Metamath Proof Explorer

Theorem erngdvlem2N

Description: Lemma for eringring . (Contributed by NM, 6-Aug-2013) (New usage is discouraged.)

Ref Expression
Hypotheses ernggrp.h 
|- H = ( LHyp ` K )
ernggrp.d 
|- D = ( ( EDRing ` K ) ` W )
erngdv.b 
|- B = ( Base ` K )
erngdv.t 
|- T = ( ( LTrn ` K ) ` W )
erngdv.e 
|- E = ( ( TEndo ` K ) ` W )
erngdv.p 
|- P = ( a e. E , b e. E |-> ( f e. T |-> ( ( a ` f ) o. ( b ` f ) ) ) )
erngdv.o 
|- .0. = ( f e. T |-> ( _I |` B ) )
erngdv.i 
|- I = ( a e. E |-> ( f e. T |-> `' ( a ` f ) ) )
Assertion erngdvlem2N 
|- ( ( K e. HL /\ W e. H ) -> D e. Abel )

Proof

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 2 3 4 5 6 7 8 erngdvlem1 
 |-  ( ( K e. HL /\ W e. H ) -> D e. Grp )
16 1 4 5 6 tendoplcom 
 |-  ( ( ( K e. HL /\ W e. H ) /\ s e. E /\ t e. E ) -> ( s P t ) = ( t P s ) )
17 11 14 15 16 isabld 
 |-  ( ( K e. HL /\ W e. H ) -> D e. Abel )