Step |
Hyp |
Ref |
Expression |
1 |
|
rhmsubcrngc.c |
⊢ 𝐶 = ( RngCat ‘ 𝑈 ) |
2 |
|
rhmsubcrngc.u |
⊢ ( 𝜑 → 𝑈 ∈ 𝑉 ) |
3 |
|
rhmsubcrngc.b |
⊢ ( 𝜑 → 𝐵 = ( Ring ∩ 𝑈 ) ) |
4 |
|
rhmsubcrngc.h |
⊢ ( 𝜑 → 𝐻 = ( RingHom ↾ ( 𝐵 × 𝐵 ) ) ) |
5 |
3
|
eleq2d |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐵 ↔ 𝑥 ∈ ( Ring ∩ 𝑈 ) ) ) |
6 |
|
elin |
⊢ ( 𝑥 ∈ ( Ring ∩ 𝑈 ) ↔ ( 𝑥 ∈ Ring ∧ 𝑥 ∈ 𝑈 ) ) |
7 |
6
|
simplbi |
⊢ ( 𝑥 ∈ ( Ring ∩ 𝑈 ) → 𝑥 ∈ Ring ) |
8 |
5 7
|
syl6bi |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐵 → 𝑥 ∈ Ring ) ) |
9 |
8
|
imp |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → 𝑥 ∈ Ring ) |
10 |
|
eqid |
⊢ ( Base ‘ 𝑥 ) = ( Base ‘ 𝑥 ) |
11 |
10
|
idrhm |
⊢ ( 𝑥 ∈ Ring → ( I ↾ ( Base ‘ 𝑥 ) ) ∈ ( 𝑥 RingHom 𝑥 ) ) |
12 |
9 11
|
syl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → ( I ↾ ( Base ‘ 𝑥 ) ) ∈ ( 𝑥 RingHom 𝑥 ) ) |
13 |
|
eqid |
⊢ ( Base ‘ 𝐶 ) = ( Base ‘ 𝐶 ) |
14 |
|
eqid |
⊢ ( Id ‘ 𝐶 ) = ( Id ‘ 𝐶 ) |
15 |
2
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → 𝑈 ∈ 𝑉 ) |
16 |
|
ringrng |
⊢ ( 𝑥 ∈ Ring → 𝑥 ∈ Rng ) |
17 |
16
|
anim2i |
⊢ ( ( 𝑥 ∈ 𝑈 ∧ 𝑥 ∈ Ring ) → ( 𝑥 ∈ 𝑈 ∧ 𝑥 ∈ Rng ) ) |
18 |
17
|
ancoms |
⊢ ( ( 𝑥 ∈ Ring ∧ 𝑥 ∈ 𝑈 ) → ( 𝑥 ∈ 𝑈 ∧ 𝑥 ∈ Rng ) ) |
19 |
6 18
|
sylbi |
⊢ ( 𝑥 ∈ ( Ring ∩ 𝑈 ) → ( 𝑥 ∈ 𝑈 ∧ 𝑥 ∈ Rng ) ) |
20 |
19
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Ring ∩ 𝑈 ) ) → ( 𝑥 ∈ 𝑈 ∧ 𝑥 ∈ Rng ) ) |
21 |
|
elin |
⊢ ( 𝑥 ∈ ( 𝑈 ∩ Rng ) ↔ ( 𝑥 ∈ 𝑈 ∧ 𝑥 ∈ Rng ) ) |
22 |
20 21
|
sylibr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Ring ∩ 𝑈 ) ) → 𝑥 ∈ ( 𝑈 ∩ Rng ) ) |
23 |
1 13 2
|
rngcbas |
⊢ ( 𝜑 → ( Base ‘ 𝐶 ) = ( 𝑈 ∩ Rng ) ) |
24 |
23
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Ring ∩ 𝑈 ) ) → ( Base ‘ 𝐶 ) = ( 𝑈 ∩ Rng ) ) |
25 |
22 24
|
eleqtrrd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Ring ∩ 𝑈 ) ) → 𝑥 ∈ ( Base ‘ 𝐶 ) ) |
26 |
25
|
ex |
⊢ ( 𝜑 → ( 𝑥 ∈ ( Ring ∩ 𝑈 ) → 𝑥 ∈ ( Base ‘ 𝐶 ) ) ) |
27 |
5 26
|
sylbid |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐵 → 𝑥 ∈ ( Base ‘ 𝐶 ) ) ) |
28 |
27
|
imp |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → 𝑥 ∈ ( Base ‘ 𝐶 ) ) |
29 |
1 13 14 15 28 10
|
rngcid |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → ( ( Id ‘ 𝐶 ) ‘ 𝑥 ) = ( I ↾ ( Base ‘ 𝑥 ) ) ) |
30 |
4
|
