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Theorem dvhgrp 31919
Description: The full vector space  U constructed from a Hilbert lattice  K (given a fiducial hyperplane 
W) is a group. (Contributed by NM, 19-Oct-2013.) (Revised by Mario Carneiro, 24-Jun-2014.)
Hypotheses
Ref Expression
dvhgrp.b  |-  B  =  ( Base `  K
)
dvhgrp.h  |-  H  =  ( LHyp `  K
)
dvhgrp.t  |-  T  =  ( ( LTrn `  K
) `  W )
dvhgrp.e  |-  E  =  ( ( TEndo `  K
) `  W )
dvhgrp.u  |-  U  =  ( ( DVecH `  K
) `  W )
dvhgrp.d  |-  D  =  (Scalar `  U )
dvhgrp.p  |-  .+^  =  ( +g  `  D )
dvhgrp.a  |-  .+  =  ( +g  `  U )
dvhgrp.o  |-  .0.  =  ( 0g `  D )
dvhgrp.i  |-  I  =  ( inv g `  D )
Assertion
Ref Expression
dvhgrp  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  U  e.  Grp )

Proof of Theorem dvhgrp
Dummy variables  f 
g  h are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dvhgrp.h . . . 4  |-  H  =  ( LHyp `  K
)
2 dvhgrp.t . . . 4  |-  T  =  ( ( LTrn `  K
) `  W )
3 dvhgrp.e . . . 4  |-  E  =  ( ( TEndo `  K
) `  W )
4 dvhgrp.u . . . 4  |-  U  =  ( ( DVecH `  K
) `  W )
5 eqid 2296 . . . 4  |-  ( Base `  U )  =  (
Base `  U )
61, 2, 3, 4, 5dvhvbase 31899 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( Base `  U
)  =  ( T  X.  E ) )
76eqcomd 2301 . 2  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( T  X.  E
)  =  ( Base `  U ) )
8 dvhgrp.a . . 3  |-  .+  =  ( +g  `  U )
98a1i 10 . 2  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  .+  =  ( +g  `  U ) )
10 dvhgrp.d . . . 4  |-  D  =  (Scalar `  U )
11 dvhgrp.p . . . 4  |-  .+^  =  ( +g  `  D )
121, 2, 3, 4, 10, 11, 8dvhvaddcl 31907 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( f  e.  ( T  X.  E
)  /\  g  e.  ( T  X.  E
) ) )  -> 
( f  .+  g
)  e.  ( T  X.  E ) )
13123impb 1147 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  ( T  X.  E )  /\  g  e.  ( T  X.  E ) )  ->  ( f  .+  g )  e.  ( T  X.  E ) )
141, 2, 3, 4, 10, 11, 8dvhvaddass 31909 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( f  e.  ( T  X.  E
)  /\  g  e.  ( T  X.  E
)  /\  h  e.  ( T  X.  E
) ) )  -> 
( ( f  .+  g )  .+  h
)  =  ( f 
.+  ( g  .+  h ) ) )
15 dvhgrp.b . . . 4  |-  B  =  ( Base `  K
)
1615, 1, 2idltrn 30961 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  (  _I  |`  B )  e.  T )
17 eqid 2296 . . . . . . . 8  |-  ( (
EDRing `  K ) `  W )  =  ( ( EDRing `  K ) `  W )
181, 17, 4, 10dvhsca 31894 . . . . . . 7  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  D  =  ( (
EDRing `  K ) `  W ) )
191, 17erngdv 31804 . . . . . . 7  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( ( EDRing `  K
) `  W )  e.  DivRing )
2018, 19eqeltrd 2370 . . . . . 6  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  D  e.  DivRing )
21 drnggrp 15536 . . . . . 6  |-  ( D  e.  DivRing  ->  D  e.  Grp )
2220, 21syl 15 . . . . 5  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  D  e.  Grp )
23 eqid 2296 . . . . . 6  |-  ( Base `  D )  =  (
Base `  D )
