Step |
Hyp |
Ref |
Expression |
1 |
|
lsspropd.b1 |
|- ( ph -> B = ( Base ` K ) ) |
2 |
|
lsspropd.b2 |
|- ( ph -> B = ( Base ` L ) ) |
3 |
|
lsspropd.w |
|- ( ph -> B C_ W ) |
4 |
|
lsspropd.p |
|- ( ( ph /\ ( x e. W /\ y e. W ) ) -> ( x ( +g ` K ) y ) = ( x ( +g ` L ) y ) ) |
5 |
|
lsspropd.s1 |
|- ( ( ph /\ ( x e. P /\ y e. B ) ) -> ( x ( .s ` K ) y ) e. W ) |
6 |
|
lsspropd.s2 |
|- ( ( ph /\ ( x e. P /\ y e. B ) ) -> ( x ( .s ` K ) y ) = ( x ( .s ` L ) y ) ) |
7 |
|
lsspropd.p1 |
|- ( ph -> P = ( Base ` ( Scalar ` K ) ) ) |
8 |
|
lsspropd.p2 |
|- ( ph -> P = ( Base ` ( Scalar ` L ) ) ) |
9 |
|
lsppropd.v1 |
|- ( ph -> K e. X ) |
10 |
|
lsppropd.v2 |
|- ( ph -> L e. Y ) |
11 |
1 2
|
eqtr3d |
|- ( ph -> ( Base ` K ) = ( Base ` L ) ) |
12 |
11
|
pweqd |
|- ( ph -> ~P ( Base ` K ) = ~P ( Base ` L ) ) |
13 |
1 2 3 4 5 6 7 8
|
lsspropd |
|- ( ph -> ( LSubSp ` K ) = ( LSubSp ` L ) ) |
14 |
13
|
rabeqdv |
|- ( ph -> { t e. ( LSubSp ` K ) | s C_ t } = { t e. ( LSubSp ` L ) | s C_ t } ) |
15 |
14
|
inteqd |
|- ( ph -> |^| { t e. ( LSubSp ` K ) | s C_ t } = |^| { t e. ( LSubSp ` L ) | s C_ t } ) |
16 |
12 15
|
mpteq12dv |
|- ( ph -> ( s e. ~P ( Base ` K ) |-> |^| { t e. ( LSubSp ` K ) | s C_ t } ) = ( s e. ~P ( Base ` L ) |-> |^| { t e. ( LSubSp ` L ) | s C_ t } ) ) |
17 |
|
eqid |
|- ( Base ` K ) = ( Base ` K ) |
18 |
|
eqid |
|- ( LSubSp ` K ) = ( LSubSp ` K ) |
19 |
|
eqid |
|- ( LSpan ` K ) = ( LSpan ` K ) |
20 |
17 18 19
|
lspfval |
|- ( K e. X -> ( LSpan ` K ) = ( s e. ~P ( Base ` K ) |-> |^| { t e. ( LSubSp ` K ) | s C_ t } ) ) |
21 |
9 20
|
syl |
|- ( ph -> ( LSpan ` K ) = ( s e. ~P ( Base ` K ) |-> |^| { t e. ( LSubSp ` K ) | s C_ t } ) ) |
22 |
|
eqid |
|- ( Base ` L ) = ( Base ` L ) |
23 |
|
eqid |
|- ( LSubSp ` L ) = ( LSubSp ` L ) |
24 |
|
eqid |
|- ( LSpan ` L ) = ( LSpan ` L ) |
25 |
22 23 24
|
lspfval |
|- ( L e. Y -> ( LSpan ` L ) = ( s e. ~P ( Base ` L ) |-> |^| { t e. ( LSubSp ` L ) | s C_ t } ) ) |
26 |
10 25
|
syl |
|- ( ph -> ( LSpan ` L ) = ( s e. ~P ( Base ` L ) |-> |^| { t e. ( LSubSp ` L ) | s C_ t } ) ) |
27 |
16 21 26
|
3eqtr4d |
|- ( ph -> ( LSpan ` K ) = ( LSpan ` L ) ) |