| Step |
Hyp |
Ref |
Expression |
| 1 |
|
df-refld |
|- RRfld = ( CCfld |`s RR ) |
| 2 |
1
|
oveq1i |
|- ( RRfld |`s NN0 ) = ( ( CCfld |`s RR ) |`s NN0 ) |
| 3 |
|
reex |
|- RR e. _V |
| 4 |
|
nn0ssre |
|- NN0 C_ RR |
| 5 |
|
ressabs |
|- ( ( RR e. _V /\ NN0 C_ RR ) -> ( ( CCfld |`s RR ) |`s NN0 ) = ( CCfld |`s NN0 ) ) |
| 6 |
3 4 5
|
mp2an |
|- ( ( CCfld |`s RR ) |`s NN0 ) = ( CCfld |`s NN0 ) |
| 7 |
2 6
|
eqtri |
|- ( RRfld |`s NN0 ) = ( CCfld |`s NN0 ) |
| 8 |
|
reofld |
|- RRfld e. oField |
| 9 |
|
isofld |
|- ( RRfld e. oField <-> ( RRfld e. Field /\ RRfld e. oRing ) ) |
| 10 |
9
|
simprbi |
|- ( RRfld e. oField -> RRfld e. oRing ) |
| 11 |
|
orngogrp |
|- ( RRfld e. oRing -> RRfld e. oGrp ) |
| 12 |
|
isogrp |
|- ( RRfld e. oGrp <-> ( RRfld e. Grp /\ RRfld e. oMnd ) ) |
| 13 |
12
|
simprbi |
|- ( RRfld e. oGrp -> RRfld e. oMnd ) |
| 14 |
10 11 13
|
3syl |
|- ( RRfld e. oField -> RRfld e. oMnd ) |
| 15 |
8 14
|
ax-mp |
|- RRfld e. oMnd |
| 16 |
|
nn0subm |
|- NN0 e. ( SubMnd ` CCfld ) |
| 17 |
|
eqid |
|- ( CCfld |`s NN0 ) = ( CCfld |`s NN0 ) |
| 18 |
17
|
submmnd |
|- ( NN0 e. ( SubMnd ` CCfld ) -> ( CCfld |`s NN0 ) e. Mnd ) |
| 19 |
16 18
|
ax-mp |
|- ( CCfld |`s NN0 ) e. Mnd |
| 20 |
7 19
|
eqeltri |
|- ( RRfld |`s NN0 ) e. Mnd |
| 21 |
|
submomnd |
|- ( ( RRfld e. oMnd /\ ( RRfld |`s NN0 ) e. Mnd ) -> ( RRfld |`s NN0 ) e. oMnd ) |
| 22 |
15 20 21
|
mp2an |
|- ( RRfld |`s NN0 ) e. oMnd |
| 23 |
7 22
|
eqeltrri |
|- ( CCfld |`s NN0 ) e. oMnd |