Step |
Hyp |
Ref |
Expression |
1 |
|
islfld.v |
|- ( ph -> V = ( Base ` W ) ) |
2 |
|
islfld.a |
|- ( ph -> .+ = ( +g ` W ) ) |
3 |
|
islfld.d |
|- ( ph -> D = ( Scalar ` W ) ) |
4 |
|
islfld.s |
|- ( ph -> .x. = ( .s ` W ) ) |
5 |
|
islfld.k |
|- ( ph -> K = ( Base ` D ) ) |
6 |
|
islfld.p |
|- ( ph -> .+^ = ( +g ` D ) ) |
7 |
|
islfld.t |
|- ( ph -> .X. = ( .r ` D ) ) |
8 |
|
islfld.f |
|- ( ph -> F = ( LFnl ` W ) ) |
9 |
|
islfld.u |
|- ( ph -> G : V --> K ) |
10 |
|
islfld.l |
|- ( ( ph /\ ( r e. K /\ x e. V /\ y e. V ) ) -> ( G ` ( ( r .x. x ) .+ y ) ) = ( ( r .X. ( G ` x ) ) .+^ ( G ` y ) ) ) |
11 |
|
islfld.w |
|- ( ph -> W e. X ) |
12 |
3
|
fveq2d |
|- ( ph -> ( Base ` D ) = ( Base ` ( Scalar ` W ) ) ) |
13 |
5 12
|
eqtrd |
|- ( ph -> K = ( Base ` ( Scalar ` W ) ) ) |
14 |
1 13
|
feq23d |
|- ( ph -> ( G : V --> K <-> G : ( Base ` W ) --> ( Base ` ( Scalar ` W ) ) ) ) |
15 |
9 14
|
mpbid |
|- ( ph -> G : ( Base ` W ) --> ( Base ` ( Scalar ` W ) ) ) |
16 |
10
|
ralrimivvva |
|- ( ph -> A. r e. K A. x e. V A. y e. V ( G ` ( ( r .x. x ) .+ y ) ) = ( ( r .X. ( G ` x ) ) .+^ ( G ` y ) ) ) |
17 |
4
|
oveqd |
|- ( ph -> ( r .x. x ) = ( r ( .s ` W ) x ) ) |
18 |
|
eqidd |
|- ( ph -> y = y ) |
19 |
2 17 18
|
oveq123d |
|- ( ph -> ( ( r .x. x ) .+ y ) = ( ( r ( .s ` W ) x ) ( +g ` W ) y ) ) |
20 |
19
|
fveq2d |
|- ( ph -> ( G ` ( ( r .x. x ) .+ y ) ) = ( G ` ( ( r ( .s ` W ) x ) ( +g ` W ) y ) ) ) |
21 |
3
|
fveq2d |
|- ( ph -> ( +g ` D ) = ( +g ` ( Scalar ` W ) ) ) |
22 |
6 21
|
eqtrd |
|- ( ph -> .+^ = ( +g ` ( Scalar ` W ) ) ) |
23 |
3
|
fveq2d |
|- ( ph -> ( .r ` D ) = ( .r ` ( Scalar ` W ) ) ) |
24 |
7 23
|
eqtrd |
|- ( ph -> .X. = ( .r ` ( Scalar ` W ) ) ) |
25 |
24
|
oveqd |
|- ( ph -> ( r .X. ( G ` x ) ) = ( r ( .r ` ( Scalar ` W ) ) ( G ` x ) ) ) |
26 |
|
eqidd |
|- ( ph -> ( G ` y ) = ( G ` y ) ) |
27 |
22 25 26
|
oveq123d |
|- ( ph -> ( ( r .X. ( G ` x ) ) .+^ ( G ` y ) ) = ( ( r ( .r ` ( Scalar ` W ) ) ( G ` x ) ) ( +g ` ( Scalar ` W ) ) ( G ` y ) ) ) |
28 |
20 27
|
eqeq12d |
|- ( ph -> ( ( G ` ( ( r .x. x ) .+ y ) ) = ( ( r .X. ( G ` x ) ) .+^ ( G ` y ) ) <-> ( G ` ( ( r ( .s ` W ) x ) ( +g ` W ) y ) ) = ( ( r ( .r ` ( Scalar ` W ) ) ( G ` x ) ) ( +g ` ( Scalar ` W ) ) ( G ` y ) ) ) ) |
29 |
1 28
|
raleqbidv |
