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Theorem dvhgrp 34474
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  =  ( invg `  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 2441 . . . 4  |-  ( Base `  U )  =  (
Base `  U )
61, 2, 3, 4, 5dvhvbase 34454 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( Base `  U
)  =  ( T  X.  E ) )
76eqcomd 2446 . 2  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( T  X.  E
)  =  ( Base `  U ) )
8 dvhgrp.a . . 3  |-  .+  =  ( +g  `  U )
98a1i 11 . 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 34462 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( f  e.  ( T  X.  E
)  /\  g  e.  ( T  X.  E
) ) )  -> 
( f  .+  g
)  e.  ( T  X.  E ) )
13123impb 1178 . 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 34464 . 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 33516 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  (  _I  |`  B )  e.  T )
17 eqid 2441 . . . . . . . 8  |-  ( (
EDRing `  K ) `  W )  =  ( ( EDRing `  K ) `  W )
181, 17, 4, 10dvhsca 34449 . . . . . . 7  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  D  =  ( (
EDRing `  K ) `  W ) )
191, 17erngdv 34359 . . . . . . 7  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( ( EDRing `  K
) `  W )  e.  DivRing )
2018, 19eqeltrd 2515 . . . . . 6  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  D  e.  DivRing )
21 drnggrp 16820 . . . . . 6  |-  ( D  e.  DivRing  ->  D  e.  Grp )
2220, 21syl 16 . . . . 5  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  D  e.  Grp )
23 eqid 2441 . . . . . 6  |-  ( Base `  D )  =  (
Base `  D )
24 dvhgrp.o . . . . . 6  |-  .0.  =  ( 0g `  D )
2523, 24grpidcl 15559 . . . . 5  |-  ( D  e.  Grp  ->  .0.  e.  ( Base `  D
) )
2622, 25syl 16 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  .0.  e.  ( Base `  D ) )
271, 3, 4, 10, 23dvhbase 34450 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( Base `  D
)  =  E )
2826, 27eleqtrd 2517 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  .0.  e.  E )
29 opelxpi 4867 . . 3  |-  ( ( (  _I  |`  B )  e.  T  /\  .0.  e.  E )  ->  <. (  _I  |`  B ) ,  .0.  >.  e.  ( T  X.  E ) )
3016, 28, 29syl2anc 656 . 2  |-  ( ( K  e.  HL  /\  W  e.  H )  -> 
<. (  _I  |`  B ) ,  .0.  >.  e.  ( T  X.  E ) )
31 simpl 454 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  ( T  X.  E ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
3216adantr 462 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  ( T  X.  E ) )  ->  (  _I  |`  B )  e.  T
)
3328adantr 462 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  ( T  X.  E ) )  ->  .0.  e.  E )
34 xp1st 6605 . . . . . 6  |-  ( f  e.  ( T  X.  E )  ->  ( 1st `  f )  e.  T )
3534adantl 463 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  ( T  X.  E ) )  ->  ( 1st `  f )  e.  T
)
36 xp2nd 6606 . . . . . 6  |-  ( f  e.  ( T  X.  E )  ->  ( 2nd `  f )  e.  E )
3736adantl 463 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  ( T  X.  E ) )  ->  ( 2nd `  f )  e.  E
)
381, 2, 3, 4, 10, 8, 11dvhopvadd 34460 . . . . 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 1222 . . . 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 33490 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( 1st `  f
)  e.  T )  ->  ( 1st `  f
) : B -1-1-onto-> B )
4135, 40syldan 467 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  ( T  X.  E ) )  ->  ( 1st `  f ) : B -1-1-onto-> B
)
42 f1of 5638 . . . . . 6  |-  ( ( 1st `  f ) : B -1-1-onto-> B  ->  ( 1st `  f ) : B --> B )
43 fcoi2 5583 . . . . . 6  |-  ( ( 1st `  f ) : B --> B  -> 
( (  _I  |`  B )  o.  ( 1st `  f
) )  =  ( 1st `  f ) )
4441, 42, 433syl 20 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  ( T  X.  E ) )  ->  ( (  _I  |`  B )  o.  ( 1st `  f
) )  =  ( 1st `  f ) )
4522adantr 462 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  ( T  X.  E ) )  ->  D  e.  Grp )
4627adantr 462 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  ( T  X.  E ) )  ->  ( Base `  D )  =  E )
4737, 46eleqtrrd 2518 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  ( T  X.  E ) )  ->  ( 2nd `  f )  e.  (
Base `  D )
)
4823, 11, 24grplid 15561 . . . . . 6  |-  ( ( D  e.  Grp  /\  ( 2nd `  f )  e.  ( Base `  D
) )  ->  (  .0.  .+^  ( 2nd `  f
) )  =  ( 2nd `  f ) )
4945, 47, 48syl2anc 656 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  ( T  X.  E ) )  ->  (  .0.  .+^  ( 2nd `  f
) )  =  ( 2nd `  f ) )
5044, 49opeq12d 4064 . . . 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 2473 . . 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 6612 . . . . 5  |-  ( f  e.  ( T  X.  E )  ->  f  =  <. ( 1st `  f
