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Theorem dvhgrp 31967
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 2438 . . . 4  |-  ( Base `  U )  =  (
Base `  U )
61, 2, 3, 4, 5dvhvbase 31947 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( Base `  U
)  =  ( T  X.  E ) )
76eqcomd 2443 . 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 31955 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( f  e.  ( T  X.  E
)  /\  g  e.  ( T  X.  E
) ) )  -> 
( f  .+  g
)  e.  ( T  X.  E ) )
13123impb 1150 . 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 31957 . 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 31009 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  (  _I  |`  B )  e.  T )
17 eqid 2438 . . . . . . . 8  |-  ( (
EDRing `  K ) `  W )  =  ( ( EDRing `  K ) `  W )
181, 17, 4, 10dvhsca 31942 . . . . . . 7  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  D  =  ( (
EDRing `  K ) `  W ) )
191, 17erngdv 31852 . . . . . . 7  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( ( EDRing `  K
) `  W )  e.  DivRing )
2018, 19eqeltrd 2512 . . . . . 6  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  D  e.  DivRing )
21 drnggrp 15845 . . . . . 6  |-  ( D  e.  DivRing  ->  D  e.  Grp )
2220, 21syl 16 . . . . 5  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  D  e.  Grp )
23 eqid 2438 . . . . . 6  |-  ( Base `  D )  =  (
Base `  D )
24 dvhgrp.o . . . . . 6  |-  .0.  =  ( 0g `  D )
2523, 24grpidcl 14835 . . . . 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 31943 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( Base `  D
)  =  E )
2826, 27eleqtrd 2514 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  .0.  e.  E )
29 opelxpi 4912 . . 3  |-  ( ( (  _I  |`  B )  e.  T  /\  .0.  e.  E )  ->  <. (  _I  |`  B ) ,  .0.  >.  e.  ( T  X.  E ) )
3016, 28, 29syl2anc 644 . 2  |-  ( ( K  e.  HL  /\  W  e.  H )  -> 
<. (  _I  |`  B ) ,  .0.  >.  e.  ( T  X.  E ) )
31 simpl 445 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  ( T  X.  E ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
3216adantr 453 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  ( T  X.  E ) )  ->  (  _I  |`  B )  e.  T
)
3328adantr 453 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  ( T  X.  E ) )  ->  .0.  e.  E )
34 xp1st 6378 . . . . . 6  |-  ( f  e.  ( T  X.  E )  ->  ( 1st `  f )  e.  T )
3534adantl 454 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  ( T  X.  E ) )  ->  ( 1st `  f )  e.  T
)
36 xp2nd 6379 . . . . . 6  |-  ( f  e.  ( T  X.  E )  ->  ( 2nd `  f )  e.  E )
3736adantl 454 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  ( T  X.  E ) )  ->  ( 2nd `  f )  e.  E
)
381, 2, 3, 4, 10, 8, 11dvhopvadd 31953 . . . . 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 1194 . . . 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 30983 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( 1st `  f
)  e.  T )  ->  ( 1st `  f
) : B -1-1-onto-> B )
4135, 40syldan 458 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  ( T  X.  E ) )  ->  ( 1st `  f ) : B -1-1-onto-> B
)
42 f1of 5676 . . . . . 6  |-  ( ( 1st `  f ) : B -1-1-onto-> B  ->  ( 1st `  f ) : B --> B )
43 fcoi2 5620 . . . . . 6  |-  ( ( 1st `  f ) : B --> B  -> 
( (  _I  |`  B )  o.  ( 1st `  f
) )  =  ( 1st `  f ) )
4441, 42, 433syl 19 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  ( T  X.  E ) )  ->  ( (  _I  |`  B )  o.  ( 1st `  f
) )  =  ( 1st `  f ) )
4522adantr 453 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  ( T  X.  E ) )  ->  D  e.  Grp )
4627adantr 453 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  ( T  X.  E ) )  ->  ( Base `  D )  =  E )
4737, 46eleqtrrd 2515 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  ( T  X.  E ) )  ->  ( 2nd `  f )  e.  (
Base `  D )
)
4823, 11, 24grplid 14837 . . . . . 6  |-  ( ( D  e.  Grp  /\  ( 2nd `  f )  e.  ( Base `  D
) )  ->  (  .0.  .+^  ( 2nd `  f
) )  =  ( 2nd `  f ) )
4945, 47, 48syl2anc 644 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  ( T  X.  E ) )  ->  (  .0.  .+^  ( 2nd `  f
) )  =  ( 2nd `  f ) )
