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Theorem dvaabl 31759
Description: The constructed partial vector space A for a lattice  K is an abelian group. (Contributed by NM, 11-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)
Hypotheses
Ref Expression
dvalvec.h  |-  H  =  ( LHyp `  K
)
dvalvec.v  |-  U  =  ( ( DVecA `  K
) `  W )
Assertion
Ref Expression
dvaabl  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  U  e.  Abel )

Proof of Theorem dvaabl
Dummy variables  f 
s  g are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dvalvec.h . . 3  |-  H  =  ( LHyp `  K
)
2 eqid 2435 . . 3  |-  ( (
LTrn `  K ) `  W )  =  ( ( LTrn `  K
) `  W )
3 eqid 2435 . . 3  |-  ( (
TEndo `  K ) `  W )  =  ( ( TEndo `  K ) `  W )
4 eqid 2435 . . 3  |-  ( (
EDRing `  K ) `  W )  =  ( ( EDRing `  K ) `  W )
5 dvalvec.v . . 3  |-  U  =  ( ( DVecA `  K
) `  W )
61, 2, 3, 4, 5dvaset 31739 . 2  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  U  =  ( {
<. ( Base `  ndx ) ,  ( ( LTrn `  K ) `  W ) >. ,  <. ( +g  `  ndx ) ,  ( f  e.  ( ( LTrn `  K
) `  W ) ,  g  e.  (
( LTrn `  K ) `  W )  |->  ( f  o.  g ) )
>. ,  <. (Scalar `  ndx ) ,  ( (
EDRing `  K ) `  W ) >. }  u.  {
<. ( .s `  ndx ) ,  ( s  e.  ( ( TEndo `  K
) `  W ) ,  f  e.  (
( LTrn `  K ) `  W )  |->  ( s `
 f ) )
>. } ) )
7 eqid 2435 . . . . 5  |-  ( (
TGrp `  K ) `  W )  =  ( ( TGrp `  K
) `  W )
81, 2, 7tgrpset 31479 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( ( TGrp `  K
) `  W )  =  { <. ( Base `  ndx ) ,  ( ( LTrn `  K ) `  W ) >. ,  <. ( +g  `  ndx ) ,  ( f  e.  ( ( LTrn `  K
) `  W ) ,  g  e.  (
( LTrn `  K ) `  W )  |->  ( f  o.  g ) )
>. } )
91, 7tgrpabl 31485 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( ( TGrp `  K
) `  W )  e.  Abel )
108, 9eqeltrrd 2510 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  { <. ( Base `  ndx ) ,  ( ( LTrn `  K ) `  W ) >. ,  <. ( +g  `  ndx ) ,  ( f  e.  ( ( LTrn `  K
) `  W ) ,  g  e.  (
( LTrn `  K ) `  W )  |->  ( f  o.  g ) )
>. }  e.  Abel )
11 fvex 5734 . . . . 5  |-  ( (
LTrn `  K ) `  W )  e.  _V
12 eqid 2435 . . . . . . 7  |-  { <. (
Base `  ndx ) ,  ( ( LTrn `  K
) `  W ) >. ,  <. ( +g  `  ndx ) ,  ( f  e.  ( ( LTrn `  K
) `  W ) ,  g  e.  (
( LTrn `  K ) `  W )  |->  ( f  o.  g ) )
>. }  =  { <. (
Base `  ndx ) ,  ( ( LTrn `  K
) `  W ) >. ,  <. ( +g  `  ndx ) ,  ( f  e.  ( ( LTrn `  K
) `  W ) ,  g  e.  (
( LTrn `  K ) `  W )  |->  ( f  o.  g ) )
>. }
1312grpbase 13561 . . . . . 6  |-  ( ( ( LTrn `  K
) `  W )  e.  _V  ->  ( ( LTrn `  K ) `  W )  =  (
Base `  { <. ( Base `  ndx ) ,  ( ( LTrn `  K
) `  W ) >. ,  <. ( +g  `  ndx ) ,  ( f  e.  ( ( LTrn `  K
) `  W ) ,  g  e.  (
