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Metamath Proof Explorer

Theorem erngdv

Description: An endomorphism ring is a division ring. TODO: fix comment. (Contributed by NM, 11-Aug-2013)

Ref Expression
Hypotheses ernggrp.h 
|- H = ( LHyp ` K )
ernggrp.d 
|- D = ( ( EDRing ` K ) ` W )
Assertion erngdv 
|- ( ( K e. HL /\ W e. H ) -> D e. DivRing )

Proof

Step Hyp Ref Expression
1 ernggrp.h 
 |-  H = ( LHyp ` K )
2 ernggrp.d 
 |-  D = ( ( EDRing ` K ) ` W )
3 eqid 
 |-  ( Base ` K ) = ( Base ` K )
4 eqid 
 |-  ( ( LTrn ` K ) ` W ) = ( ( LTrn ` K ) ` W )
5 3 1 4 cdlemftr0 
 |-  ( ( K e. HL /\ W e. H ) -> E. f e. ( ( LTrn ` K ) ` W ) f =/= ( _I |` ( Base ` K ) ) )
6 eqid 
 |-  ( ( TEndo ` K ) ` W ) = ( ( TEndo ` K ) ` W )
7 eqid 
 |-  ( a e. ( ( TEndo ` K ) ` W ) , b e. ( ( TEndo ` K ) ` W ) |-> ( f e. ( ( LTrn ` K ) ` W ) |-> ( ( a ` f ) o. ( b ` f ) ) ) ) = ( a e. ( ( TEndo ` K ) ` W ) , b e. ( ( TEndo ` K ) ` W ) |-> ( f e. ( ( LTrn ` K ) ` W ) |-> ( ( a ` f ) o. ( b ` f ) ) ) )
8 eqid 
 |-  ( f e. ( ( LTrn ` K ) ` W ) |-> ( _I |` ( Base ` K ) ) ) = ( f e. ( ( LTrn ` K ) ` W ) |-> ( _I |` ( Base ` K ) ) )
9 eqid 
 |-  ( a e. ( ( TEndo ` K ) ` W ) |-> ( f e. ( ( LTrn ` K ) ` W ) |-> `' ( a ` f ) ) ) = ( a e. ( ( TEndo ` K ) ` W ) |-> ( f e. ( ( LTrn ` K ) ` W ) |-> `' ( a ` f ) ) )
10 eqid 
 |-  ( a e. ( ( TEndo ` K ) ` W ) , b e. ( ( TEndo ` K ) ` W ) |-> ( a o. b ) ) = ( a e. ( ( TEndo ` K ) ` W ) , b e. ( ( TEndo ` K ) ` W ) |-> ( a o. b ) )
11 eqid 
 |-  ( join ` K ) = ( join ` K )
12 eqid 
 |-  ( meet ` K ) = ( meet ` K )
13 eqid 
 |-  ( ( trL ` K ) ` W ) = ( ( trL ` K ) ` W )
14 eqid 
 |-  ( ( oc ` K ) ` W ) = ( ( oc ` K ) ` W )
15 eqid 
 |-  ( ( ( ( oc ` K ) ` W ) ( join ` K ) ( ( ( trL ` K ) ` W ) ` b ) ) ( meet ` K ) ( ( f ` ( ( oc ` K ) ` W ) ) ( join ` K ) ( ( ( trL ` K ) ` W ) ` ( b o. `' ( s ` f ) ) ) ) ) = ( ( ( ( oc ` K ) ` W ) ( join ` K ) ( ( ( trL ` K ) ` W ) ` b ) ) ( meet ` K ) ( ( f ` ( ( oc ` K ) ` W ) ) ( join ` K ) ( ( ( trL ` K ) ` W ) ` ( b o. `' ( s ` f ) ) ) ) )
16 eqid 
 |-  ( ( ( ( oc ` K ) ` W ) ( join ` K ) ( ( ( trL ` K ) ` W ) ` g ) ) ( meet ` K ) ( ( ( ( ( oc ` K ) ` W ) ( join ` K ) ( ( ( trL ` K ) ` W ) ` b ) ) ( meet ` K ) ( ( f ` ( ( oc ` K ) ` W ) ) ( join ` K ) ( ( ( trL ` K ) ` W ) ` ( b o. `' ( s ` f ) ) ) ) ) ( join ` K ) ( ( ( trL ` K ) ` W ) ` ( g o. `' b ) ) ) ) = ( ( ( ( oc ` K ) ` W ) ( join ` K ) ( ( ( trL ` K ) ` W ) ` g ) ) ( meet ` K ) ( ( ( ( ( oc ` K ) ` W ) ( join ` K ) ( ( ( trL ` K ) ` W ) ` b ) ) ( meet ` K ) ( ( f ` ( ( oc ` K ) ` W ) ) ( join ` K ) ( ( ( trL ` K ) ` W ) ` ( b o. `' ( s ` f ) ) ) ) ) ( join ` K ) ( ( ( trL ` K ) ` W ) ` ( g o. `' b ) ) ) )
17 eqid 
