Step |
Hyp |
Ref |
Expression |
0 |
|
cdpj |
⊢ dProj |
1 |
|
vg |
⊢ 𝑔 |
2 |
|
cgrp |
⊢ Grp |
3 |
|
vs |
⊢ 𝑠 |
4 |
|
cdprd |
⊢ DProd |
5 |
4
|
cdm |
⊢ dom DProd |
6 |
1
|
cv |
⊢ 𝑔 |
7 |
6
|
csn |
⊢ { 𝑔 } |
8 |
5 7
|
cima |
⊢ ( dom DProd “ { 𝑔 } ) |
9 |
|
vi |
⊢ 𝑖 |
10 |
3
|
cv |
⊢ 𝑠 |
11 |
10
|
cdm |
⊢ dom 𝑠 |
12 |
9
|
cv |
⊢ 𝑖 |
13 |
12 10
|
cfv |
⊢ ( 𝑠 ‘ 𝑖 ) |
14 |
|
cpj1 |
⊢ proj1 |
15 |
6 14
|
cfv |
⊢ ( proj1 ‘ 𝑔 ) |
16 |
12
|
csn |
⊢ { 𝑖 } |
17 |
11 16
|
cdif |
⊢ ( dom 𝑠 ∖ { 𝑖 } ) |
18 |
10 17
|
cres |
⊢ ( 𝑠 ↾ ( dom 𝑠 ∖ { 𝑖 } ) ) |
19 |
6 18 4
|
co |
⊢ ( 𝑔 DProd ( 𝑠 ↾ ( dom 𝑠 ∖ { 𝑖 } ) ) ) |
20 |
13 19 15
|
co |
⊢ ( ( 𝑠 ‘ 𝑖 ) ( proj1 ‘ 𝑔 ) ( 𝑔 DProd ( 𝑠 ↾ ( dom 𝑠 ∖ { 𝑖 } ) ) ) ) |
21 |
9 11 20
|
cmpt |
⊢ ( 𝑖 ∈ dom 𝑠 ↦ ( ( 𝑠 ‘ 𝑖 ) ( proj1 ‘ 𝑔 ) ( 𝑔 DProd ( 𝑠 ↾ ( dom 𝑠 ∖ { 𝑖 } ) ) ) ) ) |
22 |
1 3 2 8 21
|
cmpo |
⊢ ( 𝑔 ∈ Grp , 𝑠 ∈ ( dom DProd “ { 𝑔 } ) ↦ ( 𝑖 ∈ dom 𝑠 ↦ ( ( 𝑠 ‘ 𝑖 ) ( proj1 ‘ 𝑔 ) ( 𝑔 DProd ( 𝑠 ↾ ( dom 𝑠 ∖ { 𝑖 } ) ) ) ) ) ) |
23 |
0 22
|
wceq |
⊢ dProj = ( 𝑔 ∈ Grp , 𝑠 ∈ ( dom DProd “ { 𝑔 } ) ↦ ( 𝑖 ∈ dom 𝑠 ↦ ( ( 𝑠 ‘ 𝑖 ) ( proj1 ‘ 𝑔 ) ( 𝑔 DProd ( 𝑠 ↾ ( dom 𝑠 ∖ { 𝑖 } ) ) ) ) ) ) |