Step |
Hyp |
Ref |
Expression |
1 |
|
srhmsubc.s |
⊢ ∀ 𝑟 ∈ 𝑆 𝑟 ∈ Ring |
2 |
|
srhmsubc.c |
⊢ 𝐶 = ( 𝑈 ∩ 𝑆 ) |
3 |
|
srhmsubc.j |
⊢ 𝐽 = ( 𝑟 ∈ 𝐶 , 𝑠 ∈ 𝐶 ↦ ( 𝑟 RingHom 𝑠 ) ) |
4 |
|
eleq1w |
⊢ ( 𝑟 = 𝑥 → ( 𝑟 ∈ Ring ↔ 𝑥 ∈ Ring ) ) |
5 |
4 1
|
vtoclri |
⊢ ( 𝑥 ∈ 𝑆 → 𝑥 ∈ Ring ) |
6 |
5
|
ssriv |
⊢ 𝑆 ⊆ Ring |
7 |
|
sslin |
⊢ ( 𝑆 ⊆ Ring → ( 𝑈 ∩ 𝑆 ) ⊆ ( 𝑈 ∩ Ring ) ) |
8 |
6 7
|
mp1i |
⊢ ( 𝑈 ∈ 𝑉 → ( 𝑈 ∩ 𝑆 ) ⊆ ( 𝑈 ∩ Ring ) ) |
9 |
2 8
|
eqsstrid |
⊢ ( 𝑈 ∈ 𝑉 → 𝐶 ⊆ ( 𝑈 ∩ Ring ) ) |
10 |
|
ssid |
⊢ ( 𝑥 RingHom 𝑦 ) ⊆ ( 𝑥 RingHom 𝑦 ) |
11 |
|
eqid |
⊢ ( RingCat ‘ 𝑈 ) = ( RingCat ‘ 𝑈 ) |
12 |
|
eqid |
⊢ ( Base ‘ ( RingCat ‘ 𝑈 ) ) = ( Base ‘ ( RingCat ‘ 𝑈 ) ) |
13 |
|
simpl |
⊢ ( ( 𝑈 ∈ 𝑉 ∧ ( 𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶 ) ) → 𝑈 ∈ 𝑉 ) |
14 |
|
eqid |
⊢ ( Hom ‘ ( RingCat ‘ 𝑈 ) ) = ( Hom ‘ ( RingCat ‘ 𝑈 ) ) |
15 |
1 2
|
srhmsubclem2 |
⊢ ( ( 𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶 ) → 𝑥 ∈ ( Base ‘ ( RingCat ‘ 𝑈 ) ) ) |
16 |
15
|
adantrr |
⊢ ( ( 𝑈 ∈ 𝑉 ∧ ( 𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶 ) ) → 𝑥 ∈ ( Base ‘ ( RingCat ‘ 𝑈 ) ) ) |
17 |
1 2
|
srhmsubclem2 |
⊢ ( ( 𝑈 ∈ 𝑉 ∧ 𝑦 ∈ 𝐶 ) → 𝑦 ∈ ( Base ‘ ( RingCat ‘ 𝑈 ) ) ) |
18 |
17
|
adantrl |
⊢ ( ( 𝑈 ∈ 𝑉 ∧ ( 𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶 ) ) → 𝑦 ∈ ( Base ‘ ( RingCat ‘ 𝑈 ) ) ) |
19 |
11 12 13 14 16 18
|
ringchom |
⊢ ( ( 𝑈 ∈ 𝑉 ∧ ( 𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶 ) ) → ( 𝑥 ( Hom ‘ ( RingCat ‘ 𝑈 ) ) 𝑦 ) = ( 𝑥 RingHom 𝑦 ) ) |
20 |
10 19
|
sseqtrrid |
⊢ ( ( 𝑈 ∈ 𝑉 ∧ ( 𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶 ) ) → ( 𝑥 RingHom 𝑦 ) ⊆ ( 𝑥 ( Hom ‘ ( RingCat ‘ 𝑈 ) ) 𝑦 ) ) |
21 |
3
|
a1i |
⊢ ( ( 𝑈 ∈ 𝑉 ∧ ( 𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶 ) ) → 𝐽 = ( 𝑟 ∈ 𝐶 , 𝑠 ∈ 𝐶 ↦ ( 𝑟 RingHom 𝑠 ) ) ) |
22 |
|
oveq12 |
⊢ ( ( 𝑟 = 𝑥 ∧ 𝑠 = 𝑦 ) → ( 𝑟 RingHom 𝑠 ) = ( 𝑥 RingHom 𝑦 ) ) |
23 |
22
|
adantl |
⊢ ( ( ( 𝑈 ∈ 𝑉 ∧ ( 𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶 ) ) ∧ ( 𝑟 = 𝑥 ∧ 𝑠 = 𝑦 ) ) → ( 𝑟 RingHom 𝑠 ) = ( 𝑥 RingHom 𝑦 ) ) |
24 |
|
simprl |
⊢ ( ( 𝑈 ∈ 𝑉 ∧ ( 𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶 ) ) → 𝑥 ∈ 𝐶 ) |
25 |
|
simprr |
⊢ ( ( 𝑈 ∈ 𝑉 ∧ ( 𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶 ) ) → 𝑦 ∈ 𝐶 ) |