 |-  ( iota_ z e. ( ( LTrn ` K ) ` W ) A. b e. ( ( LTrn ` K ) ` W ) ( ( b =/= ( _I |` ( Base ` K ) ) /\ ( ( ( trL ` K ) ` W ) ` b ) =/= ( ( ( trL ` K ) ` W ) ` ( s ` f ) ) /\ ( ( ( trL ` K ) ` W ) ` b ) =/= ( ( ( trL ` K ) ` W ) ` g ) ) -> ( z ` ( ( oc ` K ) ` W ) ) = ( ( ( ( oc ` K ) ` W ) ( join ` K ) ( ( ( trL ` K ) ` W ) ` g ) ) ( meet ` K ) ( ( ( ( ( oc ` K ) ` W ) ( join ` K ) ( ( ( trL ` K ) ` W ) ` b ) ) ( meet ` K ) ( ( f ` ( ( oc ` K ) ` W ) ) ( join ` K ) ( ( ( trL ` K ) ` W ) ` ( b o. `' ( s ` f ) ) ) ) ) ( join ` K ) ( ( ( trL ` K ) ` W ) ` ( g o. `' b ) ) ) ) ) ) = ( iota_ z e. ( ( LTrn ` K ) ` W ) A. b e. ( ( LTrn ` K ) ` W ) ( ( b =/= ( _I |` ( Base ` K ) ) /\ ( ( ( trL ` K ) ` W ) ` b ) =/= ( ( ( trL ` K ) ` W ) ` ( s ` f ) ) /\ ( ( ( trL ` K ) ` W ) ` b ) =/= ( ( ( trL ` K ) ` W ) ` g ) ) -> ( z ` ( ( oc ` K ) ` W ) ) = ( ( ( ( oc ` K ) ` W ) ( join ` K ) ( ( ( trL ` K ) ` W ) ` g ) ) ( meet ` K ) ( ( ( ( ( oc ` K ) ` W ) ( join ` K ) ( ( ( trL ` K ) ` W ) ` b ) ) ( meet ` K ) ( ( f ` ( ( oc ` K ) ` W ) ) ( join ` K ) ( ( ( trL ` K ) ` W ) ` ( b o. `' ( s ` f ) ) ) ) ) ( join ` K ) ( ( ( trL ` K ) ` W ) ` ( g o. `' b ) ) ) ) ) )
18 eqid 
 |-  ( g e. ( ( LTrn ` K ) ` W ) |-> if ( ( s ` f ) = f , g , ( iota_ z e. ( ( LTrn ` K ) ` W ) A. b e. ( ( LTrn ` K ) ` W ) ( ( b =/= ( _I |` ( Base ` K ) ) /\ ( ( ( trL ` K ) ` W ) ` b ) =/= ( ( ( trL ` K ) ` W ) ` ( s ` f ) ) /\ ( ( ( trL ` K ) ` W ) ` b ) =/= ( ( ( trL ` K ) ` W ) ` g ) ) -> ( z ` ( ( oc ` K ) ` W ) ) = ( ( ( ( oc ` K ) ` W ) ( join ` K ) ( ( ( trL ` K ) ` W ) ` g ) ) ( meet ` K ) ( ( ( ( ( oc ` K ) ` W ) ( join ` K ) ( ( ( trL ` K ) ` W ) ` b ) ) ( meet ` K ) ( ( f ` ( ( oc ` K ) ` W ) ) ( join ` K ) ( ( ( trL ` K ) ` W ) ` ( b o. `' ( s ` f ) ) ) ) ) ( join ` K ) ( ( ( trL ` K ) ` W ) ` ( g o. `' b ) ) ) ) ) ) ) ) = ( g e. ( ( LTrn ` K ) ` W ) |-> if ( ( s ` f ) = f , g , ( iota_ z e. ( ( LTrn ` K ) ` W ) A. b e. ( ( LTrn ` K ) ` W ) ( ( b =/= ( _I |` ( Base ` K ) ) /\ ( ( ( trL ` K ) ` W ) ` b ) =/= ( ( ( trL ` K ) ` W ) ` ( s ` f ) ) /\ ( ( ( trL ` K ) ` W ) ` b ) =/= ( ( ( trL ` K ) ` W ) ` g ) ) -> ( z ` ( ( oc ` K ) ` W ) ) = ( ( ( ( oc ` K ) ` W ) ( join ` K ) ( ( ( trL ` K ) ` W ) ` g ) ) ( meet ` K ) ( ( ( ( ( oc ` K ) ` W ) ( join ` K ) ( ( ( trL ` K ) ` W ) ` b ) ) ( meet ` K ) ( ( f ` ( ( oc ` K ) ` W ) ) ( join ` K ) ( ( ( trL ` K ) ` W ) ` ( b o. `' ( s ` f ) ) ) ) ) ( join ` K ) ( ( ( trL ` K ) ` W ) ` ( g o. `' b ) ) ) ) ) ) ) )
19 1 2 3 4 6 7 8 9 10 11 12 13 14 15 16 17 18 erngdvlem4 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( f e. ( ( LTrn ` K ) ` W ) /\ f =/= ( _I |` ( Base ` K ) ) ) ) -> D e. DivRing )
20 5 19 rexlimddv 
 |-  ( ( K e. HL /\ W e. H ) -> D e. DivRing )