 3
|
ringidcl |
⊢ ( 𝑅 ∈ Ring → ( 1r ‘ 𝑅 ) ∈ ( Base ‘ 𝑅 ) ) |
11 |
7 10
|
syl |
⊢ ( ( 𝑅 ∈ NzRing ∧ 0 ∈ 𝐸 ) → ( 1r ‘ 𝑅 ) ∈ ( Base ‘ 𝑅 ) ) |
12 |
|
eqid |
⊢ ( .r ‘ 𝑅 ) = ( .r ‘ 𝑅 ) |
13 |
9 12 2 7 11
|
ringlzd |
⊢ ( ( 𝑅 ∈ NzRing ∧ 0 ∈ 𝐸 ) → ( 0 ( .r ‘ 𝑅 ) ( 1r ‘ 𝑅 ) ) = 0 ) |
14 |
1 9 12 2
|
rrgeq0 |
⊢ ( ( 𝑅 ∈ Ring ∧ 0 ∈ 𝐸 ∧ ( 1r ‘ 𝑅 ) ∈ ( Base ‘ 𝑅 ) ) → ( ( 0 ( .r ‘ 𝑅 ) ( 1r ‘ 𝑅 ) ) = 0 ↔ ( 1r ‘ 𝑅 ) = 0 ) ) |
15 |
14
|
biimpa |
⊢ ( ( ( 𝑅 ∈ Ring ∧ 0 ∈ 𝐸 ∧ ( 1r ‘ 𝑅 ) ∈ ( Base ‘ 𝑅 ) ) ∧ ( 0 ( .r ‘ 𝑅 ) ( 1r ‘ 𝑅 ) ) = 0 ) → ( 1r ‘ 𝑅 ) = 0 ) |
16 |
7 8 11 13 15
|
syl31anc |
⊢ ( ( 𝑅 ∈ NzRing ∧ 0 ∈ 𝐸 ) → ( 1r ‘ 𝑅 ) = 0 ) |
17 |
5 16
|
mtand |
⊢ ( 𝑅 ∈ NzRing → ¬ 0 ∈ 𝐸 ) |