Step |
Hyp |
Ref |
Expression |
0 |
|
cmnring |
⊢ MndRing |
1 |
|
vr |
⊢ 𝑟 |
2 |
|
cvv |
⊢ V |
3 |
|
vm |
⊢ 𝑚 |
4 |
1
|
cv |
⊢ 𝑟 |
5 |
|
cfrlm |
⊢ freeLMod |
6 |
|
cbs |
⊢ Base |
7 |
3
|
cv |
⊢ 𝑚 |
8 |
7 6
|
cfv |
⊢ ( Base ‘ 𝑚 ) |
9 |
4 8 5
|
co |
⊢ ( 𝑟 freeLMod ( Base ‘ 𝑚 ) ) |
10 |
|
vv |
⊢ 𝑣 |
11 |
10
|
cv |
⊢ 𝑣 |
12 |
|
csts |
⊢ sSet |
13 |
|
cmulr |
⊢ .r |
14 |
|
cnx |
⊢ ndx |
15 |
14 13
|
cfv |
⊢ ( .r ‘ ndx ) |
16 |
|
vx |
⊢ 𝑥 |
17 |
11 6
|
cfv |
⊢ ( Base ‘ 𝑣 ) |
18 |
|
vy |
⊢ 𝑦 |
19 |
|
cgsu |
⊢ Σg |
20 |
|
va |
⊢ 𝑎 |
21 |
|
vb |
⊢ 𝑏 |
22 |
|
vi |
⊢ 𝑖 |
23 |
22
|
cv |
⊢ 𝑖 |
24 |
20
|
cv |
⊢ 𝑎 |
25 |
|
cplusg |
⊢ +g |
26 |
7 25
|
cfv |
⊢ ( +g ‘ 𝑚 ) |
27 |
21
|
cv |
⊢ 𝑏 |
28 |
24 27 26
|
co |
⊢ ( 𝑎 ( +g ‘ 𝑚 ) 𝑏 ) |
29 |
23 28
|
wceq |
⊢ 𝑖 = ( 𝑎 ( +g ‘ 𝑚 ) 𝑏 ) |
30 |
16
|
cv |
⊢ 𝑥 |
31 |
24 30
|
cfv |
⊢ ( 𝑥 ‘ 𝑎 ) |
32 |
4 13
|
cfv |
⊢ ( .r ‘ 𝑟 ) |
33 |
18
|
cv |
⊢ 𝑦 |
34 |
27 33
|
cfv |
⊢ ( 𝑦 ‘ 𝑏 ) |
35 |
31 34 32
|
co |
⊢ ( ( 𝑥 ‘ 𝑎 ) ( .r ‘ 𝑟 ) ( 𝑦 ‘ 𝑏 ) ) |
36 |
|
c0g |
⊢ 0g |
37 |
4 36
|
cfv |
⊢ ( 0g ‘ 𝑟 ) |
38 |
29 35 37
|
cif |
⊢ if ( 𝑖 = ( 𝑎 ( +g ‘ 𝑚 ) 𝑏 ) , ( ( 𝑥 ‘ 𝑎 ) ( .r ‘ 𝑟 ) ( 𝑦 ‘ 𝑏 ) ) , ( 0g ‘ 𝑟 ) ) |
39 |
22 8 38
|
cmpt |
⊢ ( 𝑖 ∈ ( Base ‘ 𝑚 ) ↦ if ( 𝑖 = ( 𝑎 ( +g ‘ 𝑚 ) 𝑏 ) , ( ( 𝑥 ‘ 𝑎 ) ( .r ‘ 𝑟 ) ( 𝑦 ‘ 𝑏 ) ) , ( 0g ‘ 𝑟 ) ) ) |
40 |
20 21 8 8 39
|
cmpo |
⊢ ( 𝑎 ∈ ( Base ‘ 𝑚 ) , 𝑏 ∈ ( Base ‘ 𝑚 ) ↦ ( 𝑖 ∈ ( Base ‘ 𝑚 ) ↦ if ( 𝑖 = ( 𝑎 ( +g ‘ 𝑚 ) 𝑏 ) , ( ( 𝑥 ‘ 𝑎 ) ( .r ‘ 𝑟 ) ( 𝑦 ‘ 𝑏 ) ) , ( 0g ‘ 𝑟 ) ) ) ) |
41 |
11 40 19
|
co |
⊢ ( 𝑣 Σg ( 𝑎 ∈ ( Base ‘ 𝑚 ) , 𝑏 ∈ ( Base ‘ 𝑚 ) ↦ ( 𝑖 ∈ ( Base ‘ 𝑚 ) ↦ if ( 𝑖 = ( 𝑎 ( +g ‘ 𝑚 ) 𝑏 ) , ( ( 𝑥 ‘ 𝑎 ) ( .r ‘ 𝑟 ) ( 𝑦 ‘ 𝑏 ) ) , ( 0g ‘ 𝑟 ) ) ) ) ) |
42 |
