Step |
Hyp |
Ref |
Expression |
1 |
|
rngcrescrhmALTV.u |
⊢ ( 𝜑 → 𝑈 ∈ 𝑉 ) |
2 |
|
rngcrescrhmALTV.c |
⊢ 𝐶 = ( RngCatALTV ‘ 𝑈 ) |
3 |
|
rngcrescrhmALTV.r |
⊢ ( 𝜑 → 𝑅 = ( Ring ∩ 𝑈 ) ) |
4 |
|
rngcrescrhmALTV.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 |
|
eqid |
⊢ ( RngCatALTV ‘ 𝑈 ) = ( RngCatALTV ‘ 𝑈 ) |
9 |
|
eqid |
⊢ ( Rng ∩ 𝑈 ) = ( Rng ∩ 𝑈 ) |
10 |
|
eqid |
⊢ ( Homf ‘ ( RngCatALTV ‘ 𝑈 ) ) = ( Homf ‘ ( RngCatALTV ‘ 𝑈 ) ) |
11 |
8 9 1 10
|
rngchomrnghmresALTV |
⊢ ( 𝜑 → ( Homf ‘ ( RngCatALTV ‘ 𝑈 ) ) = ( RngHomo ↾ ( ( Rng ∩ 𝑈 ) × ( Rng ∩ 𝑈 ) ) ) ) |
12 |
6 7 11
|
3brtr4d |
⊢ ( 𝜑 → 𝐻 ⊆cat ( Homf ‘ ( RngCatALTV ‘ 𝑈 ) ) ) |
13 |
1 2 3 4
|
rhmsubcALTVlem3 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑅 ) → ( ( Id ‘ ( RngCatALTV ‘ 𝑈 ) ) ‘ 𝑥 ) ∈ ( 𝑥 𝐻 𝑥 ) ) |
14 |
1 2 3 4
|
rhmsubcALTVlem4 |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ 𝑅 ) ∧ ( 𝑦 ∈ 𝑅 ∧ 𝑧 ∈ 𝑅 ) ) ∧ ( 𝑓 ∈ ( 𝑥 𝐻 𝑦 ) ∧ 𝑔 ∈ ( 𝑦 𝐻 𝑧 ) ) ) → ( 𝑔 ( 〈 𝑥 , 𝑦 〉 ( comp ‘ ( RngCatALTV ‘ 𝑈 ) ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 𝐻 𝑧 ) ) |
15 |
14
|
ralrimivva |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝑅 ) ∧ ( 𝑦 ∈ 𝑅 ∧ 𝑧 ∈ 𝑅 ) ) → ∀ 𝑓 ∈ ( 𝑥 𝐻 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 𝐻 𝑧 ) ( 𝑔 ( 〈 𝑥 , 𝑦 〉 ( comp ‘ ( RngCatALTV ‘ 𝑈 ) ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 𝐻 𝑧 ) ) |
16 |
15
|
ralrimivva |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑅 ) → ∀ 𝑦 ∈ 𝑅 ∀ 𝑧 ∈ 𝑅 ∀ 𝑓 ∈ ( 𝑥 𝐻 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 𝐻 𝑧 ) ( 𝑔 ( 〈 𝑥 , 𝑦 〉 ( comp ‘ ( RngCatALTV ‘ 𝑈 ) ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 𝐻 𝑧 ) ) |
17 |
13 16
|
jca |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑅 ) → ( ( ( Id ‘ ( RngCatALTV ‘ 𝑈 ) ) ‘ 𝑥 ) ∈ ( 𝑥 𝐻 𝑥 ) ∧ ∀ 𝑦 ∈ 𝑅 ∀ 𝑧 ∈ 𝑅 ∀ 𝑓 ∈ ( 𝑥 𝐻 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 𝐻 𝑧 ) ( 𝑔 ( 〈 𝑥 , 𝑦 〉 ( comp ‘ ( RngCatALTV ‘ 𝑈 ) ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 𝐻 𝑧 ) ) ) |
18 |
17
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝑅 ( ( ( Id ‘ ( RngCatALTV ‘ 𝑈 ) ) ‘ 𝑥 ) ∈ ( 𝑥 𝐻 𝑥 ) ∧ ∀ 𝑦 ∈ 𝑅 ∀ 𝑧 ∈ 𝑅 ∀ 𝑓 ∈ ( 𝑥 𝐻 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 𝐻 𝑧 ) ( 𝑔 ( 〈 𝑥 , 𝑦 〉 ( comp ‘ ( RngCatALTV ‘ 𝑈 ) ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 𝐻 𝑧 ) ) ) |
19 |
|
eqid |
⊢ ( Id ‘ ( RngCatALTV ‘ 𝑈 ) ) = ( Id ‘ ( RngCatALTV ‘ 𝑈 ) ) |
20 |
|
eqid |
⊢ ( comp ‘ ( RngCatALTV ‘ 𝑈 ) ) = ( comp ‘ ( RngCatALTV ‘ 𝑈 ) ) |
21 |
8
|
rngccatALTV |
⊢ ( 𝑈 ∈ 𝑉 → ( RngCatALTV ‘ 𝑈 ) ∈ Cat ) |
22 |
1 21
|
syl |
⊢ ( 𝜑 → ( RngCatALTV ‘ 𝑈 ) ∈ Cat ) |
23 |
1 2 3 4
|
rhmsubcALTVlem1 |
⊢ ( 𝜑 → 𝐻 Fn ( 𝑅 × 𝑅 ) ) |
24 |
10 19 20 22 23
|
issubc2 |
⊢ ( 𝜑 → ( 𝐻 ∈ ( Subcat ‘ ( RngCatALTV ‘ 𝑈 ) ) ↔ ( 𝐻 ⊆cat ( Homf ‘ ( RngCatALTV ‘ 𝑈 ) ) ∧ ∀ 𝑥 ∈ 𝑅 ( ( ( Id ‘ ( RngCatALTV ‘ 𝑈 ) ) ‘ 𝑥 ) ∈ ( 𝑥 𝐻 𝑥 ) ∧ ∀ 𝑦 ∈ 𝑅 ∀ 𝑧 ∈ 𝑅 ∀ 𝑓 ∈ ( 𝑥 𝐻 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 𝐻 𝑧 ) ( 𝑔 ( 〈 𝑥 , 𝑦 〉 ( comp ‘ ( RngCatALTV ‘ 𝑈 ) ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 𝐻 𝑧 ) ) ) ) ) |
25 |
12 18 24
|
mpbir2and |
⊢ ( 𝜑 → 𝐻 ∈ ( Subcat ‘ ( RngCatALTV ‘ 𝑈 ) ) ) |