Step |
Hyp |
Ref |
Expression |
1 |
|
psrgrp.s |
|- S = ( I mPwSer R ) |
2 |
|
psrgrp.i |
|- ( ph -> I e. V ) |
3 |
|
psrgrp.r |
|- ( ph -> R e. Grp ) |
4 |
|
psrnegcl.d |
|- D = { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin } |
5 |
|
psrnegcl.i |
|- N = ( invg ` R ) |
6 |
|
psrnegcl.b |
|- B = ( Base ` S ) |
7 |
|
psrnegcl.z |
|- ( ph -> X e. B ) |
8 |
|
psrlinv.o |
|- .0. = ( 0g ` R ) |
9 |
|
psrlinv.p |
|- .+ = ( +g ` S ) |
10 |
|
ovex |
|- ( NN0 ^m I ) e. _V |
11 |
4 10
|
rabex2 |
|- D e. _V |
12 |
11
|
a1i |
|- ( ph -> D e. _V ) |
13 |
|
fvexd |
|- ( ( ph /\ x e. D ) -> ( N ` ( X ` x ) ) e. _V ) |
14 |
|
eqid |
|- ( Base ` R ) = ( Base ` R ) |
15 |
1 14 4 6 7
|
psrelbas |
|- ( ph -> X : D --> ( Base ` R ) ) |
16 |
15
|
ffvelrnda |
|- ( ( ph /\ x e. D ) -> ( X ` x ) e. ( Base ` R ) ) |
17 |
15
|
feqmptd |
|- ( ph -> X = ( x e. D |-> ( X ` x ) ) ) |
18 |
14 5 3
|
grpinvf1o |
|- ( ph -> N : ( Base ` R ) -1-1-onto-> ( Base ` R ) ) |
19 |
|
f1of |
|- ( N : ( Base ` R ) -1-1-onto-> ( Base ` R ) -> N : ( Base ` R ) --> ( Base ` R ) ) |
20 |
18 19
|
syl |
|- ( ph -> N : ( Base ` R ) --> ( Base ` R ) ) |
21 |
20
|
feqmptd |
|- ( ph -> N = ( y e. ( Base ` R ) |-> ( N ` y ) ) ) |
22 |
|
fveq2 |
|- ( y = ( X ` x ) -> ( N ` y ) = ( N ` ( X ` x ) ) ) |
23 |
16 17 21 22
|
fmptco |
|- ( ph -> ( N o. X ) = ( x e. D |-> ( N ` ( X ` x ) ) ) ) |
24 |
12 13 16 23 17
|
offval2 |
|- ( ph -> ( ( N o. X ) oF ( +g ` R ) X ) = ( x e. D |-> ( ( N ` ( X ` x ) ) ( +g ` R ) ( X ` x ) ) ) ) |
25 |
|
eqid |
|- ( +g ` R ) = ( +g ` R ) |
26 |
1 2 3 4 5 6 7
|
psrnegcl |
|- ( ph -> ( N o. X ) e. B ) |
27 |
1 6 25 9 26 7
|
psradd |
|- ( ph -> ( ( N o. X ) .+ X ) = ( ( N o. X ) oF ( +g ` R ) X ) ) |
28 |
|
fconstmpt |
|- ( D X. { .0. } ) = ( x e. D |-> .0. ) |
29 |
14 25 8 5
|
grplinv |
|- ( ( R e. Grp /\ ( X ` x ) e. ( Base ` R ) ) -> ( ( N ` ( X ` x ) ) ( +g ` R ) ( X ` x ) ) = .0. ) |
30 |
3 16 29
|
syl2an2r |
|- ( ( ph /\ x e. D ) -> ( ( N ` ( X ` x ) ) ( +g ` R ) ( X ` x ) ) = .0. ) |
31 |
30
|
mpteq2dva |
|- ( ph -> ( x e. D |-> ( ( N ` ( X ` x ) ) ( +g ` R ) ( X ` x ) ) ) = ( x e. D |-> .0. ) ) |
32 |
28 31
|
eqtr4id |
|- ( ph -> ( D X. { .0. } ) = ( x e. D |-> ( ( N ` ( X ` x ) ) ( +g ` R ) ( X ` x ) ) ) ) |
33 |
24 27 32
|
3eqtr4d |
|- ( ph -> ( ( N o. X ) .+ X ) = ( D X. { .0. } ) ) |