16 18 17 17 41
|
cmpo |
⊢ ( 𝑥 ∈ ( Base ‘ 𝑣 ) , 𝑦 ∈ ( Base ‘ 𝑣 ) ↦ ( 𝑣 Σg ( 𝑎 ∈ ( Base ‘ 𝑚 ) , 𝑏 ∈ ( Base ‘ 𝑚 ) ↦ ( 𝑖 ∈ ( Base ‘ 𝑚 ) ↦ if ( 𝑖 = ( 𝑎 ( +g ‘ 𝑚 ) 𝑏 ) , ( ( 𝑥 ‘ 𝑎 ) ( .r ‘ 𝑟 ) ( 𝑦 ‘ 𝑏 ) ) , ( 0g ‘ 𝑟 ) ) ) ) ) ) |
43 |
15 42
|
cop |
⊢ 〈 ( .r ‘ ndx ) , ( 𝑥 ∈ ( Base ‘ 𝑣 ) , 𝑦 ∈ ( Base ‘ 𝑣 ) ↦ ( 𝑣 Σg ( 𝑎 ∈ ( Base ‘ 𝑚 ) , 𝑏 ∈ ( Base ‘ 𝑚 ) ↦ ( 𝑖 ∈ ( Base ‘ 𝑚 ) ↦ if ( 𝑖 = ( 𝑎 ( +g ‘ 𝑚 ) 𝑏 ) , ( ( 𝑥 ‘ 𝑎 ) ( .r ‘ 𝑟 ) ( 𝑦 ‘ 𝑏 ) ) , ( 0g ‘ 𝑟 ) ) ) ) ) ) 〉 |
44 |
11 43 12
|
co |
⊢ ( 𝑣 sSet 〈 ( .r ‘ ndx ) , ( 𝑥 ∈ ( Base ‘ 𝑣 ) , 𝑦 ∈ ( Base ‘ 𝑣 ) ↦ ( 𝑣 Σg ( 𝑎 ∈ ( Base ‘ 𝑚 ) , 𝑏 ∈ ( Base ‘ 𝑚 ) ↦ ( 𝑖 ∈ ( Base ‘ 𝑚 ) ↦ if ( 𝑖 = ( 𝑎 ( +g ‘ 𝑚 ) 𝑏 ) , ( ( 𝑥 ‘ 𝑎 ) ( .r ‘ 𝑟 ) ( 𝑦 ‘ 𝑏 ) ) , ( 0g ‘ 𝑟 ) ) ) ) ) ) 〉 ) |
45 |
10 9 44
|
csb |
⊢ ⦋ ( 𝑟 freeLMod ( Base ‘ 𝑚 ) ) / 𝑣 ⦌ ( 𝑣 sSet 〈 ( .r ‘ ndx ) , ( 𝑥 ∈ ( Base ‘ 𝑣 ) , 𝑦 ∈ ( Base ‘ 𝑣 ) ↦ ( 𝑣 Σg ( 𝑎 ∈ ( Base ‘ 𝑚 ) , 𝑏 ∈ ( Base ‘ 𝑚 ) ↦ ( 𝑖 ∈ ( Base ‘ 𝑚 ) ↦ if ( 𝑖 = ( 𝑎 ( +g ‘ 𝑚 ) 𝑏 ) , ( ( 𝑥 ‘ 𝑎 ) ( .r ‘ 𝑟 ) ( 𝑦 ‘ 𝑏 ) ) , ( 0g ‘ 𝑟 ) ) ) ) ) ) 〉 ) |
46 |
1 3 2 2 45
|
cmpo |
⊢ ( 𝑟 ∈ V , 𝑚 ∈ V ↦ ⦋ ( 𝑟 freeLMod ( Base ‘ 𝑚 ) ) / 𝑣 ⦌ ( 𝑣 sSet 〈 ( .r ‘ ndx ) , ( 𝑥 ∈ ( Base ‘ 𝑣 ) , 𝑦 ∈ ( Base ‘ 𝑣 ) ↦ ( 𝑣 Σg ( 𝑎 ∈ ( Base ‘ 𝑚 ) , 𝑏 ∈ ( Base ‘ 𝑚 ) ↦ ( 𝑖 ∈ ( Base ‘ 𝑚 ) ↦ if ( 𝑖 = ( 𝑎 ( +g ‘ 𝑚 ) 𝑏 ) , ( ( 𝑥 ‘ 𝑎 ) ( .r ‘ 𝑟 ) ( 𝑦 ‘ 𝑏 ) ) , ( 0g ‘ 𝑟 ) ) ) ) ) ) 〉 ) ) |
47 |
0 46
|
wceq |
⊢ MndRing = ( 𝑟 ∈ V , 𝑚 ∈ V ↦ ⦋ ( 𝑟 freeLMod ( Base ‘ 𝑚 ) ) / 𝑣 ⦌ ( 𝑣 sSet 〈 ( .r ‘ ndx ) , ( 𝑥 ∈ ( Base ‘ 𝑣 ) , 𝑦 ∈ ( Base ‘ 𝑣 ) ↦ ( 𝑣 Σg ( 𝑎 ∈ ( Base ‘ 𝑚 ) , 𝑏 ∈ ( Base ‘ 𝑚 ) ↦ ( 𝑖 ∈ ( Base ‘ 𝑚 ) ↦ if ( 𝑖 = ( 𝑎 ( +g ‘ 𝑚 ) 𝑏 ) , ( ( 𝑥 ‘ 𝑎 ) ( .r ‘ 𝑟 ) ( 𝑦 ‘ 𝑏 ) ) , ( 0g ‘ 𝑟 ) ) ) ) ) ) 〉 ) ) |