Step |
Hyp |
Ref |
Expression |
0 |
|
cir |
|- Irred |
1 |
|
vw |
|- w |
2 |
|
cvv |
|- _V |
3 |
|
cbs |
|- Base |
4 |
1
|
cv |
|- w |
5 |
4 3
|
cfv |
|- ( Base ` w ) |
6 |
|
cui |
|- Unit |
7 |
4 6
|
cfv |
|- ( Unit ` w ) |
8 |
5 7
|
cdif |
|- ( ( Base ` w ) \ ( Unit ` w ) ) |
9 |
|
vb |
|- b |
10 |
|
vz |
|- z |
11 |
9
|
cv |
|- b |
12 |
|
vx |
|- x |
13 |
|
vy |
|- y |
14 |
12
|
cv |
|- x |
15 |
|
cmulr |
|- .r |
16 |
4 15
|
cfv |
|- ( .r ` w ) |
17 |
13
|
cv |
|- y |
18 |
14 17 16
|
co |
|- ( x ( .r ` w ) y ) |
19 |
10
|
cv |
|- z |
20 |
18 19
|
wne |
|- ( x ( .r ` w ) y ) =/= z |
21 |
20 13 11
|
wral |
|- A. y e. b ( x ( .r ` w ) y ) =/= z |
22 |
21 12 11
|
wral |
|- A. x e. b A. y e. b ( x ( .r ` w ) y ) =/= z |
23 |
22 10 11
|
crab |
|- { z e. b | A. x e. b A. y e. b ( x ( .r ` w ) y ) =/= z } |
24 |
9 8 23
|
csb |
|- [_ ( ( Base ` w ) \ ( Unit ` w ) ) / b ]_ { z e. b | A. x e. b A. y e. b ( x ( .r ` w ) y ) =/= z } |
25 |
1 2 24
|
cmpt |
|- ( w e. _V |-> [_ ( ( Base ` w ) \ ( Unit ` w ) ) / b ]_ { z e. b | A. x e. b A. y e. b ( x ( .r ` w ) y ) =/= z } ) |
26 |
0 25
|
wceq |
|- Irred = ( w e. _V |-> [_ ( ( Base ` w ) \ ( Unit ` w ) ) / b ]_ { z e. b | A. x e. b A. y e. b ( x ( .r ` w ) y ) =/= z } ) |