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

Theorem lflvscl 34275
Description: Closure of a scalar product with a functional. Note that this is the scalar product for a right vector space with the scalar after the vector; reversing these fails closure. (Contributed by NM, 9-Oct-2014.) (Revised by Mario Carneiro, 22-Apr-2015.)
Hypotheses
Ref Expression
lflsccl.v  |-  V  =  ( Base `  W
)
lflsccl.d  |-  D  =  (Scalar `  W )
lflsccl.k  |-  K  =  ( Base `  D
)
lflsccl.t  |-  .x.  =  ( .r `  D )
lflsccl.f  |-  F  =  (LFnl `  W )
lflsccl.w  |-  ( ph  ->  W  e.  LMod )
lflsccl.g  |-  ( ph  ->  G  e.  F )
lflsccl.r  |-  ( ph  ->  R  e.  K )
Assertion
Ref Expression
lflvscl  |-  ( ph  ->  ( G  oF  .x.  ( V  X.  { R } ) )  e.  F )

Proof of Theorem lflvscl
Dummy variables  x  r  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lflsccl.v . . 3  |-  V  =  ( Base `  W
)
21a1i 11 . 2  |-  ( ph  ->  V  =  ( Base `  W ) )
3 eqidd 2468 . 2  |-  ( ph  ->  ( +g  `  W
)  =  ( +g  `  W ) )
4 lflsccl.d . . 3  |-  D  =  (Scalar `  W )
54a1i 11 . 2  |-  ( ph  ->  D  =  (Scalar `  W ) )
6 eqidd 2468 . 2  |-  ( ph  ->  ( .s `  W
)  =  ( .s
`  W ) )
7 lflsccl.k . . 3  |-  K  =  ( Base `  D
)
87a1i 11 . 2  |-  ( ph  ->  K  =  ( Base `  D ) )
9 eqidd 2468 . 2  |-  ( ph  ->  ( +g  `  D
)  =  ( +g  `  D ) )
10 lflsccl.t . . 3  |-  .x.  =  ( .r `  D )
1110a1i 11 . 2  |-  ( ph  ->  .x.  =  ( .r
`  D ) )
12 lflsccl.f . . 3  |-  F  =  (LFnl `  W )
1312a1i 11 . 2  |-  ( ph  ->  F  =  (LFnl `  W ) )
14 lflsccl.w . . . . 5  |-  ( ph  ->  W  e.  LMod )
154lmodring 17391 . . . . 5  |-  ( W  e.  LMod  ->  D  e. 
Ring )
1614, 15syl 16 . . . 4  |-  ( ph  ->  D  e.  Ring )
177, 10ringcl 17084 . . . . 5  |-  ( ( D  e.  Ring  /\  x  e.  K  /\  y  e.  K )  ->  (
x  .x.  y )  e.  K )
18173expb 1197 . . . 4  |-  ( ( D  e.  Ring  /\  (
x  e.  K  /\  y  e.  K )
)  ->  ( x  .x.  y )  e.  K
)
1916, 18sylan 471 . . 3  |-  ( (
ph  /\  ( x  e.  K  /\  y  e.  K ) )  -> 
( x  .x.  y
)  e.  K )
20 lflsccl.g . . . 4  |-  ( ph  ->  G  e.  F )
214, 7, 1, 12lflf 34261 . . . 4  |-  ( ( W  e.  LMod  /\  G  e.  F )  ->  G : V --> K )
2214, 20, 21syl2anc 661 . . 3  |-  ( ph  ->  G : V --> K )
23 lflsccl.r . . . 4  |-  ( ph  ->  R  e.  K )
24 fconst6g 5780 . . . 4  |-  ( R  e.  K  ->  ( V  X.  { R }
) : V --> K )
2523, 24syl 16 . . 3  |-  ( ph  ->  ( V  X.  { R } ) : V --> K )
26 fvex 5882 . . . . 5  |-  ( Base `  W )  e.  _V
271, 26eqeltri 2551 . . . 4  |-  V  e. 