) ,  ( 2nd `  f ) >. )
5352adantl 463 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  ( T  X.  E ) )  ->  f  =  <. ( 1st `  f
) ,  ( 2nd `  f ) >. )
5453oveq2d 6106 . . 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 2483 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  ( T  X.  E ) )  ->  ( <. (  _I  |`  B ) ,  .0.  >.  .+  f )  =  f )
561, 2ltrncnv 33512 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( 1st `  f
)  e.  T )  ->  `' ( 1st `  f )  e.  T
)
5735, 56syldan 467 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  ( T  X.  E ) )  ->  `' ( 1st `  f )  e.  T )
58 dvhgrp.i . . . . . 6  |-  I  =  ( invg `  D )
5923, 58grpinvcl 15576 . . . . 5  |-  ( ( D  e.  Grp  /\  ( 2nd `  f )  e.  ( Base `  D
) )  ->  (
I `  ( 2nd `  f ) )  e.  ( Base `  D
) )
6045, 47, 59syl2anc 656 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  ( T  X.  E ) )  ->  ( I `  ( 2nd `  f
) )  e.  (
Base `  D )
)
6160, 46eleqtrd 2517 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  ( T  X.  E ) )  ->  ( I `  ( 2nd `  f
) )  e.  E
)
62 opelxpi 4867 . . 3  |-  ( ( `' ( 1st `  f
)  e.  T  /\  ( I `  ( 2nd `  f ) )  e.  E )  ->  <. `' ( 1st `  f
) ,  ( I `
 ( 2nd `  f
) ) >.  e.  ( T  X.  E ) )
6357, 61, 62syl2anc 656 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  ( T  X.  E ) )  ->  <. `' ( 1st `  f ) ,  ( I `  ( 2nd `  f ) ) >.  e.  ( T  X.  E ) )
6453oveq2d 6106 . . 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 34460 . . . . 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 1222 . . . 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 5666 . . . . . 6  |-  ( ( 1st `  f ) : B -1-1-onto-> B  ->  ( `' ( 1st `  f )  o.  ( 1st `  f
) )  =  (  _I  |`  B )
)
6841, 67syl 16 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  ( T  X.  E ) )  ->  ( `' ( 1st `  f )  o.  ( 1st `  f
) )  =  (  _I  |`  B )
)
6923, 11, 24, 58grplinv 15577 . . . . . 6  |-  ( ( D  e.  Grp  /\  ( 2nd `  f )  e.  ( Base `  D
) )  ->  (
( I `  ( 2nd `  f ) ) 
.+^  ( 2nd `  f
) )  =  .0.  )
7045, 47, 69syl2anc 656 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  ( T  X.  E ) )  ->  ( (
I `  ( 2nd `  f ) )  .+^  ( 2nd `  f ) )  =  .0.  )
7168, 70opeq12d 4064 . . . 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 2473 . . 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 2473 . 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 15556 1  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  U  e.  Grp )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1364    e. wcel 1761   <.cop 3880    _I cid 4627    X. cxp 4834   `'ccnv 4835    |` cres 4838    o. ccom 4840   -->wf 5411   -1-1-onto->wf1o 5414   ` cfv 5415  (class class class)co 6090   1stc1st 6574   2ndc2nd 6575   Basecbs 14170   +g cplusg 14234  Scalarcsca 14237   0gc0g 14374   Grpcgrp 15406   invgcminusg 15407   DivRingcdr 16812   HLchlt 32717   LHypclh 33350   LTrncltrn 33467   TEndoctendo 34118   EDRingcedring 34119   DVecHcdvh 34445
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-riotaBAD 32326
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-fal 1370  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-iin 4171  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-tpos 6744  df-undef 6788  df-recs 6828  df-rdg 6862  df-1o 6916  df-oadd 6920  df-er 7097  df-map 7212  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-nn 10319  df-2 10376  df-3 10377  df-4 10378  df-5 10379  df-6 10380  df-n0 10576  df-z 10643  df-uz 10858  df-fz 11434  df-struct 14172  df-ndx 14173  df-slot 14174  df-base 14175  df-sets 14176  df-ress 14177  df-plusg 14247  df-mulr 14248  df-sca 14250  df-vsca 14251  df-0g 14376  df-poset 15112  df-plt 15124  df-lub 15140  df-glb 15141  df-join 15142  df-meet 15143  df-p0 15205  df-p1 15206  df-lat 15212  df-clat 15274  df-mnd 15411  df-grp 15538  df-minusg 15539  df-mgp 16582  df-ur 16594  df-rng 16637  df-oppr 16705  df-dvdsr 16723  df-unit 16724  df-invr 16754  df-dvr 16765  df-drng 16814  df-oposet 32543  df-ol 32545  df-oml 32546  df-covers 32633  df-ats 32634  df-atl 32665  df-cvlat 32689  df-hlat 32718  df-llines 32864  df-lplanes 32865  df-lvols 32866  df-lines 32867  df-psubsp 32869  df-pmap 32870  df-padd 33162  df-lhyp 33354  df-laut 33355  df-ldil 33470  df-ltrn 33471  df-trl 33525  df-tendo 34121  df-edring 34123  df-dvech 34446
This theorem is referenced by:  dvhlveclem  34475
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