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Mirrors > Home > MPE Home > Th. List > Mathboxes > df-rngcALTV | Structured version Visualization version Unicode version |
Description: Definition of the category Rng, relativized to a subset . This is the category of all non-unital rings in and homomorphisms between these rings. Generally, we will take to be a weak universe or Grothendieck universe, because these sets have closure properties as good as the real thing. (New usage is discouraged.) (Contributed by AV, 27-Feb-2020.) |
Ref | Expression |
---|---|
df-rngcALTV | RngCatALTV Rng RngHomo comp RngHomo RngHomo |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | crngcALTV 41958 | . 2 RngCatALTV | |
2 | vu | . . 3 | |
3 | cvv 3200 | . . 3 | |
4 | vb | . . . 4 | |
5 | 2 | cv 1482 | . . . . 5 |
6 | crng 41874 | . . . . 5 Rng | |
7 | 5, 6 | cin 3573 | . . . 4 Rng |
8 | cnx 15854 | . . . . . . 7 | |
9 | cbs 15857 | . . . . . . 7 | |
10 | 8, 9 | cfv 5888 | . . . . . 6 |
11 | 4 | cv 1482 | . . . . . 6 |
12 | 10, 11 | cop 4183 | . . . . 5 |
13 | chom 15952 | . . . . . . 7 | |
14 | 8, 13 | cfv 5888 | . . . . . 6 |
15 | vx | . . . . . . 7 | |
16 | vy | . . . . . . 7 | |
17 | 15 | cv 1482 | . . . . . . . 8 |
18 | 16 | cv 1482 | . . . . . . . 8 |
19 | crngh 41885 | . . . . . . . 8 RngHomo | |
20 | 17, 18, 19 | co 6650 | . . . . . . 7 RngHomo |
21 | 15, 16, 11, 11, 20 | cmpt2 6652 | . . . . . 6 RngHomo |
22 | 14, 21 | cop 4183 | . . . . 5 RngHomo |
23 | cco 15953 | . . . . . . 7 comp | |
24 | 8, 23 | cfv 5888 | . . . . . 6 comp |
25 | vv | . . . . . . 7 | |
26 | vz | . . . . . . 7 | |
27 | 11, 11 | cxp 5112 | . . . . . . 7 |
28 | vg | . . . . . . . 8 | |
29 | vf | . . . . . . . 8 | |
30 | 25 | cv 1482 | . . . . . . . . . 10 |
31 | c2nd 7167 | . . . . . . . . . 10 | |
32 | 30, 31 | cfv 5888 | . . . . . . . . 9 |
33 | 26 | cv 1482 | . . . . . . . . 9 |
34 | 32, 33, 19 | co 6650 | . . . . . . . 8 RngHomo |
35 | c1st 7166 | . . . . . . . . . 10 | |
36 | 30, 35 | cfv 5888 | . . . . . . . . 9 |
37 | 36, 32, 19 | co 6650 | . . . . . . . 8 RngHomo |
38 | 28 | cv 1482 | . . . . . . . . 9 |
39 | 29 | cv 1482 | . . . . . . . . 9 |
40 | 38, 39 | ccom 5118 | . . . . . . . 8 |
41 | 28, 29, 34, 37, 40 | cmpt2 6652 | . . . . . . 7 RngHomo RngHomo |
42 | 25, 26, 27, 11, 41 | cmpt2 6652 | . . . . . 6 RngHomo RngHomo |
43 | 24, 42 | cop 4183 | . . . . 5 comp RngHomo RngHomo |
44 | 12, 22, 43 | ctp 4181 | . . . 4 RngHomo comp RngHomo RngHomo |
45 | 4, 7, 44 | csb 3533 | . . 3 Rng RngHomo comp RngHomo RngHomo |
46 | 2, 3, 45 | cmpt 4729 | . 2 Rng RngHomo comp RngHomo RngHomo |
47 | 1, 46 | wceq 1483 | 1 RngCatALTV Rng RngHomo comp RngHomo RngHomo |
Colors of variables: wff setvar class |
This definition is referenced by: rngcvalALTV 41961 |
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