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Theorem dvhgrp 35071
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 2452 . . . 4  |-  ( Base `  U )  =  (
Base `  U )
61, 2, 3, 4, 5dvhvbase 35051 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( Base `  U
)  =  ( T  X.  E ) )
76eqcomd 2460 . 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 35059 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( f  e.  ( T  X.  E
)  /\  g  e.  ( T  X.  E
) ) )  -> 
( f  .+  g
)  e.  ( T  X.  E ) )
13123impb 1184 . 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 35061 . 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 34113 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  (  _I  |`  B )  e.  T )
17 eqid 2452 . . . . . . . 8  |-  ( (
EDRing `  K ) `  W )  =  ( ( EDRing `  K ) `  W )
181, 17, 4, 10dvhsca 35046 . . . . . . 7  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  D  =  ( (
EDRing `  K ) `  W ) )
191, 17erngdv 34956 . . . . . . 7  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( ( EDRing `  K
) `  W )  e.  DivRing )
2018, 19eqeltrd 2540 . . . . . 6  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  D  e.  DivRing )
21 drnggrp 16958 . . . . . 6  |-  ( D  e.  DivRing  ->  D  e.  Grp )
2220, 21syl 16 . . . . 5  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  D  e.  Grp )
23 eqid 2452 . . . . . 6  |-  ( Base `  D )  =  (
Base `  D )
24 dvhgrp.o . . . . . 6  |-  .0.  =  ( 0g `  D )
2523, 24grpidcl 15680 . . . . 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 35047 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( Base `  D
)  =  E )
2826, 27eleqtrd 2542 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  .0.  e.  E )
29 opelxpi 4974 . . 3  |-  ( ( (  _I  |`  B )  e.  T  /\  .0.  e.  E )  ->  <. (  _I  |`  B ) ,  .0.  >.  e.  ( T  X.  E ) )
3016, 28, 29syl2anc 661 . 2  |-  ( ( K  e.  HL  /\  W  e.  H )  -> 
<. (  _I  |`  B ) ,  .0.  >.  e.  ( T  X.  E ) )
31 simpl 457 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  ( T  X.  E ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
3216adantr 465 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  ( T  X.  E ) )  ->  (  _I  |`  B )  e.  T
)
3328adantr 465 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  ( T  X.  E ) )  ->  .0.  e.  E )
34 xp1st 6711 . . . . . 6  |-  ( f  e.  ( T  X.  E )  ->  ( 1st `  f )  e.  T )
3534adantl 466 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  ( T  X.  E ) )  ->  ( 1st `  f )  e.  T
)
36 xp2nd 6712 . . . . . 6  |-  ( f  e.  ( T  X.  E )  ->  ( 2nd `  f )  e.  E )
3736adantl 466 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  ( T  X.  E ) )  ->  ( 2nd `  f )  e.  E
)
381, 2, 3, 4, 10, 8, 11dvhopvadd 35057 . . . . 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 1228 . . . 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 34087 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( 1st `  f
)  e.  T )  ->  ( 1st `  f
) : B -1-1-onto-> B )
4135, 40syldan 470 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  ( T  X.  E ) )  ->  ( 1st `  f ) : B -1-1-onto-> B
)
42 f1of 5744 . . . . . 6  |-  ( ( 1st `  f ) : B -1-1-onto-> B  ->  ( 1st `  f ) : B --> B )
43 fcoi2 5689 . . . . . 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 465 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  ( T  X.  E ) )  ->  D  e.  Grp )
4627adantr 465 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  ( T  X.  E ) )  ->  ( Base `  D )  =  E )
4737, 46eleqtrrd 2543 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  ( T  X.  E ) )  ->  ( 2nd `  f )  e.  (
Base `  D )
)
4823, 11, 24grplid 15682 . . . . . 6  |-  ( ( D  e.  Grp  /\  ( 2nd `  f )  e.  ( Base `  D
) )  ->  (  .0.  .+^  ( 2nd `  f
) )  =  ( 2nd `  f ) )
4945, 47, 48syl2anc 661 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  ( T  X.  E ) )  ->  (  .0.  .+^  ( 2nd `  f
) )  =  ( 2nd `  f ) )
5044, 49opeq12d 4170 . . . 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 2493 . . 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 6718 . . . . 5  |-  ( f  e.  ( T  X.  E )  ->  f  =  <. ( 1st `  f
) ,  ( 2nd `  f ) >. )
5352adantl 466 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  ( T  X.  E ) )  ->  f  =  <. ( 1st `  f
) ,  ( 2nd `  f ) >. )
5453oveq2d 6211 . . 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 2503 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  ( T  X.  E ) )  ->  ( <. (  _I  |`  B ) ,  .0.  >.  .+  f )  =  f )
561, 2ltrncnv 34109 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( 1st `  f
