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Theorem erngdv 36862
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 2457 . . 3  |-  ( Base `  K )  =  (
Base `  K )
2 ernggrp.h . . 3  |-  H  =  ( LHyp `  K
)
3 eqid 2457 . . 3  |-  ( (
LTrn `  K ) `  W )  =  ( ( LTrn `  K
) `  W )
41, 2, 3cdlemftr0 36437 . 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 2457 . . 3  |-  ( (
TEndo `  K ) `  W )  =  ( ( TEndo `  K ) `  W )
7 eqid 2457 . . 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 2457 . . 3  |-  ( f  e.  ( ( LTrn `  K ) `  W
)  |->  (  _I  |`  ( Base `  K ) ) )  =  ( f  e.  ( ( LTrn `  K ) `  W
)  |->  (  _I  |`  ( Base `  K ) ) )
9 eqid 2457 . . 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 2457 . . 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 2457 . . 3  |-  ( join `  K )  =  (
join `  K )
12 eqid 2457 . . 3  |-  ( meet `  K )  =  (
meet `  K )
13 eqid 2457 . . 3  |-  ( ( trL `  K ) `
 W )  =  ( ( trL `  K
) `  W )
14 eqid 2457 . . 3  |-  ( ( oc `  K ) `
 W )  =  ( ( oc `  K ) `  W
)
15 eqid 2457 . . 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 2457 . . 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 2457 . . 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 2457 . . 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 36860 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( f  e.  ( ( LTrn `  K
) `  W )  /\  f  =/=  (  _I  |`  ( Base `  K
) ) ) )  ->  D  e.  DivRing )
204, 19rexlimddv 2953 1  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  D  e.  DivRing )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819    =/= wne 2652   A.wral 2807   ifcif 3944    |-> cmpt 4515    _I cid 4799   `'ccnv 5007    |` cres 5010    o. ccom 5012   ` cfv 5594   iota_crio 6257  (class class class)co 6296    |-> cmpt2 6298   Basecbs 14644   occoc 14720   joincjn 15700   meetcmee 15701   DivRingcdr 17523   HLchlt 35218   LHypclh 35851   LTrncltrn 35968   trLctrl 36026   TEndoctendo 36621   EDRingcedring 36622
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-riotaBAD 34827
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-iin 4335  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-tpos 6973  df-undef 7020  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-map 7440  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-3 10616  df-n0 10817  df-z 10886  df-uz 11107  df-fz 11698  df-struct 14646  df-ndx 14647  df-slot 14648  df-base 14649  df-sets 14650  df-ress 14651  df-plusg 14725  df-mulr 14726  df-0g 14859  df-preset 15684  df-poset 15702  df-plt 15715  df-lub 15731  df-glb 15732  df-join 15733  df-meet 15734  df-p0 15796  df-p1 15797  df-lat 15803  df-clat 15865  df-mgm 15999  df-sgrp 16038  df-mnd 16048  df-grp 16184  df-minusg 16185  df-mgp 17269  df-ur 17281  df-ring 17327  df-oppr 17399  df-dvdsr 17417  df-unit 17418  df-invr 17448  df-dvr 17459  df-drng 17525  df-oposet 35044  df-ol 35046  df-oml 35047  df-covers 35134  df-ats 35135  df-atl 35166  df-cvlat 35190  df-hlat 35219  df-llines 35365  df-lplanes 35366  df-lvols 35367  df-lines 35368  df-psubsp 35370  df-pmap 35371  df-padd 35663  df-lhyp 35855  df-laut 35856  df-ldil 35971  df-ltrn 35972  df-trl 36027  df-tendo 36624  df-edring 36626
This theorem is referenced by:  erng1r  36864  dvalveclem  36895  dvhvaddass  36967  tendoinvcl  36974  tendolinv  36975  tendorinv  36976  dvhgrp  36977  dvhlveclem  36978  cdlemn4  37068  hlhildrng  37825
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