Step |
Hyp |
Ref |
Expression |
1 |
|
rngcrescrhm.u |
⊢ ( 𝜑 → 𝑈 ∈ 𝑉 ) |
2 |
|
rngcrescrhm.c |
⊢ 𝐶 = ( RngCat ‘ 𝑈 ) |
3 |
|
rngcrescrhm.r |
⊢ ( 𝜑 → 𝑅 = ( Ring ∩ 𝑈 ) ) |
4 |
|
rngcrescrhm.h |
⊢ 𝐻 = ( RingHom ↾ ( 𝑅 × 𝑅 ) ) |
5 |
|
eqidd |
⊢ ( 𝜑 → ( Rng ∩ 𝑈 ) = ( Rng ∩ 𝑈 ) ) |
6 |
1 3 5
|
rhmsscrnghm |
⊢ ( 𝜑 → ( RingHom ↾ ( 𝑅 × 𝑅 ) ) ⊆cat ( RngHomo ↾ ( ( Rng ∩ 𝑈 ) × ( Rng ∩ 𝑈 ) ) ) ) |
7 |
4
|
a1i |
⊢ ( 𝜑 → 𝐻 = ( RingHom ↾ ( 𝑅 × 𝑅 ) ) ) |
8 |
2
|
a1i |
⊢ ( 𝜑 → 𝐶 = ( RngCat ‘ 𝑈 ) ) |
9 |
8
|
eqcomd |
⊢ ( 𝜑 → ( RngCat ‘ 𝑈 ) = 𝐶 ) |
10 |
9
|
fveq2d |
⊢ ( 𝜑 → ( Homf ‘ ( RngCat ‘ 𝑈 ) ) = ( Homf ‘ 𝐶 ) ) |
11 |
|
eqid |
⊢ ( Base ‘ 𝐶 ) = ( Base ‘ 𝐶 ) |
12 |
2 11 1
|
rngchomfeqhom |
⊢ ( 𝜑 → ( Homf ‘ 𝐶 ) = ( Hom ‘ 𝐶 ) ) |
13 |
|
eqid |
⊢ ( Hom ‘ 𝐶 ) = ( Hom ‘ 𝐶 ) |
14 |
2 11 1 13
|
rngchomfval |
⊢ ( 𝜑 → ( Hom ‘ 𝐶 ) = ( RngHomo ↾ ( ( Base ‘ 𝐶 ) × ( Base ‘ 𝐶 ) ) ) ) |
15 |
2 11 1
|
rngcbas |
⊢ ( 𝜑 → ( Base ‘ 𝐶 ) = ( 𝑈 ∩ Rng ) ) |
16 |
|
incom |
⊢ ( 𝑈 ∩ Rng ) = ( Rng ∩ 𝑈 ) |
17 |
15 16
|
eqtrdi |
⊢ ( 𝜑 → ( Base ‘ 𝐶 ) = ( Rng ∩ 𝑈 ) ) |
18 |
17
|
sqxpeqd |
⊢ ( 𝜑 → ( ( Base ‘ 𝐶 ) × ( Base ‘ 𝐶 ) ) = ( ( Rng ∩ 𝑈 ) × ( Rng ∩ 𝑈 ) ) ) |
19 |
18
|
reseq2d |
⊢ ( 𝜑 → ( RngHomo ↾ ( ( Base ‘ 𝐶 ) × ( Base ‘ 𝐶 ) ) ) = ( RngHomo ↾ ( ( Rng ∩ 𝑈 ) × ( Rng ∩ 𝑈 ) ) ) ) |
20 |
14 19
|
eqtrd |
⊢ ( 𝜑 → ( Hom ‘ 𝐶 ) = ( RngHomo ↾ ( ( Rng ∩ 𝑈 ) × ( Rng ∩ 𝑈 ) ) ) ) |
21 |
10 12 20
|
3eqtrd |
⊢ ( 𝜑 → ( Homf ‘ ( RngCat ‘ 𝑈 ) ) = ( RngHomo ↾ ( ( Rng ∩ 𝑈 ) × ( Rng ∩ 𝑈 ) ) ) ) |
22 |
6 7 21
|
3brtr4d |
⊢ ( 𝜑 → 𝐻 ⊆cat ( Homf ‘ ( RngCat ‘ 𝑈 ) ) ) |
23 |
1 2 3 4
|
rhmsubclem3 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑅 ) → ( ( Id ‘ ( RngCat ‘ 𝑈 ) ) ‘ 𝑥 ) ∈ ( 𝑥 𝐻 𝑥 ) ) |
24 |
1 2 3 4
|
rhmsubclem4 |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ 𝑅 ) ∧ ( 𝑦 ∈ 𝑅 ∧ 𝑧 ∈ 𝑅 ) ) ∧ ( 𝑓 ∈ ( 𝑥 𝐻 𝑦 ) ∧ 𝑔 ∈ ( 𝑦 𝐻 𝑧 ) ) ) → ( 𝑔 ( 〈 𝑥 , 𝑦 〉 ( comp ‘ ( RngCat ‘ 𝑈 ) ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 𝐻 𝑧 ) ) |
25 |
24
|