_V
2827a1i 11 . . 3  |-  ( ph  ->  V  e.  _V )
29 inidm 3712 . . 3  |-  ( V  i^i  V )  =  V
3019, 22, 25, 28, 28, 29off 6549 . 2  |-  ( ph  ->  ( G  oF  .x.  ( V  X.  { R } ) ) : V --> K )
3114adantr 465 . . . . . 6  |-  ( (
ph  /\  ( r  e.  K  /\  x  e.  V  /\  y  e.  V ) )  ->  W  e.  LMod )
3220adantr 465 . . . . . 6  |-  ( (
ph  /\  ( r  e.  K  /\  x  e.  V  /\  y  e.  V ) )  ->  G  e.  F )
33 simpr1 1002 . . . . . 6  |-  ( (
ph  /\  ( r  e.  K  /\  x  e.  V  /\  y  e.  V ) )  -> 
r  e.  K )
34 simpr2 1003 . . . . . 6  |-  ( (
ph  /\  ( r  e.  K  /\  x  e.  V  /\  y  e.  V ) )  ->  x  e.  V )
35 simpr3 1004 . . . . . 6  |-  ( (
ph  /\  ( r  e.  K  /\  x  e.  V  /\  y  e.  V ) )  -> 
y  e.  V )
36 eqid 2467 . . . . . . 7  |-  ( +g  `  W )  =  ( +g  `  W )
37 eqid 2467 . . . . . . 7  |-  ( .s
`  W )  =  ( .s `  W
)
38 eqid 2467 . . . . . . 7  |-  ( +g  `  D )  =  ( +g  `  D )
391, 36, 4, 37, 7, 38, 10, 12lfli 34259 . . . . . 6  |-  ( ( W  e.  LMod  /\  G  e.  F  /\  (
r  e.  K  /\  x  e.  V  /\  y  e.  V )
)  ->  ( G `  ( ( r ( .s `  W ) x ) ( +g  `  W ) y ) )  =  ( ( r  .x.  ( G `
 x ) ) ( +g  `  D
) ( G `  y ) ) )
4031, 32, 33, 34, 35, 39syl113anc 1240 . . . . 5  |-  ( (
ph  /\  ( r  e.  K  /\  x  e.  V  /\  y  e.  V ) )  -> 
( G `  (
( r ( .s
`  W ) x ) ( +g  `  W
) y ) )  =  ( ( r 
.x.  ( G `  x ) ) ( +g  `  D ) ( G `  y
) ) )
4140oveq1d 6310 . . . 4  |-  ( (
ph  /\  ( r  e.  K  /\  x  e.  V  /\  y  e.  V ) )  -> 
( ( G `  ( ( r ( .s `  W ) x ) ( +g  `  W ) y ) )  .x.  R )  =  ( ( ( r  .x.  ( G `
 x ) ) ( +g  `  D
) ( G `  y ) )  .x.  R ) )
4216adantr 465 . . . . 5  |-  ( (
ph  /\  ( r  e.  K  /\  x  e.  V  /\  y  e.  V ) )  ->  D  e.  Ring )
434, 7, 1, 12lflcl 34262 . . . . . . 7  |-  ( ( W  e.  LMod  /\  G  e.  F  /\  x  e.  V )  ->  ( G `  x )  e.  K )
4431, 32, 34, 43syl3anc 1228 . . . . . 6  |-  ( (
ph  /\  ( r  e.  K  /\  x  e.  V  /\  y  e.  V ) )  -> 
( G `  x
)  e.  K )
457, 10ringcl 17084 . . . . . 6  |-  ( ( D  e.  Ring  /\  r  e.  K  /\  ( G `  x )  e.  K )  ->  (
r  .x.  ( G `  x ) )  e.  K )
4642, 33, 44, 45syl3anc 1228 . . . . 5  |-  ( (
ph  /\  ( r  e.  K  /\  x  e.  V  /\  y  e.  V ) )  -> 
( r  .x.  ( G `  x )
)  e.  K )
474, 7, 1, 12lflcl 34262 . . . . . 6  |-  ( ( W  e.  LMod  /\  G  e.  F  /\  y  e.  V )  ->  ( G `  y )  e.  K )
4831, 32, 35, 47syl3anc 1228 . . . . 5  |-  ( (
ph  /\  ( r  e.  K  /\  x  e.  V  /\  y  e.  V ) )  -> 
( G `  y
)  e.  K )
4923adantr 465 . . . . 5  |-  ( (
ph  /\  ( r  e.  K  /\  x  e.  V  /\  y  e.  V ) )  ->  R  e.  K )
507, 38, 10ringdir 17090 . . . . 5  |-  ( ( D  e.  Ring  /\  (
( r  .x.  ( G `  x )
