Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dvhgrp Unicode version

Theorem dvhgrp 31590
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 2404 . . . 4  |-  ( Base `  U )  =  (
Base `  U )
61, 2, 3, 4, 5dvhvbase 31570 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( Base `  U
)  =  ( T  X.  E ) )
76eqcomd 2409 . 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 31578 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( f  e.  ( T  X.  E
)  /\  g  e.  ( T  X.  E
) ) )  -> 
( f  .+  g
)  e.  ( T  X.  E ) )
13123impb 1149 . 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 31580 . 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 30632 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  (  _I  |`  B )  e.  T )
17 eqid 2404 . . . . . . . 8  |-  ( (
EDRing `  K ) `  W )  =  ( ( EDRing `  K ) `  W )
181, 17, 4, 10dvhsca 31565 . . . . . . 7  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  D  =  ( (
EDRing `  K ) `  W ) )
191, 17erngdv 31475 . . . . . . 7  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( ( EDRing `  K
) `  W )  e.  DivRing )
2018, 19eqeltrd 2478 . . . . . 6  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  D  e.  DivRing )
21 drnggrp 15798 . . . . . 6  |-  ( D  e.  DivRing  ->  D  e.  Grp )
2220, 21syl 16 . . . . 5  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  D  e.  Grp )
23 eqid 2404 . . . . . 6  |-  ( Base `  D )  =  (
Base `  D )
24 dvhgrp.o . . . . . 6  |-  .0.  =  ( 0g `  D )
2523, 24grpidcl 14788 . . . . 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 31566 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( Base `  D
)  =  E )
2826, 27eleqtrd 2480 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  .0.  e.  E )
29 opelxpi 4869 . . 3  |-  ( ( (  _I  |`  B )  e.  T  /\  .0.  e.  E )  ->  <. (  _I  |`  B ) ,  .0.  >.  e.  ( T  X.  E ) )
3016, 28, 29syl2anc 643 . 2  |-  ( ( K  e.  HL  /\  W  e.  H )  -> 
<. (  _I  |`  B ) ,  .0.  >.  e.  ( T  X.  E ) )
31 simpl 444 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  ( T  X.  E ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
3216adantr 452 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  ( T  X.  E ) )  ->  (  _I  |`  B )  e.  T
)
3328adantr 452 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  ( T  X.  E ) )  ->  .0.  e.  E )
34 xp1st 6335 . . . . . 6  |-  ( f  e.  ( T  X.  E )  ->  ( 1st `  f )  e.  T )
3534adantl 453 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  ( T  X.  E ) )  ->  ( 1st `  f )  e.  T
)
36 xp2nd 6336 . . . . . 6  |-  ( f  e.  ( T  X.  E )  ->  ( 2nd `  f )  e.  E )
3736adantl 453 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  ( T  X.  E ) )  ->  ( 2nd `  f )  e.  E
)
381, 2, 3, 4, 10, 8, 11dvhopvadd 31576 . . . . 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 1193 . . . 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 30606 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( 1st `  f
)  e.  T )  ->  ( 1st `  f
) : B -1-1-onto-> B )
4135, 40syldan 457 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  ( T  X.  E ) )  ->  ( 1st `  f ) : B -1-1-onto-> B
)
42 f1of 5633 . . . . . 6  |-  ( ( 1st `  f ) : B -1-1-onto-> B  ->  ( 1st `  f ) : B --> B )
43 fcoi2 5577 . . . . . 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 452 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  ( T  X.  E ) )  ->  D  e.  Grp )
4627adantr 452 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  ( T  X.  E ) )  ->  ( Base `  D )  =  E )
4737, 46eleqtrrd 2481 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  ( T  X.  E ) )  ->  ( 2nd `  f )  e.  (
Base `  D )
)
4823, 11, 24grplid 14790 . . . . . 6  |-  ( ( D  e.  Grp  /\  ( 2nd `  f )  e.  ( Base `  D
) )  ->  (  .0.  .+^  ( 2nd `  f
) )  =  ( 2nd `  f ) )
4945, 47, 48syl2anc 643 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  ( T  X.  E ) )  ->  (  .0.  .+^  ( 2nd `  f
) )  =  ( 2nd `  f ) )
