| Step |
Hyp |
Ref |
Expression |
| 1 |
|
ldualvsass.f |
|- F = ( LFnl ` W ) |
| 2 |
|
ldualvsass.r |
|- R = ( Scalar ` W ) |
| 3 |
|
ldualvsass.k |
|- K = ( Base ` R ) |
| 4 |
|
ldualvsass.t |
|- .X. = ( .r ` R ) |
| 5 |
|
ldualvsass.d |
|- D = ( LDual ` W ) |
| 6 |
|
ldualvsass.s |
|- .x. = ( .s ` D ) |
| 7 |
|
ldualvsass.w |
|- ( ph -> W e. LMod ) |
| 8 |
|
ldualvsass.x |
|- ( ph -> X e. K ) |
| 9 |
|
ldualvsass.y |
|- ( ph -> Y e. K ) |
| 10 |
|
ldualvsass.g |
|- ( ph -> G e. F ) |
| 11 |
|
eqid |
|- ( Base ` W ) = ( Base ` W ) |
| 12 |
11 2 3 4 1 7 9 8 10
|
lflvsass |
|- ( ph -> ( G oF .X. ( ( Base ` W ) X. { ( Y .X. X ) } ) ) = ( ( G oF .X. ( ( Base ` W ) X. { Y } ) ) oF .X. ( ( Base ` W ) X. { X } ) ) ) |
| 13 |
2
|
lmodring |
|- ( W e. LMod -> R e. Ring ) |
| 14 |
7 13
|
syl |
|- ( ph -> R e. Ring ) |
| 15 |
3 4
|
ringcl |
|- ( ( R e. Ring /\ Y e. K /\ X e. K ) -> ( Y .X. X ) e. K ) |
| 16 |
14 9 8 15
|
syl3anc |
|- ( ph -> ( Y .X. X ) e. K ) |
| 17 |
1 11 2 3 4 5 6 7 16 10
|
ldualvs |
|- ( ph -> ( ( Y .X. X ) .x. G ) = ( G oF .X. ( ( Base ` W ) X. { ( Y .X. X ) } ) ) ) |
| 18 |
11 2 3 4 1 7 10 9
|
lflvscl |
|- ( ph -> ( G oF .X. ( ( Base ` W ) X. { Y } ) ) e. F ) |
| 19 |
1 11 2 3 4 5 6 7 8 18
|
ldualvs |
|- ( ph -> ( X .x. ( G oF .X. ( ( Base ` W ) X. { Y } ) ) ) = ( ( G oF .X. ( ( Base ` W ) X. { Y } ) ) oF .X. ( ( Base ` W ) X. { X } ) ) ) |
| 20 |
12 17 19
|
3eqtr4d |
|- ( ph -> ( ( Y .X. X ) .x. G ) = ( X .x. ( G oF .X. ( ( Base ` W ) X. { Y } ) ) ) ) |
| 21 |
1 11 2 3 4 5 6 7 9 10
|
ldualvs |
|- ( ph -> ( Y .x. G ) = ( G oF .X. ( ( Base ` W ) X. { Y } ) ) ) |
| 22 |
21
|
oveq2d |
|- ( ph -> ( X .x. ( Y .x. G ) ) = ( X .x. ( G oF .X. ( ( Base ` W ) X. { Y } ) ) ) ) |
| 23 |
20 22
|
eqtr4d |
|- ( ph -> ( ( Y .X. X ) .x. G ) = ( X .x. ( Y .x. G ) ) ) |