)  e.  K  /\  ( G `  y )  e.  K  /\  R  e.  K ) )  -> 
( ( ( r 
.x.  ( G `  x ) ) ( +g  `  D ) ( G `  y
) )  .x.  R
)  =  ( ( ( r  .x.  ( G `  x )
)  .x.  R )
( +g  `  D ) ( ( G `  y )  .x.  R
) ) )
5142, 46, 48, 49, 50syl13anc 1230 . . . 4  |-  ( (
ph  /\  ( r  e.  K  /\  x  e.  V  /\  y  e.  V ) )  -> 
( ( ( r 
.x.  ( G `  x ) ) ( +g  `  D ) ( G `  y
) )  .x.  R
)  =  ( ( ( r  .x.  ( G `  x )
)  .x.  R )
( +g  `  D ) ( ( G `  y )  .x.  R
) ) )
527, 10ringass 17087 . . . . . 6  |-  ( ( D  e.  Ring  /\  (
r  e.  K  /\  ( G `  x )  e.  K  /\  R  e.  K ) )  -> 
( ( r  .x.  ( G `  x ) )  .x.  R )  =  ( r  .x.  ( ( G `  x )  .x.  R
) ) )
5342, 33, 44, 49, 52syl13anc 1230 . . . . 5  |-  ( (
ph  /\  ( r  e.  K  /\  x  e.  V  /\  y  e.  V ) )  -> 
( ( r  .x.  ( G `  x ) )  .x.  R )  =  ( r  .x.  ( ( G `  x )  .x.  R
) ) )
5453oveq1d 6310 . . . 4  |-  ( (
ph  /\  ( r  e.  K  /\  x  e.  V  /\  y  e.  V ) )  -> 
( ( ( r 
.x.  ( G `  x ) )  .x.  R ) ( +g  `  D ) ( ( G `  y ) 
.x.  R ) )  =  ( ( r 
.x.  ( ( G `
 x )  .x.  R ) ) ( +g  `  D ) ( ( G `  y )  .x.  R
) ) )
5541, 51, 543eqtrd 2512 . . 3  |-  ( (
ph  /\  ( r  e.  K  /\  x  e.  V  /\  y  e.  V ) )  -> 
( ( G `  ( ( r ( .s `  W ) x ) ( +g  `  W ) y ) )  .x.  R )  =  ( ( r 
.x.  ( ( G `
 x )  .x.  R ) ) ( +g  `  D ) ( ( G `  y )  .x.  R
) ) )
561, 4, 37, 7lmodvscl 17400 . . . . . 6  |-  ( ( W  e.  LMod  /\  r  e.  K  /\  x  e.  V )  ->  (
r ( .s `  W ) x )  e.  V )
5731, 33, 34, 56syl3anc 1228 . . . . 5  |-  ( (
ph  /\  ( r  e.  K  /\  x  e.  V  /\  y  e.  V ) )  -> 
( r ( .s
`  W ) x )  e.  V )
581, 36lmodvacl 17397 . . . . 5  |-  ( ( W  e.  LMod  /\  (
r ( .s `  W ) x )  e.  V  /\  y  e.  V )  ->  (
( r ( .s
`  W ) x ) ( +g  `  W
) y )  e.  V )
5931, 57, 35, 58syl3anc 1228 . . . 4  |-  ( (
ph  /\  ( r  e.  K  /\  x  e.  V  /\  y  e.  V ) )  -> 
( ( r ( .s `  W ) x ) ( +g  `  W ) y )  e.  V )
60 ffn 5737 . . . . . 6  |-  ( G : V --> K  ->  G  Fn  V )
6122, 60syl 16 . . . . 5  |-  ( ph  ->  G  Fn  V )
62 eqidd 2468 . . . . 5  |-  ( (
ph  /\  ( (
r ( .s `  W ) x ) ( +g  `  W
) y )  e.  V )  ->  ( G `  ( (
r ( .s `  W ) x ) ( +g  `  W
) y ) )  =  ( G `  ( ( r ( .s `  W ) x ) ( +g  `  W ) y ) ) )
6328, 23, 61, 62ofc2 6559 . . . 4  |-  ( (
ph  /\  ( (
r ( .s `  W ) x ) ( +g  `  W
) y )  e.  V )  ->  (
( G  oF  .x.  ( V  X.  { R } ) ) `
 ( ( r ( .s `  W
) x ) ( +g  `  W ) y ) )  =  ( ( G `  ( ( r ( .s `  W ) x ) ( +g  `  W ) y ) )  .x.  R ) )
6459, 63syldan 470 . . 3  |-  ( (
ph  /\  ( r  e.  K  /\  x  e.  V  /\  y  e.  V ) )  -> 
( ( G  oF  .x.  ( V  X.  { R } ) ) `
 ( ( r ( .s `  W