24 dvhgrp.o . . . . . 6  |-  .0.  =  ( 0g `  D )
2523, 24grpidcl 14526 . . . . 5  |-  ( D  e.  Grp  ->  .0.  e.  ( Base `  D
) )
2622, 25syl 15 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  .0.  e.  ( Base `  D ) )
271, 3, 4, 10, 23dvhbase 31895 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( Base `  D
)  =  E )
2826, 27eleqtrd 2372 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  .0.  e.  E )
29 opelxpi 4737 . . 3  |-  ( ( (  _I  |`  B )  e.  T  /\  .0.  e.  E )  ->  <. (  _I  |`  B ) ,  .0.  >.  e.  ( T  X.  E ) )
3016, 28, 29syl2anc 642 . 2  |-  ( ( K  e.  HL  /\  W  e.  H )  -> 
<. (  _I  |`  B ) ,  .0.  >.  e.  ( T  X.  E ) )
31 simpl 443 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  ( T  X.  E ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
3216adantr 451 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  ( T  X.  E ) )  ->  (  _I  |`  B )  e.  T
)
3328adantr 451 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  ( T  X.  E ) )  ->  .0.  e.  E )
34 xp1st 6165 . . . . . 6  |-  ( f  e.  ( T  X.  E )  ->  ( 1st `  f )  e.  T )
3534adantl 452 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  ( T  X.  E ) )  ->  ( 1st `  f )  e.  T
)
36 xp2nd 6166 . . . . . 6  |-  ( f  e.  ( T  X.  E )  ->  ( 2nd `  f )  e.  E )
3736adantl 452 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  ( T  X.  E ) )  ->  ( 2nd `  f )  e.  E
)
381, 2, 3, 4, 10, 8, 11dvhopvadd 31905 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( (  _I  |`  B )  e.  T  /\  .0.  e.  E )  /\  ( ( 1st `  f )  e.  T  /\  ( 2nd `  f
)  e.  E ) )  ->  ( <. (  _I  |`  B ) ,  .0.  >.  .+  <. ( 1st `  f ) ,  ( 2nd `  f )
>. )  =  <. ( (  _I  |`  B )  o.  ( 1st `  f
) ) ,  (  .0.  .+^  ( 2nd `  f
) ) >. )
3931, 32, 33, 35, 37, 38syl122anc 1191 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  ( T  X.  E ) )  ->  ( <. (  _I  |`  B ) ,  .0.  >.  .+  <. ( 1st `  f ) ,  ( 2nd `  f )
>. )  =  <. ( (  _I  |`  B )  o.  ( 1st `  f
) ) ,  (  .0.  .+^  ( 2nd `  f
) ) >. )
4015, 1, 2ltrn1o 30935 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( 1st `  f
)  e.  T )  ->  ( 1st `  f
) : B -1-1-onto-> B )
4135, 40syldan 456 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  ( T  X.  E ) )  ->  ( 1st `  f ) : B -1-1-onto-> B
)
42 f1of 5488 . . . . . 6  |-  ( ( 1st `  f ) : B -1-1-onto-> B  ->  ( 1st `  f ) : B --> B )
43 fcoi2 5432 . . . . . 6  |-  ( ( 1st `  f ) : B --> B  -> 
( (  _I  |`  B )  o.  ( 1st `  f
) )  =  ( 1st `  f ) )
4441, 42, 433syl 18 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  ( T  X.  E ) )  ->  ( (  _I  |`  B )  o.  ( 1st `  f
) )  =  ( 1st `  f ) )
4522adantr 451 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  ( T  X.  E ) )  ->  D  e.  Grp )
4627adantr 451 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  ( T  X.  E ) )  ->  ( Base `  D )  =  E )
4737, 46eleqtrrd 2373 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  ( T  X.  E ) )  ->  ( 2nd `  f )  e.  (
Base `  D )
)
