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Theorem erngdv 34649
Description: An endomorphism ring is a division ring. Todo: fix comment. (Contributed by NM, 11-Aug-2013.)
Hypotheses
Ref Expression
ernggrp.h  |-  H  =  ( LHyp `  K
)
ernggrp.d  |-  D  =  ( ( EDRing `  K
) `  W )
Assertion
Ref Expression
erngdv  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  D  e.  DivRing )

Proof of Theorem erngdv
Dummy variables  f 
s  a  b  g  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2443 . . 3  |-  ( Base `  K )  =  (
Base `  K )
2 ernggrp.h . . 3  |-  H  =  ( LHyp `  K
)
3 eqid 2443 . . 3  |-  ( (
LTrn `  K ) `  W )  =  ( ( LTrn `  K
) `  W )
41, 2, 3cdlemftr0 34224 . 2  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. f  e.  ( ( LTrn `  K
) `  W )
f  =/=  (  _I  |`  ( Base `  K
) ) )
5 ernggrp.d . . 3  |-  D  =  ( ( EDRing `  K
) `  W )
6 eqid 2443 . . 3  |-  ( (
TEndo `  K ) `  W )  =  ( ( TEndo `  K ) `  W )
7 eqid 2443 . . 3  |-  ( 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 2443 . . 3  |-  ( f  e.  ( ( LTrn `  K ) `  W
)  |->  (  _I  |`  ( Base `  K ) ) )  =  ( f  e.  ( ( LTrn `  K ) `  W
)  |->  (  _I  |`  ( Base `  K ) ) )
9 eqid 2443 . . 3  |-  ( 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 2443 . . 3  |-  ( 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 2443 . . 3  |-  ( join `  K )  =  (
join `  K )
12 eqid 2443 . . 3  |-  ( meet `  K )  =  (
meet `  K )
13 eqid 2443 . . 3  |-  ( ( trL `  K ) `
 W )  =  ( ( trL `  K
) `  W )
14 eqid 2443 . . 3  |-  ( ( oc `  K ) `
 W )  =  ( ( oc `  K ) `  W
)
15 eqid 2443 . . 3  |-  ( ( ( ( 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 2443 . . 3  |-  ( ( ( ( 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 2443 . . 3  |-  ( 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 2443 . . 3  |-  ( 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 ) ) ) ) ) ) ) )
192, 5, 1, 3, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18erngdvlem4 34647 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( f  e.  ( ( LTrn `  K
) `  W )  /\  f  =/=  (  _I  |`  ( Base `  K
) ) ) )  ->  D  e.  DivRing )
204, 19rexlimddv 2857 1  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  D  e.  DivRing )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2618   A.wral 2727   ifcif 3803    e. cmpt 4362    _I cid 4643   `'ccnv 4851    |` cres 4854    o. ccom 4856   ` cfv 5430   iota_crio 6063  (class class class)co 6103    e. cmpt2 6105   Basecbs 14186   occoc 14258   joincjn 15126   meetcmee 15127   DivRingcdr 16844   HLchlt 33007   LHypclh 33640   LTrncltrn 33757   trLctrl 33814   TEndoctendo 34408   EDRingcedring 34409
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4415  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384  ax-cnex 9350  ax-resscn 9351  ax-1cn 9352  ax-icn 9353  ax-addcl 9354  ax-addrcl 9355  ax-mulcl 9356  ax-mulrcl 9357  ax-mulcom 9358  ax-addass 9359  ax-mulass 9360  ax-distr 9361  ax-i2m1 9362  ax-1ne0 9363  ax-1rid 9364  ax-rnegex 9365  ax-rrecex 9366  ax-cnre 9367  ax-pre-lttri 9368  ax-pre-lttrn 9369  ax-pre-ltadd 9370  ax-pre-mulgt0 9371  ax-riotaBAD 32616
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-nel 2621  df-ral 2732  df-rex 2733  df-reu 2734  df-rmo 2735  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-pss 3356  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-tp 3894  df-op 3896  df-uni 4104  df-int 4141  df-iun 4185  df-iin 4186  df-br 4305  df-opab 4363  df-mpt 4364  df-tr 4398  df-eprel 4644  df-id 4648  df-po 4653  df-so 4654  df-fr 4691  df-we 4693  df-ord 4734  df-on 4735  df-lim 4736  df-suc 4737  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-riota 6064  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-om 6489  df-1st 6589  df-2nd 6590  df-tpos 6757  df-undef 6804  df-recs 6844  df-rdg 6878  df-1o 6932  df-oadd 6936  df-er 7113  df-map 7228  df-en 7323  df-dom 7324  df-sdom 7325  df-fin 7326  df-pnf 9432  df-mnf 9433  df-xr 9434  df-ltxr 9435  df-le 9436  df-sub 9609  df-neg 9610  df-nn 10335  df-2 10392  df-3 10393  df-n0 10592  df-z 10659  df-uz 10874  df-fz 11450  df-struct 14188  df-ndx 14189  df-slot 14190  df-base 14191  df-sets 14192  df-ress 14193  df-plusg 14263  df-mulr 14264  df-0g 14392  df-poset 15128  df-plt 15140  df-lub 15156  df-glb 15157  df-join 15158  df-meet 15159  df-p0 15221  df-p1 15222  df-lat 15228  df-clat 15290  df-mnd 15427  df-grp 15557  df-minusg 15558  df-mgp 16604  df-ur 16616  df-rng 16659  df-oppr 16727  df-dvdsr 16745  df-unit 16746  df-invr 16776  df-dvr 16787  df-drng 16846  df-oposet 32833  df-ol 32835  df-oml 32836  df-covers 32923  df-ats 32924  df-atl 32955  df-cvlat 32979  df-hlat 33008  df-llines 33154  df-lplanes 33155  df-lvols 33156  df-lines 33157  df-psubsp 33159  df-pmap 33160  df-padd 33452  df-lhyp 33644  df-laut 33645  df-ldil 33760  df-ltrn 33761  df-trl 33815  df-tendo 34411  df-edring 34413
This theorem is referenced by:  erng1r  34651  dvalveclem  34682  dvhvaddass  34754  tendoinvcl  34761  tendolinv  34762  tendorinv  34763  dvhgrp  34764  dvhlveclem  34765  cdlemn4  34855  hlhildrng  35612
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