) x ) ( +g  `  W ) y ) )  =  ( ( G `  ( ( r ( .s `  W ) x ) ( +g  `  W ) y ) )  .x.  R ) )
65 eqidd 2468 . . . . . . 7  |-  ( (
ph  /\  x  e.  V )  ->  ( G `  x )  =  ( G `  x ) )
6628, 23, 61, 65ofc2 6559 . . . . . 6  |-  ( (
ph  /\  x  e.  V )  ->  (
( G  oF  .x.  ( V  X.  { R } ) ) `
 x )  =  ( ( G `  x )  .x.  R
) )
6734, 66syldan 470 . . . . 5  |-  ( (
ph  /\  ( r  e.  K  /\  x  e.  V  /\  y  e.  V ) )  -> 
( ( G  oF  .x.  ( V  X.  { R } ) ) `
 x )  =  ( ( G `  x )  .x.  R
) )
6867oveq2d 6311 . . . 4  |-  ( (
ph  /\  ( r  e.  K  /\  x  e.  V  /\  y  e.  V ) )  -> 
( r  .x.  (
( G  oF  .x.  ( V  X.  { R } ) ) `
 x ) )  =  ( r  .x.  ( ( G `  x )  .x.  R
) ) )
69 eqidd 2468 . . . . . 6  |-  ( (
ph  /\  y  e.  V )  ->  ( G `  y )  =  ( G `  y ) )
7028, 23, 61, 69ofc2 6559 . . . . 5  |-  ( (
ph  /\  y  e.  V )  ->  (
( G  oF  .x.  ( V  X.  { R } ) ) `
 y )  =  ( ( G `  y )  .x.  R
) )
7135, 70syldan 470 . . . 4  |-  ( (
ph  /\  ( r  e.  K  /\  x  e.  V  /\  y  e.  V ) )  -> 
( ( G  oF  .x.  ( V  X.  { R } ) ) `
 y )  =  ( ( G `  y )  .x.  R
) )
7268, 71oveq12d 6313 . . 3  |-  ( (
ph  /\  ( r  e.  K  /\  x  e.  V  /\  y  e.  V ) )  -> 
( ( r  .x.  ( ( G  oF  .x.  ( V  X.  { R } ) ) `
 x ) ) ( +g  `  D
) ( ( G  oF  .x.  ( V  X.  { R }
) ) `  y
) )  =  ( ( r  .x.  (
( G `  x
)  .x.  R )
) ( +g  `  D
) ( ( G `
 y )  .x.  R ) ) )
7355, 64, 723eqtr4d 2518 . 2  |-  ( (
ph  /\  ( r  e.  K  /\  x  e.  V  /\  y  e.  V ) )  -> 
( ( G  oF  .x.  ( V  X.  { R } ) ) `
 ( ( r ( .s `  W
) x ) ( +g  `  W ) y ) )  =  ( ( r  .x.  ( ( G  oF  .x.  ( V  X.  { R } ) ) `
 x ) ) ( +g  `  D
) ( ( G  oF  .x.  ( V  X.  { R }
) ) `  y
) ) )
742, 3, 5, 6, 8, 9, 11, 13, 30, 73, 14islfld 34260 1  |-  ( ph  ->  ( G  oF  .x.  ( V  X.  { R } ) )  e.  F )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   _Vcvv 3118   {csn 4033    X. cxp 5003    Fn wfn 5589   -->wf 5590   ` cfv 5594  (class class class)co 6295    oFcof 6533   Basecbs 14507   +g cplusg 14572   .rcmulr 14573  Scalarcsca 14575   .scvsca 14576   Ringcrg 17070   LModclmod 17383  LFnlclfn 34255
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-of 6535  df-om 6696  df-recs 7054  df-rdg 7088  df-er 7323  df-map 7434  df-en 7529  df-dom 7530  df-sdom 7531  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-2 10606  df-ndx 14510  df-slot 14511  df-base 14512  df-sets 14513  df-plusg 14585  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-grp 15929  df-mgp 17014  df-ring 17072  df-lmod 17385  df-lfl 34256
This theorem is referenced by:  lkrsc  34295  lfl1dim  34319  ldualvscl  34337  ldualvsass  34339
  Copyright terms: Public domain W3C validator