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Theorem erngdv 31790
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 2436 . . 3  |-  ( Base `  K )  =  (
Base `  K )
2 ernggrp.h . . 3  |-  H  =  ( LHyp `  K
)
3 eqid 2436 . . 3  |-  ( (
LTrn `  K ) `  W )  =  ( ( LTrn `  K
) `  W )
41, 2, 3cdlemftr0 31365 . 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 2436 . . 3  |-  ( (
TEndo `  K ) `  W )  =  ( ( TEndo `  K ) `  W )
7 eqid 2436 . . 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 2436 . . 3  |-  ( f  e.  ( ( LTrn `  K ) `  W
)  |->  (  _I  |`  ( Base `  K ) ) )  =  ( f  e.  ( ( LTrn `  K ) `  W
)  |->  (  _I  |`  ( Base `  K ) ) )
9 eqid 2436 . . 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 2436 . . 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 2436 . . 3  |-  ( join `  K )  =  (
join `  K )
12 eqid 2436 . . 3  |-  ( meet `  K )  =  (
meet `  K )
13 eqid 2436 . . 3  |-  ( ( trL `  K ) `
 W )  =  ( ( trL `  K
) `  W )
14 eqid 2436 . . 3  |-  ( ( oc `  K ) `
 W )  =  ( ( oc `  K ) `  W
)
15 eqid 2436 . . 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 2436 . . 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 2436 . . 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 2436 . . 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 31788 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( f  e.  ( ( LTrn `  K
) `  W )  /\  f  =/=  (  _I  |`  ( Base `  K
) ) ) )  ->  D  e.  DivRing )
204, 19rexlimddv 2834 1  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  D  e.  DivRing )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2599   A.wral 2705   ifcif 3739    e. cmpt 4266    _I cid 4493   `'ccnv 4877    |` cres 4880    o. ccom 4882   ` cfv 5454  (class class class)co 6081    e. cmpt2 6083   iota_crio 6542   Basecbs 13469   occoc 13537   joincjn 14401   meetcmee 14402   DivRingcdr 15835   HLchlt 30148   LHypclh 30781   LTrncltrn 30898   trLctrl 30955   TEndoctendo 31549   EDRingcedring 31550
This theorem is referenced by:  erng1r  31792  dvalveclem  31823  dvhvaddass  31895  tendoinvcl  31902  tendolinv  31903  tendorinv  31904  dvhgrp  31905  dvhlveclem  31906  cdlemn4  31996  hlhildrng  32753
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-fal 1329  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-iin 4096  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-tpos 6479  df-undef 6543  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-er 6905  df-map 7020  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-2 10058  df-3 10059  df-n0 10222  df-z 10283  df-uz 10489  df-fz 11044  df-struct 13471  df-ndx 13472  df-slot 13473  df-base 13474  df-sets 13475  df-ress 13476  df-plusg 13542  df-mulr 13543  df-0g 13727  df-poset 14403  df-plt 14415  df-lub 14431  df-glb 14432  df-join 14433  df-meet 14434  df-p0 14468  df-p1 14469  df-lat 14475  df-clat 14537  df-mnd 14690  df-grp 14812  df-minusg 14813  df-mgp 15649  df-rng 15663  df-ur 15665  df-oppr 15728  df-dvdsr 15746  df-unit 15747  df-invr 15777  df-dvr 15788  df-drng 15837  df-oposet 29974  df-ol 29976  df-oml 29977  df-covers 30064  df-ats 30065  df-atl 30096  df-cvlat 30120  df-hlat 30149  df-llines 30295  df-lplanes 30296  df-lvols 30297  df-lines 30298  df-psubsp 30300  df-pmap 30301  df-padd 30593  df-lhyp 30785  df-laut 30786  df-ldil 30901  df-ltrn 30902  df-trl 30956  df-tendo 31552  df-edring 31554
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