| Step |
Hyp |
Ref |
Expression |
| 1 |
|
ernggrp.h |
|- H = ( LHyp ` K ) |
| 2 |
|
ernggrp.d |
|- D = ( ( EDRing ` K ) ` W ) |
| 3 |
|
erngdv.b |
|- B = ( Base ` K ) |
| 4 |
|
erngdv.t |
|- T = ( ( LTrn ` K ) ` W ) |
| 5 |
|
erngdv.e |
|- E = ( ( TEndo ` K ) ` W ) |
| 6 |
|
erngdv.p |
|- P = ( a e. E , b e. E |-> ( f e. T |-> ( ( a ` f ) o. ( b ` f ) ) ) ) |
| 7 |
|
erngdv.o |
|- .0. = ( f e. T |-> ( _I |` B ) ) |
| 8 |
|
erngdv.i |
|- I = ( a e. E |-> ( f e. T |-> `' ( a ` f ) ) ) |
| 9 |
|
eqid |
|- ( Base ` D ) = ( Base ` D ) |
| 10 |
1 4 5 2 9
|
erngbase |
|- ( ( K e. HL /\ W e. H ) -> ( Base ` D ) = E ) |
| 11 |
10
|
eqcomd |
|- ( ( K e. HL /\ W e. H ) -> E = ( Base ` D ) ) |
| 12 |
|
eqid |
|- ( +g ` D ) = ( +g ` D ) |
| 13 |
1 4 5 2 12
|
erngfplus |
|- ( ( K e. HL /\ W e. H ) -> ( +g ` D ) = ( a e. E , b e. E |-> ( f e. T |-> ( ( a ` f ) o. ( b ` f ) ) ) ) ) |
| 14 |
6 13
|
eqtr4id |
|- ( ( K e. HL /\ W e. H ) -> P = ( +g ` D ) ) |
| 15 |
1 4 5 6
|
tendoplcl |
|- ( ( ( K e. HL /\ W e. H ) /\ s e. E /\ t e. E ) -> ( s P t ) e. E ) |
| 16 |
1 4 5 6
|
tendoplass |
|- ( ( ( K e. HL /\ W e. H ) /\ ( s e. E /\ t e. E /\ u e. E ) ) -> ( ( s P t ) P u ) = ( s P ( t P u ) ) ) |
| 17 |
3 1 4 5 7
|
tendo0cl |
|- ( ( K e. HL /\ W e. H ) -> .0. e. E ) |
| 18 |
3 1 4 5 7 6
|
tendo0pl |
|- ( ( ( K e. HL /\ W e. H ) /\ s e. E ) -> ( .0. P s ) = s ) |
| 19 |
1 4 5 8
|
tendoicl |
|- ( ( ( K e. HL /\ W e. H ) /\ s e. E ) -> ( I ` s ) e. E ) |
| 20 |
1 4 5 8 3 6 7
|
tendoipl |
|- ( ( ( K e. HL /\ W e. H ) /\ s e. E ) -> ( ( I ` s ) P s ) = .0. ) |
| 21 |
11 14 15 16 17 18 19 20
|
isgrpd |
|- ( ( K e. HL /\ W e. H ) -> D e. Grp ) |