26 |
|
ovexd |
⊢ ( ( 𝑈 ∈ 𝑉 ∧ ( 𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶 ) ) → ( 𝑥 RingHom 𝑦 ) ∈ V ) |
27 |
21 23 24 25 26
|
ovmpod |
⊢ ( ( 𝑈 ∈ 𝑉 ∧ ( 𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶 ) ) → ( 𝑥 𝐽 𝑦 ) = ( 𝑥 RingHom 𝑦 ) ) |
28 |
|
eqid |
⊢ ( Homf ‘ ( RingCat ‘ 𝑈 ) ) = ( Homf ‘ ( RingCat ‘ 𝑈 ) ) |
29 |
28 12 14 16 18
|
homfval |
⊢ ( ( 𝑈 ∈ 𝑉 ∧ ( 𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶 ) ) → ( 𝑥 ( Homf ‘ ( RingCat ‘ 𝑈 ) ) 𝑦 ) = ( 𝑥 ( Hom ‘ ( RingCat ‘ 𝑈 ) ) 𝑦 ) ) |
30 |
20 27 29
|
3sstr4d |
⊢ ( ( 𝑈 ∈ 𝑉 ∧ ( 𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶 ) ) → ( 𝑥 𝐽 𝑦 ) ⊆ ( 𝑥 ( Homf ‘ ( RingCat ‘ 𝑈 ) ) 𝑦 ) ) |
31 |
30
|
ralrimivva |
⊢ ( 𝑈 ∈ 𝑉 → ∀ 𝑥 ∈ 𝐶 ∀ 𝑦 ∈ 𝐶 ( 𝑥 𝐽 𝑦 ) ⊆ ( 𝑥 ( Homf ‘ ( RingCat ‘ 𝑈 ) ) 𝑦 ) ) |
32 |
|
ovex |
⊢ ( 𝑟 RingHom 𝑠 ) ∈ V |
33 |
3 32
|
fnmpoi |
⊢ 𝐽 Fn ( 𝐶 × 𝐶 ) |
34 |
33
|
a1i |
⊢ ( 𝑈 ∈ 𝑉 → 𝐽 Fn ( 𝐶 × 𝐶 ) ) |
35 |
28 12
|
homffn |
⊢ ( Homf ‘ ( RingCat ‘ 𝑈 ) ) Fn ( ( Base ‘ ( RingCat ‘ 𝑈 ) ) × ( Base ‘ ( RingCat ‘ 𝑈 ) ) ) |
36 |
|
id |
⊢ ( 𝑈 ∈ 𝑉 → 𝑈 ∈ 𝑉 ) |
37 |
11 12 36
|
ringcbas |
⊢ ( 𝑈 ∈ 𝑉 → ( Base ‘ ( RingCat ‘ 𝑈 ) ) = ( 𝑈 ∩ Ring ) ) |
38 |
37
|
eqcomd |
⊢ ( 𝑈 ∈ 𝑉 → ( 𝑈 ∩ Ring ) = ( Base ‘ ( RingCat ‘ 𝑈 ) ) ) |
39 |
38
|
sqxpeqd |
⊢ ( 𝑈 ∈ 𝑉 → ( ( 𝑈 ∩ Ring ) × ( 𝑈 ∩ Ring ) ) = ( ( Base ‘ ( RingCat ‘ 𝑈 ) ) × ( Base ‘ ( RingCat ‘ 𝑈 ) ) ) ) |
40 |
39
|
fneq2d |
⊢ ( 𝑈 ∈ 𝑉 → ( ( Homf ‘ ( RingCat ‘ 𝑈 ) ) Fn ( ( 𝑈 ∩ Ring ) × ( 𝑈 ∩ Ring ) ) ↔ ( Homf ‘ ( RingCat ‘ 𝑈 ) ) Fn ( ( Base ‘ ( RingCat ‘ 𝑈 ) ) × ( Base ‘ ( RingCat ‘ 𝑈 ) ) ) ) ) |
41 |
35 40
|
mpbiri |
⊢ ( 𝑈 ∈ 𝑉 → ( Homf ‘ ( RingCat ‘ 𝑈 ) ) Fn ( ( 𝑈 ∩ Ring ) × ( 𝑈 ∩ Ring ) ) ) |
42 |
|
inex1g |
⊢ ( 𝑈 ∈ 𝑉 → ( 𝑈 ∩ Ring ) ∈ V ) |
43 |
34 41 42
|
isssc |
⊢ ( 𝑈 ∈ 𝑉 → ( 𝐽 ⊆cat ( Homf ‘ ( RingCat ‘ 𝑈 ) ) ↔ ( 𝐶 ⊆ ( 𝑈 ∩ Ring ) ∧ ∀ 𝑥 ∈ 𝐶 ∀ 𝑦 ∈ 𝐶 ( 𝑥 𝐽 𝑦 ) ⊆ ( 𝑥 ( Homf ‘ ( RingCat ‘ 𝑈 ) ) 𝑦 ) ) ) ) |
44 |
9 31 43
|
mpbir2and |
⊢ ( 𝑈 ∈ 𝑉 → 𝐽 ⊆cat ( Homf ‘ ( RingCat ‘ 𝑈 ) ) ) |
45 |
2
|
elin2 |
⊢ ( 𝑥 ∈ 𝐶 ↔ ( 𝑥 ∈ 𝑈 ∧ 𝑥 ∈ 𝑆 ) ) |
46 |
5
|
adantl |
⊢ ( ( 𝑥 ∈ 𝑈 ∧ 𝑥 ∈ 𝑆 ) → 𝑥 ∈ Ring ) |
47 |
45 46
|
sylbi |
⊢ ( 𝑥 ∈ 𝐶 → 𝑥 ∈ Ring ) |
48 |
47
|
adantl |
⊢ ( ( 𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶 ) → 𝑥 ∈ Ring ) |
