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Theorem dvaabl 34044
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 2402 . . 3  |-  ( (
LTrn `  K ) `  W )  =  ( ( LTrn `  K
) `  W )
3 eqid 2402 . . 3  |-  ( (
TEndo `  K ) `  W )  =  ( ( TEndo `  K ) `  W )
4 eqid 2402 . . 3  |-  ( (
EDRing `  K ) `  W )  =  ( ( EDRing `  K ) `  W )
5 dvalvec.v . . 3  |-  U  =  ( ( DVecA `  K
) `  W )
61, 2, 3, 4, 5dvaset 34024 . 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 2402 . . . . 5  |-  ( (
TGrp `  K ) `  W )  =  ( ( TGrp `  K
) `  W )
81, 2, 7tgrpset 33764 . . . 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 33770 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( ( TGrp `  K
) `  W )  e.  Abel )
108, 9eqeltrrd 2491 . . 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 5859 . . . . 5  |-  ( (
LTrn `  K ) `  W )  e.  _V
12 eqid 2402 . . . . . . 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 14953 . . . . . 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 2402 . . . . . . 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 14978 . . . . . 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 2445 . . . . 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 5 . . . 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 6861 . . . . 5  |-  ( f  e.  ( ( LTrn `  K ) `  W
) ,  g  e.  ( ( LTrn `  K
) `  W )  |->  ( f  o.  g
) )  e.  _V
1912grpplusg 14954 . . . . . 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 14979 . . . . . 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 2445 . . . . 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 5 . . . 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 17133 . . 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 196 . 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 2490 1  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  U  e.  Abel )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1405    e. wcel 1842   _Vcvv 3059    u. cun 3412   {csn 3972   {cpr 3974   {ctp 3976   <.cop 3978    o. ccom 4827   ` cfv 5569    |-> cmpt2 6280   ndxcnx 14838   Basecbs 14841   +g cplusg 14909  Scalarcsca 14912   .scvsca 14913   Abelcabl 17123   HLchlt 32368   LHypclh 33001   LTrncltrn 33118   TGrpctgrp 33761   TEndoctendo 33771   EDRingcedring 33772   DVecAcdveca 34021
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599  ax-riotaBAD 31977
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-int 4228  df-iun 4273  df-iin 4274  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-om 6684  df-1st 6784  df-2nd 6785  df-undef 7005  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-1o 7167  df-oadd 7171  df-er 7348  df-map 7459  df-en 7555  df-dom 7556  df-sdom 7557  df-fin 7558  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-nn 10577  df-2 10635  df-3 10636  df-4 10637  df-5 10638  df-6 10639  df-n0 10837  df-z 10906  df-uz 11128  df-fz 11727  df-struct 14843  df-ndx 14844  df-slot 14845  df-base 14846  df-plusg 14922  df-sca 14925  df-vsca 14926  df-0g 15056  df-preset 15881  df-poset 15899  df-plt 15912  df-lub 15928  df-glb 15929  df-join 15930  df-meet 15931  df-p0 15993  df-p1 15994  df-lat 16000  df-clat 16062  df-mgm 16196  df-sgrp 16235  df-mnd 16245  df-grp 16381  df-cmn 17124  df-abl 17125  df-oposet 32194  df-ol 32196  df-oml 32197  df-covers 32284  df-ats 32285  df-atl 32316  df-cvlat 32340  df-hlat 32369  df-llines 32515  df-lplanes 32516  df-lvols 32517  df-lines 32518  df-psubsp 32520  df-pmap 32521  df-padd 32813  df-lhyp 33005  df-laut 33006  df-ldil 33121  df-ltrn 33122  df-trl 33177  df-tgrp 33762  df-dveca 34022
This theorem is referenced by:  dvalveclem  34045
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