ralrimivva |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝑅 ) ∧ ( 𝑦 ∈ 𝑅 ∧ 𝑧 ∈ 𝑅 ) ) → ∀ 𝑓 ∈ ( 𝑥 𝐻 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 𝐻 𝑧 ) ( 𝑔 ( 〈 𝑥 , 𝑦 〉 ( comp ‘ ( RngCat ‘ 𝑈 ) ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 𝐻 𝑧 ) ) |
26 |
25
|
ralrimivva |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑅 ) → ∀ 𝑦 ∈ 𝑅 ∀ 𝑧 ∈ 𝑅 ∀ 𝑓 ∈ ( 𝑥 𝐻 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 𝐻 𝑧 ) ( 𝑔 ( 〈 𝑥 , 𝑦 〉 ( comp ‘ ( RngCat ‘ 𝑈 ) ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 𝐻 𝑧 ) ) |
27 |
23 26
|
jca |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑅 ) → ( ( ( Id ‘ ( RngCat ‘ 𝑈 ) ) ‘ 𝑥 ) ∈ ( 𝑥 𝐻 𝑥 ) ∧ ∀ 𝑦 ∈ 𝑅 ∀ 𝑧 ∈ 𝑅 ∀ 𝑓 ∈ ( 𝑥 𝐻 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 𝐻 𝑧 ) ( 𝑔 ( 〈 𝑥 , 𝑦 〉 ( comp ‘ ( RngCat ‘ 𝑈 ) ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 𝐻 𝑧 ) ) ) |
28 |
27
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝑅 ( ( ( Id ‘ ( RngCat ‘ 𝑈 ) ) ‘ 𝑥 ) ∈ ( 𝑥 𝐻 𝑥 ) ∧ ∀ 𝑦 ∈ 𝑅 ∀ 𝑧 ∈ 𝑅 ∀ 𝑓 ∈ ( 𝑥 𝐻 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 𝐻 𝑧 ) ( 𝑔 ( 〈 𝑥 , 𝑦 〉 ( comp ‘ ( RngCat ‘ 𝑈 ) ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 𝐻 𝑧 ) ) ) |
29 |
|
eqid |
⊢ ( Homf ‘ ( RngCat ‘ 𝑈 ) ) = ( Homf ‘ ( RngCat ‘ 𝑈 ) ) |
30 |
|
eqid |
⊢ ( Id ‘ ( RngCat ‘ 𝑈 ) ) = ( Id ‘ ( RngCat ‘ 𝑈 ) ) |
31 |
|
eqid |
⊢ ( comp ‘ ( RngCat ‘ 𝑈 ) ) = ( comp ‘ ( RngCat ‘ 𝑈 ) ) |
32 |
|
eqid |
⊢ ( RngCat ‘ 𝑈 ) = ( RngCat ‘ 𝑈 ) |
33 |
32
|
rngccat |
⊢ ( 𝑈 ∈ 𝑉 → ( RngCat ‘ 𝑈 ) ∈ Cat ) |
34 |
1 33
|
syl |
⊢ ( 𝜑 → ( RngCat ‘ 𝑈 ) ∈ Cat ) |
35 |
1 2 3 4
|
rhmsubclem1 |
⊢ ( 𝜑 → 𝐻 Fn ( 𝑅 × 𝑅 ) ) |
36 |
29 30 31 34 35
|
issubc2 |
⊢ ( 𝜑 → ( 𝐻 ∈ ( Subcat ‘ ( RngCat ‘ 𝑈 ) ) ↔ ( 𝐻 ⊆cat ( Homf ‘ ( RngCat ‘ 𝑈 ) ) ∧ ∀ 𝑥 ∈ 𝑅 ( ( ( Id ‘ ( RngCat ‘ 𝑈 ) ) ‘ 𝑥 ) ∈ ( 𝑥 𝐻 𝑥 ) ∧ ∀ 𝑦 ∈ 𝑅 ∀ 𝑧 ∈ 𝑅 ∀ 𝑓 ∈ ( 𝑥 𝐻 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 𝐻 𝑧 ) ( 𝑔 ( 〈 𝑥 , 𝑦 〉 ( comp ‘ ( RngCat ‘ 𝑈 ) ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 𝐻 𝑧 ) ) ) ) ) |
37 |
22 28 36
|
mpbir2and |
⊢ ( 𝜑 → 𝐻 ∈ ( Subcat ‘ ( RngCat ‘ 𝑈 ) ) ) |