oveqdr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → ( 𝑥 𝐻 𝑥 ) = ( 𝑥 ( RingHom ↾ ( 𝐵 × 𝐵 ) ) 𝑥 ) ) |
31 |
|
eqid |
⊢ ( RingCat ‘ 𝑈 ) = ( RingCat ‘ 𝑈 ) |
32 |
|
eqid |
⊢ ( Base ‘ ( RingCat ‘ 𝑈 ) ) = ( Base ‘ ( RingCat ‘ 𝑈 ) ) |
33 |
|
eqid |
⊢ ( Hom ‘ ( RingCat ‘ 𝑈 ) ) = ( Hom ‘ ( RingCat ‘ 𝑈 ) ) |
34 |
31 32 2 33
|
ringchomfval |
⊢ ( 𝜑 → ( Hom ‘ ( RingCat ‘ 𝑈 ) ) = ( RingHom ↾ ( ( Base ‘ ( RingCat ‘ 𝑈 ) ) × ( Base ‘ ( RingCat ‘ 𝑈 ) ) ) ) ) |
35 |
31 32 2
|
ringcbas |
⊢ ( 𝜑 → ( Base ‘ ( RingCat ‘ 𝑈 ) ) = ( 𝑈 ∩ Ring ) ) |
36 |
|
incom |
⊢ ( Ring ∩ 𝑈 ) = ( 𝑈 ∩ Ring ) |
37 |
3 36
|
eqtrdi |
⊢ ( 𝜑 → 𝐵 = ( 𝑈 ∩ Ring ) ) |
38 |
37
|
eqcomd |
⊢ ( 𝜑 → ( 𝑈 ∩ Ring ) = 𝐵 ) |
39 |
35 38
|
eqtrd |
⊢ ( 𝜑 → ( Base ‘ ( RingCat ‘ 𝑈 ) ) = 𝐵 ) |
40 |
39
|
sqxpeqd |
⊢ ( 𝜑 → ( ( Base ‘ ( RingCat ‘ 𝑈 ) ) × ( Base ‘ ( RingCat ‘ 𝑈 ) ) ) = ( 𝐵 × 𝐵 ) ) |
41 |
40
|
reseq2d |
⊢ ( 𝜑 → ( RingHom ↾ ( ( Base ‘ ( RingCat ‘ 𝑈 ) ) × ( Base ‘ ( RingCat ‘ 𝑈 ) ) ) ) = ( RingHom ↾ ( 𝐵 × 𝐵 ) ) ) |
42 |
34 41
|
eqtrd |
⊢ ( 𝜑 → ( Hom ‘ ( RingCat ‘ 𝑈 ) ) = ( RingHom ↾ ( 𝐵 × 𝐵 ) ) ) |
43 |
42
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → ( Hom ‘ ( RingCat ‘ 𝑈 ) ) = ( RingHom ↾ ( 𝐵 × 𝐵 ) ) ) |
44 |
43
|
eqcomd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → ( RingHom ↾ ( 𝐵 × 𝐵 ) ) = ( Hom ‘ ( RingCat ‘ 𝑈 ) ) ) |
45 |
44
|
oveqd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → ( 𝑥 ( RingHom ↾ ( 𝐵 × 𝐵 ) ) 𝑥 ) = ( 𝑥 ( Hom ‘ ( RingCat ‘ 𝑈 ) ) 𝑥 ) ) |
46 |
37
|
eleq2d |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐵 ↔ 𝑥 ∈ ( 𝑈 ∩ Ring ) ) ) |
47 |
46
|
biimpa |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → 𝑥 ∈ ( 𝑈 ∩ Ring ) ) |
48 |
35
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → ( Base ‘ ( RingCat ‘ 𝑈 ) ) = ( 𝑈 ∩ Ring ) ) |
49 |
47 48
|
eleqtrrd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → 𝑥 ∈ ( Base ‘ ( RingCat ‘ 𝑈 ) ) ) |
50 |
31 32 15 33 49 49
|
ringchom |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → ( 𝑥 ( Hom ‘ ( RingCat ‘ 𝑈 ) ) 𝑥 ) = ( 𝑥 RingHom 𝑥 ) ) |
51 |
30 45 50
|
3eqtrd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → ( 𝑥 𝐻 𝑥 ) = ( 𝑥 RingHom 𝑥 ) ) |
52 |
12 29 51
|
3eltr4d |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → ( ( Id ‘ 𝐶 ) ‘ 𝑥 ) ∈ ( 𝑥 𝐻 𝑥 ) ) |