)  e.  T )  ->  `' ( 1st `  f )  e.  T
)
5735, 56syldan 470 . . 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 15697 . . . . 5  |-  ( ( D  e.  Grp  /\  ( 2nd `  f )  e.  ( Base `  D
) )  ->  (
I `  ( 2nd `  f ) )  e.  ( Base `  D
) )
6045, 47, 59syl2anc 661 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  ( T  X.  E ) )  ->  ( I `  ( 2nd `  f
) )  e.  (
Base `  D )
)
6160, 46eleqtrd 2542 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  ( T  X.  E ) )  ->  ( I `  ( 2nd `  f
) )  e.  E
)
62 opelxpi 4974 . . 3  |-  ( ( `' ( 1st `  f
)  e.  T  /\  ( I `  ( 2nd `  f ) )  e.  E )  ->  <. `' ( 1st `  f
) ,  ( I `
 ( 2nd `  f
) ) >.  e.  ( T  X.  E ) )
6357, 61, 62syl2anc 661 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  ( T  X.  E ) )  ->  <. `' ( 1st `  f ) ,  ( I `  ( 2nd `  f ) ) >.  e.  ( T  X.  E ) )
6453oveq2d 6211 . . 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 35057 . . . . 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 1228 . . . 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 5772 . . . . . 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 15698 . . . . . 6  |-  ( ( D  e.  Grp  /\  ( 2nd `  f )  e.  ( Base `  D
) )  ->  (
( I `  ( 2nd `  f ) ) 
.+^  ( 2nd `  f
) )  =  .0.  )
7045, 47, 69syl2anc 661 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  ( T  X.  E ) )  ->  ( (
I `  ( 2nd `  f ) )  .+^  ( 2nd `  f ) )  =  .0.  )
7168, 70opeq12d 4170 . . . 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 2493 . . 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 2493 . 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 15677 1  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  U  e.  Grp )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   <.cop 3986    _I cid 4734    X. cxp 4941   `'ccnv 4942    |` cres 4945    o. ccom 4947   -->wf 5517   -1-1-onto->wf1o 5520   ` cfv 5521  (class class class)co 6195   1stc1st 6680   2ndc2nd 6681   Basecbs 14287   +g cplusg 14352  Scalarcsca 14355   0gc0g 14492   Grpcgrp 15524   invgcminusg 15525   DivRingcdr 16950   HLchlt 33314   LHypclh 33947   LTrncltrn 34064   TEndoctendo 34715   EDRingcedring 34716   DVecHcdvh 35042
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477  ax-cnex 9444  ax-resscn 9445  ax-1cn 9446  ax-icn 9447  ax-addcl 9448  ax-addrcl 9449  ax-mulcl 9450  ax-mulrcl 9451  ax-mulcom 9452  ax-addass 9453  ax-mulass 9454  ax-distr 9455  ax-i2m1 9456  ax-1ne0 9457  ax-1rid 9458  ax-rnegex 9459  ax-rrecex 9460  ax-cnre 9461  ax-pre-lttri 9462  ax-pre-lttrn 9463  ax-pre-ltadd 9464  ax-pre-mulgt0 9465  ax-riotaBAD 32923
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-nel 2648  df-ral 2801  df-rex 2802  df-reu 2803  df-rmo 2804  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-pss 3447  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4195  df-int 4232  df-iun 4276  df-iin 4277  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4489  df-eprel 4735  df-id 4739  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-ord 4825  df-on 4826  df-lim 4827  df-suc 4828  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-riota 6156  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-om 6582  df-1st 6682  df-2nd 6683  df-tpos 6850  df-undef 6897  df-recs 6937  df-rdg 6971  df-1o 7025  df-oadd 7029  df-er 7206  df-map 7321  df-en 7416  df-dom 7417  df-sdom 7418  df-fin 7419  df-pnf 9526  df-mnf 9527  df-xr 9528  df-ltxr 9529  df-le 9530  df-sub 9703  df-neg 9704  df-nn 10429  df-2 10486  df-3 10487  df-4 10488  df-5 10489  df-6 10490  df-n0 10686  df-z 10753  df-uz 10968  df-fz 11550  df-struct 14289  df-ndx 14290  df-slot 14291  df-base 14292  df-sets 14293  df-ress 14294  df-plusg 14365  df-mulr 14366  df-sca 14368  df-vsca 14369  df-0g 14494  df-poset 15230  df-plt 15242  df-lub 15258  df-glb 15259  df-join 15260  df-meet 15261  df-p0 15323  df-p1 15324  df-lat 15330  df-clat 15392  df-mnd 15529  df-grp 15659  df-minusg 15660  df-mgp 16709  df-ur 16721  df-rng 16765  df-oppr 16833  df-dvdsr 16851  df-unit 16852  df-invr 16882  df-dvr 16893  df-drng 16952  df-oposet 33140  df-ol 33142  df-oml 33143  df-covers 33230  df-ats 33231  df-atl 33262  df-cvlat 33286  df-hlat 33315  df-llines 33461  df-lplanes 33462  df-lvols 33463  df-lines 33464  df-psubsp 33466  df-pmap 33467  df-padd 33759  df-lhyp 33951  df-laut 33952  df-ldil 34067  df-ltrn 34068  df-trl 34122  df-tendo 34718  df-edring 34720  df-dvech 35043
This theorem is referenced by:  dvhlveclem  35072
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