( LTrn `  K ) `  W )  |->  ( f  o.  g ) )
>. } ) )
14 eqid 2435 . . . . . . 7  |-  ( {
<. ( Base `  ndx ) ,  ( ( LTrn `  K ) `  W ) >. ,  <. ( +g  `  ndx ) ,  ( f  e.  ( ( LTrn `  K
) `  W ) ,  g  e.  (
( LTrn `  K ) `  W )  |->  ( f  o.  g ) )
>. ,  <. (Scalar `  ndx ) ,  ( (
EDRing `  K ) `  W ) >. }  u.  {
<. ( .s `  ndx ) ,  ( s  e.  ( ( TEndo `  K
) `  W ) ,  f  e.  (
( LTrn `  K ) `  W )  |->  ( s `
 f ) )
>. } )  =  ( { <. ( Base `  ndx ) ,  ( ( LTrn `  K ) `  W ) >. ,  <. ( +g  `  ndx ) ,  ( f  e.  ( ( LTrn `  K
) `  W ) ,  g  e.  (
( LTrn `  K ) `  W )  |->  ( f  o.  g ) )
>. ,  <. (Scalar `  ndx ) ,  ( (
EDRing `  K ) `  W ) >. }  u.  {
<. ( .s `  ndx ) ,  ( s  e.  ( ( TEndo `  K
) `  W ) ,  f  e.  (
( LTrn `  K ) `  W )  |->  ( s `
 f ) )
>. } )
1514lmodbase 13586 . . . . . 6  |-  ( ( ( LTrn `  K
) `  W )  e.  _V  ->  ( ( LTrn `  K ) `  W )  =  (
Base `  ( { <. ( Base `  ndx ) ,  ( ( LTrn `  K ) `  W ) >. ,  <. ( +g  `  ndx ) ,  ( f  e.  ( ( LTrn `  K
) `  W ) ,  g  e.  (
( LTrn `  K ) `  W )  |->  ( f  o.  g ) )
>. ,  <. (Scalar `  ndx ) ,  ( (
EDRing `  K ) `  W ) >. }  u.  {
<. ( .s `  ndx ) ,  ( s  e.  ( ( TEndo `  K
) `  W ) ,  f  e.  (
( LTrn `  K ) `  W )  |->  ( s `
 f ) )
>. } ) ) )
1613, 15eqtr3d 2469 . . . . 5  |-  ( ( ( LTrn `  K
) `  W )  e.  _V  ->  ( Base `  { <. ( Base `  ndx ) ,  ( ( LTrn `  K ) `  W ) >. ,  <. ( +g  `  ndx ) ,  ( f  e.  ( ( LTrn `  K
) `  W ) ,  g  e.  (
( LTrn `  K ) `  W )  |->  ( f  o.  g ) )
>. } )  =  (
Base `  ( { <. ( Base `  ndx ) ,  ( ( LTrn `  K ) `  W ) >. ,  <. ( +g  `  ndx ) ,  ( f  e.  ( ( LTrn `  K
) `  W ) ,  g  e.  (
( LTrn `  K ) `  W )  |->  ( f  o.  g ) )
>. ,  <. (Scalar `  ndx ) ,  ( (
EDRing `  K ) `  W ) >. }  u.  {
<. ( .s `  ndx ) ,  ( s  e.  ( ( TEndo `  K
) `  W ) ,  f  e.  (
( LTrn `  K ) `  W )  |->  ( s `
 f ) )
>. } ) ) )
1711, 16ax-mp 8 . . . 4  |-  ( Base `  { <. ( Base `  ndx ) ,  ( ( LTrn `  K ) `  W ) >. ,  <. ( +g  `  ndx ) ,  ( f  e.  ( ( LTrn `  K
) `  W ) ,  g  e.  (
( LTrn `  K ) `  W )  |->  ( f  o.  g ) )
>. } )  =  (
Base `  ( { <. ( Base `  ndx ) ,  ( ( LTrn `  K ) `  W ) >. ,  <. ( +g  `  ndx ) ,  ( f  e.  ( ( LTrn `  K
) `  W ) ,  g  e.  (
( LTrn `  K ) `  W )  |->  ( f  o.  g ) )
>. ,  <. (Scalar `  ndx ) ,  ( (
EDRing `  K ) `  W ) >. }  u.  {
<. ( .s `  ndx ) ,  ( s  e.  ( ( TEndo `  K
) `  W ) ,  f  e.  (
( LTrn `  K ) `  W )  |->  ( s `
 f ) )
>. } ) )
1811, 11mpt2ex 6417 . . . . 5  |-  ( f  e.  ( ( LTrn `  K ) `  W
) ,  g  e.  ( ( LTrn `  K
) `  W )  |->  ( f  o.  g
) )  e.  _V
1912grpplusg 13562 . . . . . 6  |-  ( ( f  e.  ( (
LTrn `  K ) `  W ) ,  g  e.  ( ( LTrn `  K ) `  W
)  |->  ( f  o.  g ) )  e. 
