| Step |
Hyp |
Ref |
Expression |
| 1 |
|
cphsubrglem.k |
|- K = ( Base ` F ) |
| 2 |
|
cphsubrglem.1 |
|- ( ph -> F = ( CCfld |`s A ) ) |
| 3 |
|
cphsubrglem.2 |
|- ( ph -> F e. DivRing ) |
| 4 |
1 2 3
|
cphsubrglem |
|- ( ph -> ( F = ( CCfld |`s K ) /\ K = ( A i^i CC ) /\ K e. ( SubRing ` CCfld ) ) ) |
| 5 |
4
|
simp3d |
|- ( ph -> K e. ( SubRing ` CCfld ) ) |
| 6 |
5
|
3ad2ant1 |
|- ( ( ph /\ X e. K /\ X =/= 0 ) -> K e. ( SubRing ` CCfld ) ) |
| 7 |
|
cnfldbas |
|- CC = ( Base ` CCfld ) |
| 8 |
7
|
subrgss |
|- ( K e. ( SubRing ` CCfld ) -> K C_ CC ) |
| 9 |
6 8
|
syl |
|- ( ( ph /\ X e. K /\ X =/= 0 ) -> K C_ CC ) |
| 10 |
|
simp2 |
|- ( ( ph /\ X e. K /\ X =/= 0 ) -> X e. K ) |
| 11 |
9 10
|
sseldd |
|- ( ( ph /\ X e. K /\ X =/= 0 ) -> X e. CC ) |
| 12 |
|
simp3 |
|- ( ( ph /\ X e. K /\ X =/= 0 ) -> X =/= 0 ) |
| 13 |
|
cnfldinv |
|- ( ( X e. CC /\ X =/= 0 ) -> ( ( invr ` CCfld ) ` X ) = ( 1 / X ) ) |
| 14 |
11 12 13
|
syl2anc |
|- ( ( ph /\ X e. K /\ X =/= 0 ) -> ( ( invr ` CCfld ) ` X ) = ( 1 / X ) ) |
| 15 |
|
eqid |
|- ( CCfld |`s K ) = ( CCfld |`s K ) |
| 16 |
|
cnfld0 |
|- 0 = ( 0g ` CCfld ) |
| 17 |
15 16
|
subrg0 |
|- ( K e. ( SubRing ` CCfld ) -> 0 = ( 0g ` ( CCfld |`s K ) ) ) |
| 18 |
6 17
|
syl |
|- ( ( ph /\ X e. K /\ X =/= 0 ) -> 0 = ( 0g ` ( CCfld |`s K ) ) ) |
| 19 |
4
|
simp1d |
|- ( ph -> F = ( CCfld |`s K ) ) |
| 20 |
19
|
3ad2ant1 |
|- ( ( ph /\ X e. K /\ X =/= 0 ) -> F = ( CCfld |`s K ) ) |
| 21 |
20
|
fveq2d |
|- ( ( ph /\ X e. K /\ X =/= 0 ) -> ( 0g ` F ) = ( 0g ` ( CCfld |`s K ) ) ) |
| 22 |
18 21
|
eqtr4d |
|- ( ( ph /\ X e. K /\ X =/= 0 ) -> 0 = ( 0g ` F ) ) |
| 23 |
12 22
|
neeqtrd |
|- ( ( ph /\ X e. K /\ X =/= 0 ) -> X =/= ( 0g ` F ) ) |
| 24 |
|
eldifsn |
|- ( X e. ( K \ { ( 0g ` F ) } ) <-> ( X e. K /\ X =/= ( 0g ` F ) ) ) |
| 25 |
10 23 24
|
sylanbrc |
|- ( ( ph /\ X e. K /\ X =/= 0 ) -> X e. ( K \ { ( 0g ` F ) } ) ) |
| 26 |
3
|
3ad2ant1 |
|- ( ( ph /\ X e. K /\ X =/= 0 ) -> F e. DivRing ) |
| 27 |
|
eqid |
|- ( Unit ` F ) = ( Unit ` F ) |
| 28 |
|
eqid |
|- ( 0g ` F ) = ( 0g ` F ) |
| 29 |
1 27 28
|
isdrng |
|- ( F e. DivRing <-> ( F e. Ring /\ ( Unit ` F ) = ( K \ { ( 0g ` F ) } ) ) ) |
| 30 |
29
|
simprbi |
|- ( F e. DivRing -> ( Unit ` F ) = ( K \ { ( 0g ` F ) } ) ) |
| 31 |
26 30
|
syl |
|- ( ( ph /\ X e. K /\ X =/= 0 ) -> ( Unit ` F ) = ( K \ { ( 0g ` F ) } ) ) |
| 32 |
20
|
fveq2d |
|- ( ( ph /\ X e. K /\ X =/= 0 ) -> ( Unit ` F ) = ( Unit ` ( CCfld |`s K ) ) ) |
| 33 |
31 32
|
eqtr3d |
|- ( ( ph /\ X e. K /\ X =/= 0 ) -> ( K \ { ( 0g ` F ) } ) = ( Unit ` ( CCfld |`s K ) ) ) |
| 34 |
25 33
|
eleqtrd |
|- ( ( ph /\ X e. K /\ X =/= 0 ) -> X e. ( Unit ` ( CCfld |`s K ) ) ) |
| 35 |
|
eqid |
|- ( Unit ` CCfld ) = ( Unit ` CCfld ) |
| 36 |
|
eqid |
|- ( Unit ` ( CCfld |`s K ) ) = ( Unit ` ( CCfld |`s K ) ) |
| 37 |
|
eqid |
|- ( invr ` CCfld ) = ( invr ` CCfld ) |
| 38 |
15 35 36 37
|
subrgunit |
|- ( K e. ( SubRing ` CCfld ) -> ( X e. ( Unit ` ( CCfld |`s K ) ) <-> ( X e. ( Unit ` CCfld ) /\ X e. K /\ ( ( invr ` CCfld ) ` X ) e. K ) ) ) |
| 39 |
6 38
|
syl |
|- ( ( ph /\ X e. K /\ X =/= 0 ) -> ( X e. ( Unit ` ( CCfld |`s K ) ) <-> ( X e. ( Unit ` CCfld ) /\ X e. K /\ ( ( invr ` CCfld ) ` X ) e. K ) ) ) |
| 40 |
34 39
|
mpbid |
|- ( ( ph /\ X e. K /\ X =/= 0 ) -> ( X e. ( Unit ` CCfld ) /\ X e. K /\ ( ( invr ` CCfld ) ` X ) e. K ) ) |
| 41 |
40
|
simp3d |
|- ( ( ph /\ X e. K /\ X =/= 0 ) -> ( ( invr ` CCfld ) ` X ) e. K ) |
| 42 |
14 41
|
eqeltrrd |
|- ( ( ph /\ X e. K /\ X =/= 0 ) -> ( 1 / X ) e. K ) |