4823, 11, 24grplid 14528 . . . . . 6  |-  ( ( D  e.  Grp  /\  ( 2nd `  f )  e.  ( Base `  D
) )  ->  (  .0.  .+^  ( 2nd `  f
) )  =  ( 2nd `  f ) )
4945, 47, 48syl2anc 642 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  ( T  X.  E ) )  ->  (  .0.  .+^  ( 2nd `  f
) )  =  ( 2nd `  f ) )
5044, 49opeq12d 3820 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  ( T  X.  E ) )  ->  <. ( (  _I  |`  B )  o.  ( 1st `  f
) ) ,  (  .0.  .+^  ( 2nd `  f
) ) >.  =  <. ( 1st `  f ) ,  ( 2nd `  f
) >. )
5139, 50eqtrd 2328 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  ( T  X.  E ) )  ->  ( <. (  _I  |`  B ) ,  .0.  >.  .+  <. ( 1st `  f ) ,  ( 2nd `  f )
>. )  =  <. ( 1st `  f ) ,  ( 2nd `  f
) >. )
52 1st2nd2 6175 . . . . 5  |-  ( f  e.  ( T  X.  E )  ->  f  =  <. ( 1st `  f
) ,  ( 2nd `  f ) >. )
5352adantl 452 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  ( T  X.  E ) )  ->  f  =  <. ( 1st `  f
) ,  ( 2nd `  f ) >. )
5453oveq2d 5890 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  ( T  X.  E ) )  ->  ( <. (  _I  |`  B ) ,  .0.  >.  .+  f )  =  ( <. (  _I  |`  B ) ,  .0.  >.  .+  <. ( 1st `  f ) ,  ( 2nd `  f )
>. ) )
5551, 54, 533eqtr4d 2338 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  ( T  X.  E ) )  ->  ( <. (  _I  |`  B ) ,  .0.  >.  .+  f )  =  f )
561, 2ltrncnv 30957 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( 1st `  f
)  e.  T )  ->  `' ( 1st `  f )  e.  T
)
5735, 56syldan 456 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  ( T  X.  E ) )  ->  `' ( 1st `  f )  e.  T )
58 dvhgrp.i . . . . . 6  |-  I  =  ( inv g `  D )
5923, 58grpinvcl 14543 . . . . 5  |-  ( ( D  e.  Grp  /\  ( 2nd `  f )  e.  ( Base `  D
) )  ->  (
I `  ( 2nd `  f ) )  e.  ( Base `  D
) )
6045, 47, 59syl2anc 642 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  ( T  X.  E ) )  ->  ( I `  ( 2nd `  f
) )  e.  (
Base `  D )
)
6160, 46eleqtrd 2372 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  ( T  X.  E ) )  ->  ( I `  ( 2nd `  f
) )  e.  E
)
62 opelxpi 4737 . . 3  |-  ( ( `' ( 1st `  f
)  e.  T  /\  ( I `  ( 2nd `  f ) )  e.  E )  ->  <. `' ( 1st `  f
) ,  ( I `
 ( 2nd `  f
) ) >.  e.  ( T  X.  E ) )
6357, 61, 62syl2anc 642 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  ( T  X.  E ) )  ->  <. `' ( 1st `  f ) ,  ( I `  ( 2nd `  f ) ) >.  e.  ( T  X.  E ) )
6453oveq2d 5890 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  ( T  X.  E ) )  ->  ( <. `' ( 1st `  f
) ,  ( I `
 ( 2nd `  f
) ) >.  .+  f
)  =  ( <. `' ( 1st `  f
) ,  ( I `
 ( 2nd `  f
) ) >.  .+  <. ( 1st `  f ) ,  ( 2nd `  f
) >. ) )
651, 2, 3, 4, 10, 8, 11dvhopvadd 31905 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( `' ( 1st `  f )  e.  T  /\  (
I `  ( 2nd `  f ) )  e.  E )  /\  (
( 1st `  f
)  e.  T  /\  ( 2nd `  f )  e.  E ) )  ->  ( <. `' ( 1st `  f ) ,  ( I `  ( 2nd `  f ) ) >.  .+  <. ( 1st `  f ) ,  ( 2nd `  f
) >. )  =  <. ( `' ( 1st `  f
)  o.  ( 1st `  f ) ) ,  ( ( I `  ( 2nd `  f ) )  .+^  ( 2nd `  f ) ) >.