_V  ->  ( f  e.  ( ( LTrn `  K
) `  W ) ,  g  e.  (
( LTrn `  K ) `  W )  |->  ( f  o.  g ) )  =  ( +g  `  { <. ( Base `  ndx ) ,  ( ( LTrn `  K ) `  W ) >. ,  <. ( +g  `  ndx ) ,  ( f  e.  ( ( LTrn `  K
) `  W ) ,  g  e.  (
( LTrn `  K ) `  W )  |->  ( f  o.  g ) )
>. } ) )
2014lmodplusg 13587 . . . . . 6  |-  ( ( f  e.  ( (
LTrn `  K ) `  W ) ,  g  e.  ( ( LTrn `  K ) `  W
)  |->  ( f  o.  g ) )  e. 
_V  ->  ( f  e.  ( ( LTrn `  K
) `  W ) ,  g  e.  (
( LTrn `  K ) `  W )  |->  ( f  o.  g ) )  =  ( +g  `  ( { <. ( Base `  ndx ) ,  ( ( LTrn `  K ) `  W ) >. ,  <. ( +g  `  ndx ) ,  ( f  e.  ( ( LTrn `  K
) `  W ) ,  g  e.  (
( LTrn `  K ) `  W )  |->  ( f  o.  g ) )
>. ,  <. (Scalar `  ndx ) ,  ( (
EDRing `  K ) `  W ) >. }  u.  {
<. ( .s `  ndx ) ,  ( s  e.  ( ( TEndo `  K
) `  W ) ,  f  e.  (
( LTrn `  K ) `  W )  |->  ( s `
 f ) )
>. } ) ) )
2119, 20eqtr3d 2469 . . . . 5  |-  ( ( f  e.  ( (
LTrn `  K ) `  W ) ,  g  e.  ( ( LTrn `  K ) `  W
)  |->  ( f  o.  g ) )  e. 
_V  ->  ( +g  `  { <. ( Base `  ndx ) ,  ( ( LTrn `  K ) `  W ) >. ,  <. ( +g  `  ndx ) ,  ( f  e.  ( ( LTrn `  K
) `  W ) ,  g  e.  (
( LTrn `  K ) `  W )  |->  ( f  o.  g ) )
>. } )  =  ( +g  `  ( {
<. ( Base `  ndx ) ,  ( ( LTrn `  K ) `  W ) >. ,  <. ( +g  `  ndx ) ,  ( f  e.  ( ( LTrn `  K
) `  W ) ,  g  e.  (
( LTrn `  K ) `  W )  |->  ( f  o.  g ) )
>. ,  <. (Scalar `  ndx ) ,  ( (
EDRing `  K ) `  W ) >. }  u.  {
<. ( .s `  ndx ) ,  ( s  e.  ( ( TEndo `  K
) `  W ) ,  f  e.  (
( LTrn `  K ) `  W )  |->  ( s `
 f ) )
>. } ) ) )
2218, 21ax-mp 8 . . . 4  |-  ( +g  `  { <. ( Base `  ndx ) ,  ( ( LTrn `  K ) `  W ) >. ,  <. ( +g  `  ndx ) ,  ( f  e.  ( ( LTrn `  K
) `  W ) ,  g  e.  (
( LTrn `  K ) `  W )  |->  ( f  o.  g ) )
>. } )  =  ( +g  `  ( {
<. ( Base `  ndx ) ,  ( ( LTrn `  K ) `  W ) >. ,  <. ( +g  `  ndx ) ,  ( f  e.  ( ( LTrn `  K
) `  W ) ,  g  e.  (
( LTrn `  K ) `  W )  |->  ( f  o.  g ) )
>. ,  <. (Scalar `  ndx ) ,  ( (
EDRing `  K ) `  W ) >. }  u.  {
<. ( .s `  ndx ) ,  ( s  e.  ( ( TEndo `  K
) `  W ) ,  f  e.  (
( LTrn `  K ) `  W )  |->  ( s `
 f ) )
>. } ) )
2317, 22ablprop 15415 . . 3  |-  ( {
<. ( Base `  ndx ) ,  ( ( LTrn `  K ) `  W ) >. ,  <. ( +g  `  ndx ) ,  ( f  e.  ( ( LTrn `  K
) `  W ) ,  g  e.  (
( LTrn `  K ) `  W )  |->  ( f  o.  g ) )
>. }  e.  Abel  <->  ( { <. ( Base `  ndx ) ,  ( ( LTrn `  K ) `  W ) >. ,  <. ( +g  `  ndx ) ,  ( f  e.  ( ( LTrn `  K
