Step |
Hyp |
Ref |
Expression |
1 |
|
zrinitorngc.u |
⊢ ( 𝜑 → 𝑈 ∈ 𝑉 ) |
2 |
|
zrinitorngc.c |
⊢ 𝐶 = ( RngCat ‘ 𝑈 ) |
3 |
|
zrinitorngc.z |
⊢ ( 𝜑 → 𝑍 ∈ ( Ring ∖ NzRing ) ) |
4 |
|
zrinitorngc.e |
⊢ ( 𝜑 → 𝑍 ∈ 𝑈 ) |
5 |
1 2 3 4
|
zrinitorngc |
⊢ ( 𝜑 → 𝑍 ∈ ( InitO ‘ 𝐶 ) ) |
6 |
1 2 3 4
|
zrtermorngc |
⊢ ( 𝜑 → 𝑍 ∈ ( TermO ‘ 𝐶 ) ) |
7 |
|
eqid |
⊢ ( Base ‘ 𝐶 ) = ( Base ‘ 𝐶 ) |
8 |
|
eqid |
⊢ ( Hom ‘ 𝐶 ) = ( Hom ‘ 𝐶 ) |
9 |
2
|
rngccat |
⊢ ( 𝑈 ∈ 𝑉 → 𝐶 ∈ Cat ) |
10 |
1 9
|
syl |
⊢ ( 𝜑 → 𝐶 ∈ Cat ) |
11 |
3
|
eldifad |
⊢ ( 𝜑 → 𝑍 ∈ Ring ) |
12 |
|
ringrng |
⊢ ( 𝑍 ∈ Ring → 𝑍 ∈ Rng ) |
13 |
11 12
|
syl |
⊢ ( 𝜑 → 𝑍 ∈ Rng ) |
14 |
4 13
|
elind |
⊢ ( 𝜑 → 𝑍 ∈ ( 𝑈 ∩ Rng ) ) |
15 |
2 7 1
|
rngcbas |
⊢ ( 𝜑 → ( Base ‘ 𝐶 ) = ( 𝑈 ∩ Rng ) ) |
16 |
14 15
|
eleqtrrd |
⊢ ( 𝜑 → 𝑍 ∈ ( Base ‘ 𝐶 ) ) |
17 |
7 8 10 16
|
iszeroo |
⊢ ( 𝜑 → ( 𝑍 ∈ ( ZeroO ‘ 𝐶 ) ↔ ( 𝑍 ∈ ( InitO ‘ 𝐶 ) ∧ 𝑍 ∈ ( TermO ‘ 𝐶 ) ) ) ) |
18 |
5 6 17
|
mpbir2and |
⊢ ( 𝜑 → 𝑍 ∈ ( ZeroO ‘ 𝐶 ) ) |