5044, 49opeq12d 3952 . . . 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 2436 . . 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 6345 . . . . 5  |-  ( f  e.  ( T  X.  E )  ->  f  =  <. ( 1st `  f
) ,  ( 2nd `  f ) >. )
5352adantl 453 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  ( T  X.  E ) )  ->  f  =  <. ( 1st `  f
) ,  ( 2nd `  f ) >. )
5453oveq2d 6056 . . 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 2446 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  ( T  X.  E ) )  ->  ( <. (  _I  |`  B ) ,  .0.  >.  .+  f )  =  f )
561, 2ltrncnv 30628 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( 1st `  f
)  e.  T )  ->  `' ( 1st `  f )  e.  T
)
5735, 56syldan 457 . . 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 14805 . . . . 5  |-  ( ( D  e.  Grp  /\  ( 2nd `  f )  e.  ( Base `  D
) )  ->  (
I `  ( 2nd `  f ) )  e.  ( Base `  D
) )
6045, 47, 59syl2anc 643 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  ( T  X.  E ) )  ->  ( I `  ( 2nd `  f
) )  e.  (
Base `  D )
)
6160, 46eleqtrd 2480 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  ( T  X.  E ) )  ->  ( I `  ( 2nd `  f
) )  e.  E
)
62 opelxpi 4869 . . 3  |-  ( ( `' ( 1st `  f
)  e.  T  /\  ( I `  ( 2nd `  f ) )  e.  E )  ->  <. `' ( 1st `  f
) ,  ( I `
 ( 2nd `  f
) ) >.  e.  ( T  X.  E ) )
6357, 61, 62syl2anc 643 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  ( T  X.  E ) )  ->  <. `' ( 1st `  f ) ,  ( I `  ( 2nd `  f ) ) >.  e.  ( T  X.  E ) )
6453oveq2d 6056 . . 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 31576 . . . . 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 1193 . . . 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 5663 . . . . . 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 14806 . . . . . 6  |-  ( ( D  e.  Grp  /\  ( 2nd `  f )  e.  ( Base `  D
) )  ->  (
( I `  ( 2nd `  f ) ) 
.+^  ( 2nd `  f
) )  =  .0.  )
7045, 47, 69syl2anc 643 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  ( T  X.  E ) )  ->  ( (
I `  ( 2nd `  f ) )  .+^  ( 2nd `  f ) )  =  .0.  )
7168, 70opeq12d 3952 . . . 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 2436 . . 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 2436 . 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 14785 1  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  U  e.  Grp )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   <.cop 3777    _I cid 4453    X. cxp 4835   `'ccnv 4836    |` cres 4839    o. ccom 4841   -->wf 5409   -1-1-onto->wf1o 5412   ` cfv 5413  (class class class)co 6040   1stc1st 6306   2ndc2nd 6307   Basecbs 13424   +g cplusg 13484  Scalarcsca 13487   0gc0g 13678   Grpcgrp 14640   inv gcminusg 14641   DivRingcdr 15790   HLchlt 29833   LHypclh 30466   LTrncltrn 30583   TEndoctendo 31234   EDRingcedring 31235   DVecHcdvh 31561
This theorem is referenced by:  dvhlveclem  31591
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-fal 1326  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-tpos 6438  df-undef 6502  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-map 6979  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-n0 10178  df-z 10239  df-uz 10445  df-fz 11000  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-sca 13500  df-vsca 13501  df-0g 13682  df-poset 14358  df-plt 14370  df-lub 14386  df-glb 14387  df-join 14388  df-meet 14389  df-p0 14423  df-p1 14424  df-lat 14430  df-clat 14492  df-mnd 14645  df-grp 14767  df-minusg 14768  df-mgp 15604  df-rng 15618  df-ur 15620  df-oppr 15683  df-dvdsr 15701  df-unit 15702  df-invr 15732  df-dvr 15743  df-drng 15792  df-oposet 29659  df-ol 29661  df-oml 29662  df-covers 29749  df-ats 29750  df-atl 29781  df-cvlat 29805  df-hlat 29834  df-llines 29980  df-lplanes 29981  df-lvols 29982  df-lines 29983  df-psubsp 29985  df-pmap 29986  df-padd 30278  df-lhyp 30470  df-laut 30471  df-ldil 30586  df-ltrn 30587  df-trl 30641  df-tendo 31237  df-edring 31239  df-dvech 31562
  Copyright terms: Public domain W3C validator