49 |
|
eqid |
⊢ ( Base ‘ 𝑥 ) = ( Base ‘ 𝑥 ) |
50 |
49
|
idrhm |
⊢ ( 𝑥 ∈ Ring → ( I ↾ ( Base ‘ 𝑥 ) ) ∈ ( 𝑥 RingHom 𝑥 ) ) |
51 |
48 50
|
syl |
⊢ ( ( 𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶 ) → ( I ↾ ( Base ‘ 𝑥 ) ) ∈ ( 𝑥 RingHom 𝑥 ) ) |
52 |
|
eqid |
⊢ ( Id ‘ ( RingCat ‘ 𝑈 ) ) = ( Id ‘ ( RingCat ‘ 𝑈 ) ) |
53 |
|
simpl |
⊢ ( ( 𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶 ) → 𝑈 ∈ 𝑉 ) |
54 |
11 12 52 53 15 49
|
ringcid |
⊢ ( ( 𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶 ) → ( ( Id ‘ ( RingCat ‘ 𝑈 ) ) ‘ 𝑥 ) = ( I ↾ ( Base ‘ 𝑥 ) ) ) |
55 |
3
|
a1i |
⊢ ( ( 𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶 ) → 𝐽 = ( 𝑟 ∈ 𝐶 , 𝑠 ∈ 𝐶 ↦ ( 𝑟 RingHom 𝑠 ) ) ) |
56 |
|
oveq12 |
⊢ ( ( 𝑟 = 𝑥 ∧ 𝑠 = 𝑥 ) → ( 𝑟 RingHom 𝑠 ) = ( 𝑥 RingHom 𝑥 ) ) |
57 |
56
|
adantl |
⊢ ( ( ( 𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶 ) ∧ ( 𝑟 = 𝑥 ∧ 𝑠 = 𝑥 ) ) → ( 𝑟 RingHom 𝑠 ) = ( 𝑥 RingHom 𝑥 ) ) |
58 |
|
simpr |
⊢ ( ( 𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶 ) → 𝑥 ∈ 𝐶 ) |
59 |
|
ovexd |
⊢ ( ( 𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶 ) → ( 𝑥 RingHom 𝑥 ) ∈ V ) |
60 |
55 57 58 58 59
|
ovmpod |
⊢ ( ( 𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶 ) → ( 𝑥 𝐽 𝑥 ) = ( 𝑥 RingHom 𝑥 ) ) |
61 |
51 54 60
|
3eltr4d |
⊢ ( ( 𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶 ) → ( ( Id ‘ ( RingCat ‘ 𝑈 ) ) ‘ 𝑥 ) ∈ ( 𝑥 𝐽 𝑥 ) ) |
62 |
|
eqid |
⊢ ( comp ‘ ( RingCat ‘ 𝑈 ) ) = ( comp ‘ ( RingCat ‘ 𝑈 ) ) |
63 |
11
|
ringccat |
⊢ ( 𝑈 ∈ 𝑉 → ( RingCat ‘ 𝑈 ) ∈ Cat ) |
64 |
63
|
ad3antrrr |
⊢ ( ( ( ( 𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶 ) ∧ ( 𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶 ) ) ∧ ( 𝑓 ∈ ( 𝑥 𝐽 𝑦 ) ∧ 𝑔 ∈ ( 𝑦 𝐽 𝑧 ) ) ) → ( RingCat ‘ 𝑈 ) ∈ Cat ) |
65 |
15
|
adantr |
⊢ ( ( ( 𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶 ) ∧ ( 𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶 ) ) → 𝑥 ∈ ( Base ‘ ( RingCat ‘ 𝑈 ) ) ) |
66 |
65
|
adantr |
⊢ ( ( ( ( 𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶 ) ∧ ( 𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶 ) ) ∧ ( 𝑓 ∈ ( 𝑥 𝐽 𝑦 ) ∧ 𝑔 ∈ ( 𝑦 𝐽 𝑧 ) ) ) → 𝑥 ∈ ( Base ‘ ( RingCat ‘ 𝑈 ) ) ) |
67 |
17
|
ad2ant2r |
