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Theorem erngdv-rN 31495
Description: An endomorphism ring is a division ring. Todo: fix comment. (Contributed by NM, 11-Aug-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
ernggrp.h-r  |-  H  =  ( LHyp `  K
)
ernggrp.d-r  |-  D  =  ( ( EDRing R `  K ) `  W
)
Assertion
Ref Expression
erngdv-rN  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  D  e.  DivRing )

Proof of Theorem erngdv-rN
Dummy variables  f 
s  a  b  g  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2412 . . 3  |-  ( Base `  K )  =  (
Base `  K )
2 ernggrp.h-r . . 3  |-  H  =  ( LHyp `  K
)
3 eqid 2412 . . 3  |-  ( (
LTrn `  K ) `  W )  =  ( ( LTrn `  K
) `  W )
41, 2, 3cdlemftr0 31062 . 2  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. f  e.  ( ( LTrn `  K
) `  W )
f  =/=  (  _I  |`  ( Base `  K
) ) )
5 ernggrp.d-r . . 3  |-  D  =  ( ( EDRing R `  K ) `  W
)
6 eqid 2412 . . 3  |-  ( (
TEndo `  K ) `  W )  =  ( ( TEndo `  K ) `  W )
7 eqid 2412 . . 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 2412 . . 3  |-  ( f  e.  ( ( LTrn `  K ) `  W
)  |->  (  _I  |`  ( Base `  K ) ) )  =  ( f  e.  ( ( LTrn `  K ) `  W
)  |->  (  _I  |`  ( Base `  K ) ) )
9 eqid 2412 . . 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 2412 . . 3  |-  ( a  e.  ( ( TEndo `  K ) `  W
) ,  b  e.  ( ( TEndo `  K
) `  W )  |->  ( b  o.  a
) )  =  ( a  e.  ( (
TEndo `  K ) `  W ) ,  b  e.  ( ( TEndo `  K ) `  W
)  |->  ( b  o.  a ) )
11 eqid 2412 . . 3  |-  ( join `  K )  =  (
join `  K )
12 eqid 2412 . . 3  |-  ( meet `  K )  =  (
meet `  K )
13 eqid 2412 . . 3  |-  ( ( trL `  K ) `
 W )  =  ( ( trL `  K
) `  W )
14 eqid 2412 . . 3  |-  ( ( oc `  K ) `
 W )  =  ( ( oc `  K ) `  W
)
15 eqid 2412 . . 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 2412 . . 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 2412 . . 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 2412 . . 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-rN 31493 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( f  e.  ( ( LTrn `  K
) `  W )  /\  f  =/=  (  _I  |`  ( Base `  K
) ) ) )  ->  D  e.  DivRing )
204, 19rexlimddv 2802 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 1649    e. wcel 1721    =/= wne 2575   A.wral 2674   ifcif 3707    e. cmpt 4234    _I cid 4461   `'ccnv 4844    |` cres 4847    o. ccom 4849   ` cfv 5421  (class class class)co 6048    e. cmpt2 6050   iota_crio 6509   Basecbs 13432   occoc 13500   joincjn 14364   meetcmee 14365   DivRingcdr 15798   HLchlt 29845   LHypclh 30478   LTrncltrn 30595   trLctrl 30652   TEndoctendo 31246   EDRing Rcedring-rN 31248
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-fal 1326  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-int 4019  df-iun 4063  df-iin 4064  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-tpos 6446  df-undef 6510  df-riota 6516  df-recs 6600  df-rdg 6635  df-1o 6691  df-oadd 6695  df-er 6872  df-map 6987  df-en 7077  df-dom 7078  df-sdom 7079  df-fin 7080  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-nn 9965  df-2 10022  df-3 10023  df-n0 10186  df-z 10247  df-uz 10453  df-fz 11008  df-struct 13434  df-ndx 13435  df-slot 13436  df-base 13437  df-sets 13438  df-ress 13439  df-plusg 13505  df-mulr 13506  df-0g 13690  df-poset 14366  df-plt 14378  df-lub 14394  df-glb 14395  df-join 14396  df-meet 14397  df-p0 14431  df-p1 14432  df-lat 14438  df-clat 14500  df-mnd 14653  df-grp 14775  df-minusg 14776  df-mgp 15612  df-rng 15626  df-ur 15628  df-oppr 15691  df-dvdsr 15709  df-unit 15710  df-invr 15740  df-dvr 15751  df-drng 15800  df-oposet 29671  df-ol 29673  df-oml 29674  df-covers 29761  df-ats 29762  df-atl 29793  df-cvlat 29817  df-hlat 29846  df-llines 29992  df-lplanes 29993  df-lvols 29994  df-lines 29995  df-psubsp 29997  df-pmap 29998  df-padd 30290  df-lhyp 30482  df-laut 30483  df-ldil 30598  df-ltrn 30599  df-trl 30653  df-tendo 31249  df-edring-rN 31250
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