Step |
Hyp |
Ref |
Expression |
0 |
|
cgex |
|- gEx |
1 |
|
vg |
|- g |
2 |
|
cvv |
|- _V |
3 |
|
vn |
|- n |
4 |
|
cn |
|- NN |
5 |
|
vx |
|- x |
6 |
|
cbs |
|- Base |
7 |
1
|
cv |
|- g |
8 |
7 6
|
cfv |
|- ( Base ` g ) |
9 |
3
|
cv |
|- n |
10 |
|
cmg |
|- .g |
11 |
7 10
|
cfv |
|- ( .g ` g ) |
12 |
5
|
cv |
|- x |
13 |
9 12 11
|
co |
|- ( n ( .g ` g ) x ) |
14 |
|
c0g |
|- 0g |
15 |
7 14
|
cfv |
|- ( 0g ` g ) |
16 |
13 15
|
wceq |
|- ( n ( .g ` g ) x ) = ( 0g ` g ) |
17 |
16 5 8
|
wral |
|- A. x e. ( Base ` g ) ( n ( .g ` g ) x ) = ( 0g ` g ) |
18 |
17 3 4
|
crab |
|- { n e. NN | A. x e. ( Base ` g ) ( n ( .g ` g ) x ) = ( 0g ` g ) } |
19 |
|
vi |
|- i |
20 |
19
|
cv |
|- i |
21 |
|
c0 |
|- (/) |
22 |
20 21
|
wceq |
|- i = (/) |
23 |
|
cc0 |
|- 0 |
24 |
|
cr |
|- RR |
25 |
|
clt |
|- < |
26 |
20 24 25
|
cinf |
|- inf ( i , RR , < ) |
27 |
22 23 26
|
cif |
|- if ( i = (/) , 0 , inf ( i , RR , < ) ) |
28 |
19 18 27
|
csb |
|- [_ { n e. NN | A. x e. ( Base ` g ) ( n ( .g ` g ) x ) = ( 0g ` g ) } / i ]_ if ( i = (/) , 0 , inf ( i , RR , < ) ) |
29 |
1 2 28
|
cmpt |
|- ( g e. _V |-> [_ { n e. NN | A. x e. ( Base ` g ) ( n ( .g ` g ) x ) = ( 0g ` g ) } / i ]_ if ( i = (/) , 0 , inf ( i , RR , < ) ) ) |
30 |
0 29
|
wceq |
|- gEx = ( g e. _V |-> [_ { n e. NN | A. x e. ( Base ` g ) ( n ( .g ` g ) x ) = ( 0g ` g ) } / i ]_ if ( i = (/) , 0 , inf ( i , RR , < ) ) ) |