| Step |
Hyp |
Ref |
Expression |
| 1 |
|
qcvs.q |
|- Q = ( ringLMod ` ( CCfld |`s QQ ) ) |
| 2 |
|
qsubdrg |
|- ( QQ e. ( SubRing ` CCfld ) /\ ( CCfld |`s QQ ) e. DivRing ) |
| 3 |
|
drngring |
|- ( ( CCfld |`s QQ ) e. DivRing -> ( CCfld |`s QQ ) e. Ring ) |
| 4 |
3
|
adantl |
|- ( ( QQ e. ( SubRing ` CCfld ) /\ ( CCfld |`s QQ ) e. DivRing ) -> ( CCfld |`s QQ ) e. Ring ) |
| 5 |
2 4
|
ax-mp |
|- ( CCfld |`s QQ ) e. Ring |
| 6 |
|
rlmlmod |
|- ( ( CCfld |`s QQ ) e. Ring -> ( ringLMod ` ( CCfld |`s QQ ) ) e. LMod ) |
| 7 |
5 6
|
ax-mp |
|- ( ringLMod ` ( CCfld |`s QQ ) ) e. LMod |
| 8 |
2
|
simpri |
|- ( CCfld |`s QQ ) e. DivRing |
| 9 |
|
rlmsca |
|- ( ( CCfld |`s QQ ) e. DivRing -> ( CCfld |`s QQ ) = ( Scalar ` ( ringLMod ` ( CCfld |`s QQ ) ) ) ) |
| 10 |
9
|
eqcomd |
|- ( ( CCfld |`s QQ ) e. DivRing -> ( Scalar ` ( ringLMod ` ( CCfld |`s QQ ) ) ) = ( CCfld |`s QQ ) ) |
| 11 |
8 10
|
ax-mp |
|- ( Scalar ` ( ringLMod ` ( CCfld |`s QQ ) ) ) = ( CCfld |`s QQ ) |
| 12 |
2
|
simpli |
|- QQ e. ( SubRing ` CCfld ) |
| 13 |
|
eqid |
|- ( Scalar ` ( ringLMod ` ( CCfld |`s QQ ) ) ) = ( Scalar ` ( ringLMod ` ( CCfld |`s QQ ) ) ) |
| 14 |
13
|
isclmi |
|- ( ( ( ringLMod ` ( CCfld |`s QQ ) ) e. LMod /\ ( Scalar ` ( ringLMod ` ( CCfld |`s QQ ) ) ) = ( CCfld |`s QQ ) /\ QQ e. ( SubRing ` CCfld ) ) -> ( ringLMod ` ( CCfld |`s QQ ) ) e. CMod ) |
| 15 |
7 11 12 14
|
mp3an |
|- ( ringLMod ` ( CCfld |`s QQ ) ) e. CMod |
| 16 |
|
rlmlvec |
|- ( ( CCfld |`s QQ ) e. DivRing -> ( ringLMod ` ( CCfld |`s QQ ) ) e. LVec ) |
| 17 |
8 16
|
ax-mp |
|- ( ringLMod ` ( CCfld |`s QQ ) ) e. LVec |
| 18 |
15 17
|
elini |
|- ( ringLMod ` ( CCfld |`s QQ ) ) e. ( CMod i^i LVec ) |
| 19 |
|
df-cvs |
|- CVec = ( CMod i^i LVec ) |
| 20 |
18 1 19
|
3eltr4i |
|- Q e. CVec |