) `  W ) ,  g  e.  (
( LTrn `  K ) `  W )  |->  ( f  o.  g ) )
>. ,  <. (Scalar `  ndx ) ,  ( (
EDRing `  K ) `  W ) >. }  u.  {
<. ( .s `  ndx ) ,  ( s  e.  ( ( TEndo `  K
) `  W ) ,  f  e.  (
( LTrn `  K ) `  W )  |->  ( s `
 f ) )
>. } )  e.  Abel )
2410, 23sylib 189 . 2  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( { <. ( Base `  ndx ) ,  ( ( LTrn `  K
) `  W ) >. ,  <. ( +g  `  ndx ) ,  ( f  e.  ( ( LTrn `  K
) `  W ) ,  g  e.  (
( LTrn `  K ) `  W )  |->  ( f  o.  g ) )
>. ,  <. (Scalar `  ndx ) ,  ( (
EDRing `  K ) `  W ) >. }  u.  {
<. ( .s `  ndx ) ,  ( s  e.  ( ( TEndo `  K
) `  W ) ,  f  e.  (
( LTrn `  K ) `  W )  |->  ( s `
 f ) )
>. } )  e.  Abel )
256, 24eqeltrd 2509 1  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  U  e.  Abel )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   _Vcvv 2948    u. cun 3310   {csn 3806   {cpr 3807   {ctp 3808   <.cop 3809    o. ccom 4874   ` cfv 5446    e. cmpt2 6075   ndxcnx 13458   Basecbs 13461   +g cplusg 13521  Scalarcsca 13524   .scvsca 13525   Abelcabel 15405   HLchlt 30085   LHypclh 30718   LTrncltrn 30835   TGrpctgrp 31476   TEndoctendo 31486   EDRingcedring 31487   DVecAcdveca 31736
This theorem is referenced by:  dvalveclem  31760
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 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-undef 6535  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-map 7012  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-2 10050  df-3 10051  df-4 10052  df-5 10053  df-6 10054  df-n0 10214  df-z 10275  df-uz 10481  df-fz 11036  df-struct 13463  df-ndx 13464  df-slot 13465  df-base 13466  df-plusg 13534  df-sca 13537  df-vsca 13538  df-0g 13719  df-poset 14395  df-plt 14407  df-lub 14423  df-glb 14424  df-join 14425  df-meet 14426  df-p0 14460  df-p1 14461  df-lat 14467  df-clat 14529  df-mnd 14682  df-grp 14804  df-cmn 15406  df-abl 15407  df-oposet 29911  df-ol 29913  df-oml 29914  df-covers 30001  df-ats 30002  df-atl 30033  df-cvlat 30057  df-hlat 30086  df-llines 30232  df-lplanes 30233  df-lvols 30234  df-lines 30235  df-psubsp 30237  df-pmap 30238  df-padd 30530  df-lhyp 30722  df-laut 30723  df-ldil 30838  df-ltrn 30839  df-trl 30893  df-tgrp 31477  df-dveca 31737
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