⊢ ( ( ( 𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶 ) ∧ ( 𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶 ) ) → 𝑦 ∈ ( Base ‘ ( RingCat ‘ 𝑈 ) ) ) |
68 |
67
|
adantr |
⊢ ( ( ( ( 𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶 ) ∧ ( 𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶 ) ) ∧ ( 𝑓 ∈ ( 𝑥 𝐽 𝑦 ) ∧ 𝑔 ∈ ( 𝑦 𝐽 𝑧 ) ) ) → 𝑦 ∈ ( Base ‘ ( RingCat ‘ 𝑈 ) ) ) |
69 |
1 2
|
srhmsubclem2 |
⊢ ( ( 𝑈 ∈ 𝑉 ∧ 𝑧 ∈ 𝐶 ) → 𝑧 ∈ ( Base ‘ ( RingCat ‘ 𝑈 ) ) ) |
70 |
69
|
ad2ant2rl |
⊢ ( ( ( 𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶 ) ∧ ( 𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶 ) ) → 𝑧 ∈ ( Base ‘ ( RingCat ‘ 𝑈 ) ) ) |
71 |
70
|
adantr |
⊢ ( ( ( ( 𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶 ) ∧ ( 𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶 ) ) ∧ ( 𝑓 ∈ ( 𝑥 𝐽 𝑦 ) ∧ 𝑔 ∈ ( 𝑦 𝐽 𝑧 ) ) ) → 𝑧 ∈ ( Base ‘ ( RingCat ‘ 𝑈 ) ) ) |
72 |
53
|
adantr |
⊢ ( ( ( 𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶 ) ∧ ( 𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶 ) ) → 𝑈 ∈ 𝑉 ) |
73 |
|
simpl |
⊢ ( ( 𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶 ) → 𝑦 ∈ 𝐶 ) |
74 |
58 73
|
anim12i |
⊢ ( ( ( 𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶 ) ∧ ( 𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶 ) ) → ( 𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶 ) ) |
75 |
72 74
|
jca |
⊢ ( ( ( 𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶 ) ∧ ( 𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶 ) ) → ( 𝑈 ∈ 𝑉 ∧ ( 𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶 ) ) ) |
76 |
1 2 3
|
srhmsubclem3 |
⊢ ( ( 𝑈 ∈ 𝑉 ∧ ( 𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶 ) ) → ( 𝑥 𝐽 𝑦 ) = ( 𝑥 ( Hom ‘ ( RingCat ‘ 𝑈 ) ) 𝑦 ) ) |
77 |
75 76
|
syl |
⊢ ( ( ( 𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶 ) ∧ ( 𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶 ) ) → ( 𝑥 𝐽 𝑦 ) = ( 𝑥 ( Hom ‘ ( RingCat ‘ 𝑈 ) ) 𝑦 ) ) |
78 |
77
|
eleq2d |
⊢ ( ( ( 𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶 ) ∧ ( 𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶 ) ) → ( 𝑓 ∈ ( 𝑥 𝐽 𝑦 ) ↔ 𝑓 ∈ ( 𝑥 ( Hom ‘ ( RingCat ‘ 𝑈 ) ) 𝑦 ) ) ) |
79 |
78
|
biimpcd |