5044, 49opeq12d 3994 . . . 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 2470 . . 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 6388 . . . . 5  |-  ( f  e.  ( T  X.  E )  ->  f  =  <. ( 1st `  f
) ,  ( 2nd `  f ) >. )
5352adantl 454 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  ( T  X.  E ) )  ->  f  =  <. ( 1st `  f
) ,  ( 2nd `  f ) >. )
5453oveq2d 6099 . . 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 2480 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  ( T  X.  E ) )  ->  ( <. (  _I  |`  B ) ,  .0.  >.  .+  f )  =  f )
561, 2ltrncnv 31005 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( 1st `  f
)  e.  T )  ->  `' ( 1st `  f )  e.  T
)
5735, 56syldan 458 . . 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 14852 . . . . 5  |-  ( ( D  e.  Grp  /\  ( 2nd `  f )  e.  ( Base `  D
) )  ->  (
I `  ( 2nd `  f ) )  e.  ( Base `  D
) )
6045, 47, 59syl2anc 644 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  ( T  X.  E ) )  ->  ( I `  ( 2nd `  f
) )  e.  (
Base `  D )
)
6160, 46eleqtrd 2514 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  ( T  X.  E ) )  ->  ( I `  ( 2nd `  f
) )  e.  E
)
62 opelxpi 4912 . . 3  |-  ( ( `' ( 1st `  f
)  e.  T  /\  ( I `  ( 2nd `  f ) )  e.  E )  ->  <. `' ( 1st `  f
) ,  ( I `
 ( 2nd `  f
) ) >.  e.  ( T  X.  E ) )
6357, 61, 62syl2anc 644 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  ( T  X.  E ) )  ->  <. `' ( 1st `  f ) ,  ( I `  ( 2nd `  f ) ) >.  e.  ( T  X.  E ) )
6453oveq2d 6099 . . 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 31953 . . . . 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 1194 . . . 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 5706 . . . . . 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 14853 . . . . . 6  |-  ( ( D  e.  Grp  /\  ( 2nd `  f )  e.  ( Base `  D
) )  ->  (
( I `  ( 2nd `  f ) ) 
.+^  ( 2nd `  f
) )  =  .0.  )
7045, 47, 69syl2anc 644 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  ( T  X.  E ) )  ->  ( (
I `  ( 2nd `  f ) )  .+^  ( 2nd `  f ) )  =  .0.  )
7168, 70opeq12d 3994 . . . 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 2470 . . 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 2470 . 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 14832 1  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  U  e.  Grp )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   <.cop 3819    _I cid 4495    X. cxp 4878   `'ccnv 4879    |` cres 4882    o. ccom 4884   -->wf 5452   -1-1-onto->wf1o 5455   ` cfv 5456  (class class class)co 6083   1stc1st 6349   2ndc2nd 6350   Basecbs 13471   +g cplusg 13531  Scalarcsca 13534   0gc0g 13725   Grpcgrp 14687   inv gcminusg 14688   DivRingcdr 15837   HLchlt 30210   LHypclh 30843   LTrncltrn 30960   TEndoctendo 31611   EDRingcedring 31612   DVecHcdvh 31938
This theorem is referenced by:  dvhlveclem  31968
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-fal 1330  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-iin 4098  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-tpos 6481  df-undef 6545  df-riota 6551  df-recs 6635  df-rdg 6670  df-1o 6726  df-oadd 6730  df-er 6907  df-map 7022  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-nn 10003  df-2 10060  df-3 10061  df-4 10062  df-5 10063  df-6 10064  df-n0 10224  df-z 10285  df-uz 10491  df-fz 11046  df-struct 13473  df-ndx 13474  df-slot 13475  df-base 13476  df-sets 13477  df-ress 13478  df-plusg 13544  df-mulr 13545  df-sca 13547  df-vsca 13548  df-0g 13729  df-poset 14405  df-plt 14417  df-lub 14433  df-glb 14434  df-join 14435  df-meet 14436  df-p0 14470  df-p1 14471  df-lat 14477  df-clat 14539  df-mnd 14692  df-grp 14814  df-minusg 14815  df-mgp 15651  df-rng 15665  df-ur 15667  df-oppr 15730  df-dvdsr 15748  df-unit 15749  df-invr 15779  df-dvr 15790  df-drng 15839  df-oposet 30036  df-ol 30038  df-oml 30039  df-covers 30126  df-ats 30127  df-atl 30158  df-cvlat 30182  df-hlat 30211  df-llines 30357  df-lplanes 30358  df-lvols 30359  df-lines 30360  df-psubsp 30362  df-pmap 30363  df-padd 30655  df-lhyp 30847  df-laut 30848  df-ldil 30963  df-ltrn 30964  df-trl 31018  df-tendo 31614  df-edring 31616  df-dvech 31939
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