|- ( ph -> ( A. y e. V ( G ` ( ( r .x. x ) .+ y ) ) = ( ( r .X. ( G ` x ) ) .+^ ( G ` y ) ) <-> A. y e. ( Base ` W ) ( G ` ( ( r ( .s ` W ) x ) ( +g ` W ) y ) ) = ( ( r ( .r ` ( Scalar ` W ) ) ( G ` x ) ) ( +g ` ( Scalar ` W ) ) ( G ` y ) ) ) ) |
30 |
1 29
|
raleqbidv |
|- ( ph -> ( A. x e. V A. y e. V ( G ` ( ( r .x. x ) .+ y ) ) = ( ( r .X. ( G ` x ) ) .+^ ( G ` y ) ) <-> A. x e. ( Base ` W ) A. y e. ( Base ` W ) ( G ` ( ( r ( .s ` W ) x ) ( +g ` W ) y ) ) = ( ( r ( .r ` ( Scalar ` W ) ) ( G ` x ) ) ( +g ` ( Scalar ` W ) ) ( G ` y ) ) ) ) |
31 |
13 30
|
raleqbidv |
|- ( ph -> ( A. r e. K A. x e. V A. y e. V ( G ` ( ( r .x. x ) .+ y ) ) = ( ( r .X. ( G ` x ) ) .+^ ( G ` y ) ) <-> A. r e. ( Base ` ( Scalar ` W ) ) A. x e. ( Base ` W ) A. y e. ( Base ` W ) ( G ` ( ( r ( .s ` W ) x ) ( +g ` W ) y ) ) = ( ( r ( .r ` ( Scalar ` W ) ) ( G ` x ) ) ( +g ` ( Scalar ` W ) ) ( G ` y ) ) ) ) |
32 |
16 31
|
mpbid |
|- ( ph -> A. r e. ( Base ` ( Scalar ` W ) ) A. x e. ( Base ` W ) A. y e. ( Base ` W ) ( G ` ( ( r ( .s ` W ) x ) ( +g ` W ) y ) ) = ( ( r ( .r ` ( Scalar ` W ) ) ( G ` x ) ) ( +g ` ( Scalar ` W ) ) ( G ` y ) ) ) |
33 |
|
eqid |
|- ( Base ` W ) = ( Base ` W ) |
34 |
|
eqid |
|- ( +g ` W ) = ( +g ` W ) |
35 |
|
eqid |
|- ( Scalar ` W ) = ( Scalar ` W ) |
36 |
|
eqid |
|- ( .s ` W ) = ( .s ` W ) |
37 |
|
eqid |
|- ( Base ` ( Scalar ` W ) ) = ( Base ` ( Scalar ` W ) ) |
38 |
|
eqid |
|- ( +g ` ( Scalar ` W ) ) = ( +g ` ( Scalar ` W ) ) |
39 |
|
eqid |
|- ( .r ` ( Scalar ` W ) ) = ( .r ` ( Scalar ` W ) ) |
40 |
|
eqid |
|- ( LFnl ` W ) = ( LFnl ` W ) |
41 |
33 34 35 36 37 38 39 40
|
islfl |
|- ( W e. X -> ( G e. ( LFnl ` W ) <-> ( G : ( Base ` W ) --> ( Base ` ( Scalar ` W ) ) /\ A. r e. ( Base ` ( Scalar ` W ) ) A. x e. ( Base ` W ) A. y e. ( Base ` W ) ( G ` ( ( r ( .s ` W ) x ) ( +g ` W ) y ) ) = ( ( r ( .r ` ( Scalar ` W ) ) ( G ` x ) ) ( +g ` ( Scalar ` W ) ) ( G ` y ) ) ) ) ) |
42 |
41
|
biimpar |
|- ( ( W e. X /\ ( G : ( Base ` W ) --> ( Base ` ( Scalar ` W ) ) /\ A. r e. ( Base ` ( Scalar ` W ) ) A. x e. ( Base ` W ) A. y e. ( Base ` W ) ( G ` ( ( r ( .s ` W ) x ) ( +g ` W ) y ) ) = ( ( r ( .r ` ( Scalar ` W ) ) ( G ` x ) ) ( +g ` ( Scalar ` W ) ) ( G ` y ) ) ) ) -> G e. ( LFnl ` W ) ) |
43 |
11 15 32 42
|
syl12anc |
|- ( ph -> G e. ( LFnl ` W ) ) |
44 |
43 8
|
eleqtrrd |
|- ( ph -> G e. F ) |