⊢ ( 𝑓 ∈ ( 𝑥 𝐽 𝑦 ) → ( ( ( 𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶 ) ∧ ( 𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶 ) ) → 𝑓 ∈ ( 𝑥 ( Hom ‘ ( RingCat ‘ 𝑈 ) ) 𝑦 ) ) ) |
80 |
79
|
adantr |
⊢ ( ( 𝑓 ∈ ( 𝑥 𝐽 𝑦 ) ∧ 𝑔 ∈ ( 𝑦 𝐽 𝑧 ) ) → ( ( ( 𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶 ) ∧ ( 𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶 ) ) → 𝑓 ∈ ( 𝑥 ( Hom ‘ ( RingCat ‘ 𝑈 ) ) 𝑦 ) ) ) |
81 |
80
|
impcom |
⊢ ( ( ( ( 𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶 ) ∧ ( 𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶 ) ) ∧ ( 𝑓 ∈ ( 𝑥 𝐽 𝑦 ) ∧ 𝑔 ∈ ( 𝑦 𝐽 𝑧 ) ) ) → 𝑓 ∈ ( 𝑥 ( Hom ‘ ( RingCat ‘ 𝑈 ) ) 𝑦 ) ) |
82 |
1 2 3
|
srhmsubclem3 |
⊢ ( ( 𝑈 ∈ 𝑉 ∧ ( 𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶 ) ) → ( 𝑦 𝐽 𝑧 ) = ( 𝑦 ( Hom ‘ ( RingCat ‘ 𝑈 ) ) 𝑧 ) ) |
83 |
82
|
adantlr |
⊢ ( ( ( 𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶 ) ∧ ( 𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶 ) ) → ( 𝑦 𝐽 𝑧 ) = ( 𝑦 ( Hom ‘ ( RingCat ‘ 𝑈 ) ) 𝑧 ) ) |
84 |
83
|
eleq2d |
⊢ ( ( ( 𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶 ) ∧ ( 𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶 ) ) → ( 𝑔 ∈ ( 𝑦 𝐽 𝑧 ) ↔ 𝑔 ∈ ( 𝑦 ( Hom ‘ ( RingCat ‘ 𝑈 ) ) 𝑧 ) ) ) |
85 |
84
|
biimpd |
⊢ ( ( ( 𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶 ) ∧ ( 𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶 ) ) → ( 𝑔 ∈ ( 𝑦 𝐽 𝑧 ) → 𝑔 ∈ ( 𝑦 ( Hom ‘ ( RingCat ‘ 𝑈 ) ) 𝑧 ) ) ) |
86 |
85
|
adantld |
⊢ ( ( ( 𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶 ) ∧ ( 𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶 ) ) → ( ( 𝑓 ∈ ( 𝑥 𝐽 𝑦 ) ∧ 𝑔 ∈ ( 𝑦 𝐽 𝑧 ) ) → 𝑔 ∈ ( 𝑦 ( Hom ‘ ( RingCat ‘ 𝑈 ) ) 𝑧 ) ) ) |
87 |
86
|
imp |
⊢ ( ( ( ( 𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶 ) ∧ ( 𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶 ) ) ∧ ( 𝑓 ∈ ( 𝑥 𝐽 𝑦 ) ∧ 𝑔 ∈ ( 𝑦 𝐽 𝑧 ) ) ) → 𝑔 ∈ ( 𝑦 ( Hom ‘ ( RingCat ‘ 𝑈 ) ) 𝑧 ) ) |
88 |
12 14 62 64 66 68 71 81 87
|
catcocl |
⊢ ( ( ( ( 𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶 ) ∧ ( 𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶 ) ) ∧ ( 𝑓 ∈ ( 𝑥 𝐽 𝑦 ) ∧ 𝑔 ∈ ( 𝑦 𝐽 𝑧 ) ) ) → ( 𝑔 ( 〈 𝑥 , 𝑦 〉 ( comp ‘ ( RingCat ‘ 𝑈 ) ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ ( RingCat ‘ 𝑈 ) ) 𝑧 ) ) |