)
6631, 57, 61, 35, 37, 65syl122anc 1191 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  ( T  X.  E ) )  ->  ( <. `' ( 1st `  f
) ,  ( I `
 ( 2nd `  f
) ) >.  .+  <. ( 1st `  f ) ,  ( 2nd `  f
) >. )  =  <. ( `' ( 1st `  f
)  o.  ( 1st `  f ) ) ,  ( ( I `  ( 2nd `  f ) )  .+^  ( 2nd `  f ) ) >.
)
67 f1ococnv1 5518 . . . . . 6  |-  ( ( 1st `  f ) : B -1-1-onto-> B  ->  ( `' ( 1st `  f )  o.  ( 1st `  f
) )  =  (  _I  |`  B )
)
6841, 67syl 15 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  ( T  X.  E ) )  ->  ( `' ( 1st `  f )  o.  ( 1st `  f
) )  =  (  _I  |`  B )
)
6923, 11, 24, 58grplinv 14544 . . . . . 6  |-  ( ( D  e.  Grp  /\  ( 2nd `  f )  e.  ( Base `  D
) )  ->  (
( I `  ( 2nd `  f ) ) 
.+^  ( 2nd `  f
) )  =  .0.  )
7045, 47, 69syl2anc 642 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  ( T  X.  E ) )  ->  ( (
I `  ( 2nd `  f ) )  .+^  ( 2nd `  f ) )  =  .0.  )
7168, 70opeq12d 3820 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  ( T  X.  E ) )  ->  <. ( `' ( 1st `  f
)  o.  ( 1st `  f ) ) ,  ( ( I `  ( 2nd `  f ) )  .+^  ( 2nd `  f ) ) >.  =  <. (  _I  |`  B ) ,  .0.  >. )
7266, 71eqtrd 2328 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  ( T  X.  E ) )  ->  ( <. `' ( 1st `  f
) ,  ( I `
 ( 2nd `  f
) ) >.  .+  <. ( 1st `  f ) ,  ( 2nd `  f
) >. )  =  <. (  _I  |`  B ) ,  .0.  >. )
7364, 72eqtrd 2328 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  ( T  X.  E ) )  ->  ( <. `' ( 1st `  f
) ,  ( I `
 ( 2nd `  f
) ) >.  .+  f
)  =  <. (  _I  |`  B ) ,  .0.  >. )
747, 9, 13, 14, 30, 55, 63, 73isgrpd 14523 1  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  U  e.  Grp )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   <.cop 3656    _I cid 4320    X. cxp 4703   `'ccnv 4704    |` cres 4707    o. ccom 4709   -->wf 5267   -1-1-onto->wf1o 5270   ` cfv 5271  (class class class)co 5874   1stc1st 6136   2ndc2nd 6137   Basecbs 13164   +g cplusg 13224  Scalarcsca 13227   0gc0g 13416   Grpcgrp 14378   inv gcminusg 14379   DivRingcdr 15528   HLchlt 30162   LHypclh 30795   LTrncltrn 30912   TEndoctendo 31563   EDRingcedring 31564   DVecHcdvh 31890
This theorem is referenced by:  dvhlveclem  31920
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-fal 1311  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-tpos 6250  df-undef 6314  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-n0 9982  df-z 10041  df-uz 10247  df-fz 10799  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-sca 13240  df-vsca 13241  df-0g 13420  df-poset 14096  df-plt 14108  df-lub 14124  df-glb 14125  df-join 14126  df-meet 14127  df-p0 14161  df-p1 14162  df-lat 14168  df-clat 14230  df-mnd 14383  df-grp 14505  df-minusg 14506  df-mgp 15342  df-rng 15356  df-ur 15358  df-oppr 15421  df-dvdsr 15439  df-unit 15440  df-invr 15470  df-dvr 15481  df-drng 15530  df-oposet 29988  df-ol 29990  df-oml 29991  df-covers 30078  df-ats 30079  df-atl 30110  df-cvlat 30134  df-hlat 30163  df-llines 30309  df-lplanes 30310  df-lvols 30311  df-lines 30312  df-psubsp 30314  df-pmap 30315  df-padd 30607  df-lhyp 30799  df-laut 30800  df-ldil 30915  df-ltrn 30916  df-trl 30970  df-tendo 31566  df-edring 31568  df-dvech 31891
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