Step |
Hyp |
Ref |
Expression |
1 |
|
mulgnn0z.b |
|- B = ( Base ` G ) |
2 |
|
mulgnn0z.t |
|- .x. = ( .g ` G ) |
3 |
|
mulgnn0z.o |
|- .0. = ( 0g ` G ) |
4 |
|
grpmnd |
|- ( G e. Grp -> G e. Mnd ) |
5 |
4
|
adantr |
|- ( ( G e. Grp /\ N e. ZZ ) -> G e. Mnd ) |
6 |
1 2 3
|
mulgnn0z |
|- ( ( G e. Mnd /\ N e. NN0 ) -> ( N .x. .0. ) = .0. ) |
7 |
5 6
|
sylan |
|- ( ( ( G e. Grp /\ N e. ZZ ) /\ N e. NN0 ) -> ( N .x. .0. ) = .0. ) |
8 |
|
simpll |
|- ( ( ( G e. Grp /\ N e. ZZ ) /\ -u N e. NN0 ) -> G e. Grp ) |
9 |
|
nn0z |
|- ( -u N e. NN0 -> -u N e. ZZ ) |
10 |
9
|
adantl |
|- ( ( ( G e. Grp /\ N e. ZZ ) /\ -u N e. NN0 ) -> -u N e. ZZ ) |
11 |
1 3
|
grpidcl |
|- ( G e. Grp -> .0. e. B ) |
12 |
11
|
ad2antrr |
|- ( ( ( G e. Grp /\ N e. ZZ ) /\ -u N e. NN0 ) -> .0. e. B ) |
13 |
|
eqid |
|- ( invg ` G ) = ( invg ` G ) |
14 |
1 2 13
|
mulgneg |
|- ( ( G e. Grp /\ -u N e. ZZ /\ .0. e. B ) -> ( -u -u N .x. .0. ) = ( ( invg ` G ) ` ( -u N .x. .0. ) ) ) |
15 |
8 10 12 14
|
syl3anc |
|- ( ( ( G e. Grp /\ N e. ZZ ) /\ -u N e. NN0 ) -> ( -u -u N .x. .0. ) = ( ( invg ` G ) ` ( -u N .x. .0. ) ) ) |
16 |
|
zcn |
|- ( N e. ZZ -> N e. CC ) |
17 |
16
|
ad2antlr |
|- ( ( ( G e. Grp /\ N e. ZZ ) /\ -u N e. NN0 ) -> N e. CC ) |
18 |
17
|
negnegd |
|- ( ( ( G e. Grp /\ N e. ZZ ) /\ -u N e. NN0 ) -> -u -u N = N ) |
19 |
18
|
oveq1d |
|- ( ( ( G e. Grp /\ N e. ZZ ) /\ -u N e. NN0 ) -> ( -u -u N .x. .0. ) = ( N .x. .0. ) ) |
20 |
1 2 3
|
mulgnn0z |
|- ( ( G e. Mnd /\ -u N e. NN0 ) -> ( -u N .x. .0. ) = .0. ) |
21 |
5 20
|
sylan |
|- ( ( ( G e. Grp /\ N e. ZZ ) /\ -u N e. NN0 ) -> ( -u N .x. .0. ) = .0. ) |
22 |
21
|
fveq2d |
|- ( ( ( G e. Grp /\ N e. ZZ ) /\ -u N e. NN0 ) -> ( ( invg ` G ) ` ( -u N .x. .0. ) ) = ( ( invg ` G ) ` .0. ) ) |
23 |
3 13
|
grpinvid |
|- ( G e. Grp -> ( ( invg ` G ) ` .0. ) = .0. ) |
24 |
23
|
ad2antrr |
|- ( ( ( G e. Grp /\ N e. ZZ ) /\ -u N e. NN0 ) -> ( ( invg ` G ) ` .0. ) = .0. ) |
25 |
22 24
|
eqtrd |
|- ( ( ( G e. Grp /\ N e. ZZ ) /\ -u N e. NN0 ) -> ( ( invg ` G ) ` ( -u N .x. .0. ) ) = .0. ) |
26 |
15 19 25
|
3eqtr3d |
|- ( ( ( G e. Grp /\ N e. ZZ ) /\ -u N e. NN0 ) -> ( N .x. .0. ) = .0. ) |
27 |
|
elznn0 |
|- ( N e. ZZ <-> ( N e. RR /\ ( N e. NN0 \/ -u N e. NN0 ) ) ) |
28 |
27
|
simprbi |
|- ( N e. ZZ -> ( N e. NN0 \/ -u N e. NN0 ) ) |
29 |
28
|
adantl |
|- ( ( G e. Grp /\ N e. ZZ ) -> ( N e. NN0 \/ -u N e. NN0 ) ) |
30 |
7 26 29
|
mpjaodan |
|- ( ( G e. Grp /\ N e. ZZ ) -> ( N .x. .0. ) = .0. ) |