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Theorem erngdv-rN 31872
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 2438 . . 3  |-  ( Base `  K )  =  (
Base `  K )
2 ernggrp.h-r . . 3  |-  H  =  ( LHyp `  K
)
3 eqid 2438 . . 3  |-  ( (
LTrn `  K ) `  W )  =  ( ( LTrn `  K
) `  W )
41, 2, 3cdlemftr0 31439 . 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 2438 . . 3  |-  ( (
TEndo `  K ) `  W )  =  ( ( TEndo `  K ) `  W )
7 eqid 2438 . . 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 2438 . . 3  |-  ( f  e.  ( ( LTrn `  K ) `  W
)  |->  (  _I  |`  ( Base `  K ) ) )  =  ( f  e.  ( ( LTrn `  K ) `  W
)  |->  (  _I  |`  ( Base `  K ) ) )
9 eqid 2438 . . 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 2438 . . 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 2438 . . 3  |-  ( join `  K )  =  (
join `  K )
12 eqid 2438 . . 3  |-  ( meet `  K )  =  (
meet `  K )
13 eqid 2438 . . 3  |-  ( ( trL `  K ) `
 W )  =  ( ( trL `  K
) `  W )
14 eqid 2438 . . 3  |-  ( ( oc `  K ) `
 W )  =  ( ( oc `  K ) `  W
)
15 eqid 2438 . . 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 2438 . . 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 2438 . . 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 2438 . . 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 31870 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( f  e.  ( ( LTrn `  K
) `  W )  /\  f  =/=  (  _I  |`  ( Base `  K
) ) ) )  ->  D  e.  DivRing )
204, 19rexlimddv 2836 1  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  D  e.  DivRing )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601   A.wral 2707   ifcif 3741    e. cmpt 4269    _I cid 4496   `'ccnv 4880    |` cres 4883    o. ccom 4885   ` cfv 5457  (class class class)co 6084    e. cmpt2 6086   iota_crio 6545   Basecbs 13474   occoc 13542   joincjn 14406   meetcmee 14407   DivRingcdr 15840   HLchlt 30222   LHypclh 30855   LTrncltrn 30972   trLctrl 31029   TEndoctendo 31623   EDRing Rcedring-rN 31625
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-fal 1330  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-iin 4098  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-tpos 6482  df-undef 6546  df-riota 6552  df-recs 6636  df-rdg 6671  df-1o 6727  df-oadd 6731  df-er 6908  df-map 7023  df-en 7113  df-dom 7114  df-sdom 7115  df-fin 7116  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-nn 10006  df-2 10063  df-3 10064  df-n0 10227  df-z 10288  df-uz 10494  df-fz 11049  df-struct 13476  df-ndx 13477  df-slot 13478  df-base 13479  df-sets 13480  df-ress 13481  df-plusg 13547  df-mulr 13548  df-0g 13732  df-poset 14408  df-plt 14420  df-lub 14436  df-glb 14437  df-join 14438  df-meet 14439  df-p0 14473  df-p1 14474  df-lat 14480  df-clat 14542  df-mnd 14695  df-grp 14817  df-minusg 14818  df-mgp 15654  df-rng 15668  df-ur 15670  df-oppr 15733  df-dvdsr 15751  df-unit 15752  df-invr 15782  df-dvr 15793  df-drng 15842  df-oposet 30048  df-ol 30050  df-oml 30051  df-covers 30138  df-ats 30139  df-atl 30170  df-cvlat 30194  df-hlat 30223  df-llines 30369  df-lplanes 30370  df-lvols 30371  df-lines 30372  df-psubsp 30374  df-pmap 30375  df-padd 30667  df-lhyp 30859  df-laut 30860  df-ldil 30975  df-ltrn 30976  df-trl 31030  df-tendo 31626  df-edring-rN 31627
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