Step |
Hyp |
Ref |
Expression |
1 |
|
elprnq |
|- ( ( A e. P. /\ g e. A ) -> g e. Q. ) |
2 |
|
ltrnq |
|- ( x ( *Q ` ( g +Q h ) ) |
3 |
|
ltmnq |
|- ( x e. Q. -> ( ( *Q ` ( g +Q h ) ) ( x .Q ( *Q ` ( g +Q h ) ) ) |
4 |
|
ovex |
|- ( x .Q ( *Q ` ( g +Q h ) ) ) e. _V |
5 |
|
ovex |
|- ( x .Q ( *Q ` x ) ) e. _V |
6 |
|
ltmnq |
|- ( w e. Q. -> ( y ( w .Q y ) |
7 |
|
vex |
|- g e. _V |
8 |
|
mulcomnq |
|- ( y .Q z ) = ( z .Q y ) |
9 |
4 5 6 7 8
|
caovord2 |
|- ( g e. Q. -> ( ( x .Q ( *Q ` ( g +Q h ) ) ) ( ( x .Q ( *Q ` ( g +Q h ) ) ) .Q g ) |
10 |
3 9
|
sylan9bbr |
|- ( ( g e. Q. /\ x e. Q. ) -> ( ( *Q ` ( g +Q h ) ) ( ( x .Q ( *Q ` ( g +Q h ) ) ) .Q g ) |
11 |
2 10
|
syl5bb |
|- ( ( g e. Q. /\ x e. Q. ) -> ( x ( ( x .Q ( *Q ` ( g +Q h ) ) ) .Q g ) |
12 |
|
recidnq |
|- ( x e. Q. -> ( x .Q ( *Q ` x ) ) = 1Q ) |
13 |
12
|
oveq1d |
|- ( x e. Q. -> ( ( x .Q ( *Q ` x ) ) .Q g ) = ( 1Q .Q g ) ) |
14 |
|
mulcomnq |
|- ( 1Q .Q g ) = ( g .Q 1Q ) |
15 |
|
mulidnq |
|- ( g e. Q. -> ( g .Q 1Q ) = g ) |
16 |
14 15
|
eqtrid |
|- ( g e. Q. -> ( 1Q .Q g ) = g ) |
17 |
13 16
|
sylan9eqr |
|- ( ( g e. Q. /\ x e. Q. ) -> ( ( x .Q ( *Q ` x ) ) .Q g ) = g ) |
18 |
17
|
breq2d |
|- ( ( g e. Q. /\ x e. Q. ) -> ( ( ( x .Q ( *Q ` ( g +Q h ) ) ) .Q g ) ( ( x .Q ( *Q ` ( g +Q h ) ) ) .Q g ) |
19 |
11 18
|
bitrd |
|- ( ( g e. Q. /\ x e. Q. ) -> ( x ( ( x .Q ( *Q ` ( g +Q h ) ) ) .Q g ) |
20 |
1 19
|
sylan |
|- ( ( ( A e. P. /\ g e. A ) /\ x e. Q. ) -> ( x ( ( x .Q ( *Q ` ( g +Q h ) ) ) .Q g ) |
21 |
|
prcdnq |
|- ( ( A e. P. /\ g e. A ) -> ( ( ( x .Q ( *Q ` ( g +Q h ) ) ) .Q g ) ( ( x .Q ( *Q ` ( g +Q h ) ) ) .Q g ) e. A ) ) |
22 |
21
|
adantr |
|- ( ( ( A e. P. /\ g e. A ) /\ x e. Q. ) -> ( ( ( x .Q ( *Q ` ( g +Q h ) ) ) .Q g ) ( ( x .Q ( *Q ` ( g +Q h ) ) ) .Q g ) e. A ) ) |
23 |
20 22
|
sylbid |
|- ( ( ( A e. P. /\ g e. A ) /\ x e. Q. ) -> ( x ( ( x .Q ( *Q ` ( g +Q h ) ) ) .Q g ) e. A ) ) |