89 |
11 12 72 14 65 70
|
ringchom |
⊢ ( ( ( 𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶 ) ∧ ( 𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶 ) ) → ( 𝑥 ( Hom ‘ ( RingCat ‘ 𝑈 ) ) 𝑧 ) = ( 𝑥 RingHom 𝑧 ) ) |
90 |
89
|
eqcomd |
⊢ ( ( ( 𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶 ) ∧ ( 𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶 ) ) → ( 𝑥 RingHom 𝑧 ) = ( 𝑥 ( Hom ‘ ( RingCat ‘ 𝑈 ) ) 𝑧 ) ) |
91 |
90
|
adantr |
⊢ ( ( ( ( 𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶 ) ∧ ( 𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶 ) ) ∧ ( 𝑓 ∈ ( 𝑥 𝐽 𝑦 ) ∧ 𝑔 ∈ ( 𝑦 𝐽 𝑧 ) ) ) → ( 𝑥 RingHom 𝑧 ) = ( 𝑥 ( Hom ‘ ( RingCat ‘ 𝑈 ) ) 𝑧 ) ) |
92 |
88 91
|
eleqtrrd |
⊢ ( ( ( ( 𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶 ) ∧ ( 𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶 ) ) ∧ ( 𝑓 ∈ ( 𝑥 𝐽 𝑦 ) ∧ 𝑔 ∈ ( 𝑦 𝐽 𝑧 ) ) ) → ( 𝑔 ( 〈 𝑥 , 𝑦 〉 ( comp ‘ ( RingCat ‘ 𝑈 ) ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 RingHom 𝑧 ) ) |
93 |
3
|
a1i |
⊢ ( ( ( 𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶 ) ∧ ( 𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶 ) ) → 𝐽 = ( 𝑟 ∈ 𝐶 , 𝑠 ∈ 𝐶 ↦ ( 𝑟 RingHom 𝑠 ) ) ) |
94 |
|
oveq12 |
⊢ ( ( 𝑟 = 𝑥 ∧ 𝑠 = 𝑧 ) → ( 𝑟 RingHom 𝑠 ) = ( 𝑥 RingHom 𝑧 ) ) |
95 |
94
|
adantl |
⊢ ( ( ( ( 𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶 ) ∧ ( 𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶 ) ) ∧ ( 𝑟 = 𝑥 ∧ 𝑠 = 𝑧 ) ) → ( 𝑟 RingHom 𝑠 ) = ( 𝑥 RingHom 𝑧 ) ) |
96 |
58
|
adantr |
⊢ ( ( ( 𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶 ) ∧ ( 𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶 ) ) → 𝑥 ∈ 𝐶 ) |
97 |
|
simprr |
⊢ ( ( ( 𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶 ) ∧ ( 𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶 ) ) → 𝑧 ∈ 𝐶 ) |
98 |
|
ovexd |
⊢ ( ( ( 𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶 ) ∧ ( 𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶 ) ) → ( 𝑥 RingHom 𝑧 ) ∈ V ) |
99 |
93 95 96 97 98
|
ovmpod |
⊢ ( ( ( 𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶 ) ∧ ( 𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶 ) ) → ( 𝑥 𝐽 𝑧 ) = ( 𝑥 RingHom 𝑧 ) ) |
100 |
99
|
adantr |
