Step |
Hyp |
Ref |
Expression |
0 |
|
cedring-rN |
|- EDRingR |
1 |
|
vk |
|- k |
2 |
|
cvv |
|- _V |
3 |
|
vw |
|- w |
4 |
|
clh |
|- LHyp |
5 |
1
|
cv |
|- k |
6 |
5 4
|
cfv |
|- ( LHyp ` k ) |
7 |
|
cbs |
|- Base |
8 |
|
cnx |
|- ndx |
9 |
8 7
|
cfv |
|- ( Base ` ndx ) |
10 |
|
ctendo |
|- TEndo |
11 |
5 10
|
cfv |
|- ( TEndo ` k ) |
12 |
3
|
cv |
|- w |
13 |
12 11
|
cfv |
|- ( ( TEndo ` k ) ` w ) |
14 |
9 13
|
cop |
|- <. ( Base ` ndx ) , ( ( TEndo ` k ) ` w ) >. |
15 |
|
cplusg |
|- +g |
16 |
8 15
|
cfv |
|- ( +g ` ndx ) |
17 |
|
vs |
|- s |
18 |
|
vt |
|- t |
19 |
|
vf |
|- f |
20 |
|
cltrn |
|- LTrn |
21 |
5 20
|
cfv |
|- ( LTrn ` k ) |
22 |
12 21
|
cfv |
|- ( ( LTrn ` k ) ` w ) |
23 |
17
|
cv |
|- s |
24 |
19
|
cv |
|- f |
25 |
24 23
|
cfv |
|- ( s ` f ) |
26 |
18
|
cv |
|- t |
27 |
24 26
|
cfv |
|- ( t ` f ) |
28 |
25 27
|
ccom |
|- ( ( s ` f ) o. ( t ` f ) ) |
29 |
19 22 28
|
cmpt |
|- ( f e. ( ( LTrn ` k ) ` w ) |-> ( ( s ` f ) o. ( t ` f ) ) ) |
30 |
17 18 13 13 29
|
cmpo |
|- ( s e. ( ( TEndo ` k ) ` w ) , t e. ( ( TEndo ` k ) ` w ) |-> ( f e. ( ( LTrn ` k ) ` w ) |-> ( ( s ` f ) o. ( t ` f ) ) ) ) |
31 |
16 30
|
cop |
|- <. ( +g ` ndx ) , ( s e. ( ( TEndo ` k ) ` w ) , t e. ( ( TEndo ` k ) ` w ) |-> ( f e. ( ( LTrn ` k ) ` w ) |-> ( ( s ` f ) o. ( t ` f ) ) ) ) >. |
32 |
|
cmulr |
|- .r |
33 |
8 32
|
cfv |
|- ( .r ` ndx ) |
34 |
26 23
|
ccom |
|- ( t o. s ) |
35 |
17 18 13 13 34
|
cmpo |
|- ( s e. ( ( TEndo ` k ) ` w ) , t e. ( ( TEndo ` k ) ` w ) |-> ( t o. s ) ) |
36 |
33 35
|
cop |
|- <. ( .r ` ndx ) , ( s e. ( ( TEndo ` k ) ` w ) , t e. ( ( TEndo ` k ) ` w ) |-> ( t o. s ) ) >. |
37 |
14 31 36
|
ctp |
|- { <. ( Base ` ndx ) , ( ( TEndo ` k ) ` w ) >. , <. ( +g ` ndx ) , ( s e. ( ( TEndo ` k ) ` w ) , t e. ( ( TEndo ` k ) ` w ) |-> ( f e. ( ( LTrn ` k ) ` w ) |-> ( ( s ` f ) o. ( t ` f ) ) ) ) >. , <. ( .r ` ndx ) , ( s e. ( ( TEndo ` k ) ` w ) , t e. ( ( TEndo ` k ) ` w ) |-> ( t o. s ) ) >. } |
38 |
3 6 37
|
cmpt |
|- ( w e. ( LHyp ` k ) |-> { <. ( Base ` ndx ) , ( ( TEndo ` k ) ` w ) >. , <. ( +g ` ndx ) , ( s e. ( ( TEndo ` k ) ` w ) , t e. ( ( TEndo ` k ) ` w ) |-> ( f e. ( ( LTrn ` k ) ` w ) |-> ( ( s ` f ) o. ( t ` f ) ) ) ) >. , <. ( .r ` ndx ) , ( s e. ( ( TEndo ` k ) ` w ) , t e. ( ( TEndo ` k ) ` w ) |-> ( t o. s ) ) >. } ) |
39 |
1 2 38
|
cmpt |
|- ( k e. _V |-> ( w e. ( LHyp ` k ) |-> { <. ( Base ` ndx ) , ( ( TEndo ` k ) ` w ) >. , <. ( +g ` ndx ) , ( s e. ( ( TEndo ` k ) ` w ) , t e. ( ( TEndo ` k ) ` w ) |-> ( f e. ( ( LTrn ` k ) ` w ) |-> ( ( s ` f ) o. ( t ` f ) ) ) ) >. , <. ( .r ` ndx ) , ( s e. ( ( TEndo ` k ) ` w ) , t e. ( ( TEndo ` k ) ` w ) |-> ( t o. s ) ) >. } ) ) |
40 |
0 39
|
wceq |
|- EDRingR = ( k e. _V |-> ( w e. ( LHyp ` k ) |-> { <. ( Base ` ndx ) , ( ( TEndo ` k ) ` w ) >. , <. ( +g ` ndx ) , ( s e. ( ( TEndo ` k ) ` w ) , t e. ( ( TEndo ` k ) ` w ) |-> ( f e. ( ( LTrn ` k ) ` w ) |-> ( ( s ` f ) o. ( t ` f ) ) ) ) >. , <. ( .r ` ndx ) , ( s e. ( ( TEndo ` k ) ` w ) , t e. ( ( TEndo ` k ) ` w ) |-> ( t o. s ) ) >. } ) ) |