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Metamath Proof Explorer

Theorem erngdvlem1-rN

Description: Lemma for eringring . (Contributed by NM, 4-Aug-2013) (New usage is discouraged.)

Ref Expression
Hypotheses ernggrp.h-r 
|- H = ( LHyp ` K )
ernggrp.d-r 
|- D = ( ( EDRingR ` K ) ` W )
ernggrplem.b-r 
|- B = ( Base ` K )
ernggrplem.t-r 
|- T = ( ( LTrn ` K ) ` W )
ernggrplem.e-r 
|- E = ( ( TEndo ` K ) ` W )
ernggrplem.p-r 
|- P = ( a e. E , b e. E |-> ( f e. T |-> ( ( a ` f ) o. ( b ` f ) ) ) )
ernggrplem.o-r 
|- O = ( f e. T |-> ( _I |` B ) )
ernggrplem.i-r 
|- I = ( a e. E |-> ( f e. T |-> `' ( a ` f ) ) )
Assertion erngdvlem1-rN 
|- ( ( K e. HL /\ W e. H ) -> D e. Grp )

Proof

Step Hyp Ref Expression
1 ernggrp.h-r 
 |-  H = ( LHyp ` K )
2 ernggrp.d-r 
 |-  D = ( ( EDRingR ` K ) ` W )
3 ernggrplem.b-r 
 |-  B = ( Base ` K )
4 ernggrplem.t-r 
 |-  T = ( ( LTrn ` K ) ` W )
5 ernggrplem.e-r 
 |-  E = ( ( TEndo ` K ) ` W )
6 ernggrplem.p-r 
 |-  P = ( a e. E , b e. E |-> ( f e. T |-> ( ( a ` f ) o. ( b ` f ) ) ) )
7 ernggrplem.o-r 
 |-  O = ( f e. T |-> ( _I |` B ) )
8 ernggrplem.i-r 
 |-  I = ( a e. E |-> ( f e. T |-> `' ( a ` f ) ) )
9 eqid 
 |-  ( Base ` D ) = ( Base ` D )
10 1 4 5 2 9 erngbase-rN 
 |-  ( ( 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-rN 
 |-  ( ( 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 ) -> O e. E )
18 3 1 4 5 7 6 tendo0pl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ s e. E ) -> ( O 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 ) = O )
21 11 14 15 16 17 18 19 20 isgrpd 
 |-  ( ( K e. HL /\ W e. H ) -> D e. Grp )