⊢ ( ( ( ( 𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶 ) ∧ ( 𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶 ) ) ∧ ( 𝑓 ∈ ( 𝑥 𝐽 𝑦 ) ∧ 𝑔 ∈ ( 𝑦 𝐽 𝑧 ) ) ) → ( 𝑥 𝐽 𝑧 ) = ( 𝑥 RingHom 𝑧 ) ) |
101 |
92 100
|
eleqtrrd |
⊢ ( ( ( ( 𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶 ) ∧ ( 𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶 ) ) ∧ ( 𝑓 ∈ ( 𝑥 𝐽 𝑦 ) ∧ 𝑔 ∈ ( 𝑦 𝐽 𝑧 ) ) ) → ( 𝑔 ( 〈 𝑥 , 𝑦 〉 ( comp ‘ ( RingCat ‘ 𝑈 ) ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 𝐽 𝑧 ) ) |
102 |
101
|
ralrimivva |
⊢ ( ( ( 𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶 ) ∧ ( 𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶 ) ) → ∀ 𝑓 ∈ ( 𝑥 𝐽 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 𝐽 𝑧 ) ( 𝑔 ( 〈 𝑥 , 𝑦 〉 ( comp ‘ ( RingCat ‘ 𝑈 ) ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 𝐽 𝑧 ) ) |
103 |
102
|
ralrimivva |
⊢ ( ( 𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶 ) → ∀ 𝑦 ∈ 𝐶 ∀ 𝑧 ∈ 𝐶 ∀ 𝑓 ∈ ( 𝑥 𝐽 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 𝐽 𝑧 ) ( 𝑔 ( 〈 𝑥 , 𝑦 〉 ( comp ‘ ( RingCat ‘ 𝑈 ) ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 𝐽 𝑧 ) ) |
104 |
61 103
|
jca |
⊢ ( ( 𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶 ) → ( ( ( Id ‘ ( RingCat ‘ 𝑈 ) ) ‘ 𝑥 ) ∈ ( 𝑥 𝐽 𝑥 ) ∧ ∀ 𝑦 ∈ 𝐶 ∀ 𝑧 ∈ 𝐶 ∀ 𝑓 ∈ ( 𝑥 𝐽 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 𝐽 𝑧 ) ( 𝑔 ( 〈 𝑥 , 𝑦 〉 ( comp ‘ ( RingCat ‘ 𝑈 ) ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 𝐽 𝑧 ) ) ) |
105 |
104
|
ralrimiva |
⊢ ( 𝑈 ∈ 𝑉 → ∀ 𝑥 ∈ 𝐶 ( ( ( Id ‘ ( RingCat ‘ 𝑈 ) ) ‘ 𝑥 ) ∈ ( 𝑥 𝐽 𝑥 ) ∧ ∀ 𝑦 ∈ 𝐶 ∀ 𝑧 ∈ 𝐶 ∀ 𝑓 ∈ ( 𝑥 𝐽 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 𝐽 𝑧 ) ( 𝑔 ( 〈 𝑥 , 𝑦 〉 ( comp ‘ ( RingCat ‘ 𝑈 ) ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 𝐽 𝑧 ) ) ) |
106 |
28 52 62 63 34
|
issubc2 |
⊢ ( 𝑈 ∈ 𝑉 → ( 𝐽 ∈ ( Subcat ‘ ( RingCat ‘ 𝑈 ) ) ↔ ( 𝐽 ⊆cat ( Homf ‘ ( RingCat ‘ 𝑈 ) ) ∧ ∀ 𝑥 ∈ 𝐶 ( ( ( Id ‘ ( RingCat ‘ 𝑈 ) ) ‘ 𝑥 ) ∈ ( 𝑥 𝐽 𝑥 ) ∧ ∀ 𝑦 ∈ 𝐶 ∀ 𝑧 ∈ 𝐶 ∀ 𝑓 ∈ ( 𝑥 𝐽 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 𝐽 𝑧 ) ( 𝑔 ( 〈 𝑥 , 𝑦 〉 ( comp ‘ ( RingCat ‘ 𝑈 ) ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 𝐽 𝑧 ) ) ) ) ) |
107 |
44 105 106
|
mpbir2and |
⊢ ( 𝑈 ∈ 𝑉 → 𝐽 ∈ ( Subcat ‘ ( RingCat ‘ 𝑈 ) ) ) |