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Theorem lflvscl 29267
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  o F 
.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 10 . 2  |-  ( ph  ->  V  =  ( Base `  W ) )
3 eqidd 2284 . 2  |-  ( ph  ->  ( +g  `  W
)  =  ( +g  `  W ) )
4 lflsccl.d . . 3  |-  D  =  (Scalar `  W )
54a1i 10 . 2  |-  ( ph  ->  D  =  (Scalar `  W ) )
6 eqidd 2284 . 2  |-  ( ph  ->  ( .s `  W
)  =  ( .s
`  W ) )
7 lflsccl.k . . 3  |-  K  =  ( Base `  D
)
87a1i 10 . 2  |-  ( ph  ->  K  =  ( Base `  D ) )
9 eqidd 2284 . 2  |-  ( ph  ->  ( +g  `  D
)  =  ( +g  `  D ) )
10 lflsccl.t . . 3  |-  .x.  =  ( .r `  D )
1110a1i 10 . 2  |-  ( ph  ->  .x.  =  ( .r
`  D ) )
12 lflsccl.f . . 3  |-  F  =  (LFnl `  W )
1312a1i 10 . 2  |-  ( ph  ->  F  =  (LFnl `  W ) )
14 lflsccl.w . . . . 5  |-  ( ph  ->  W  e.  LMod )
154lmodrng 15635 . . . . 5  |-  ( W  e.  LMod  ->  D  e. 
Ring )
1614, 15syl 15 . . . 4  |-  ( ph  ->  D  e.  Ring )
177, 10rngcl 15354 . . . . 5  |-  ( ( D  e.  Ring  /\  x  e.  K  /\  y  e.  K )  ->  (
x  .x.  y )  e.  K )
18173expb 1152 . . . 4  |-  ( ( D  e.  Ring  /\  (
x  e.  K  /\  y  e.  K )
)  ->  ( x  .x.  y )  e.  K
)
1916, 18sylan 457 . . 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 29253 . . . 4  |-  ( ( W  e.  LMod  /\  G  e.  F )  ->  G : V --> K )
2214, 20, 21syl2anc 642 . . 3  |-  ( ph  ->  G : V --> K )
23 lflsccl.r . . . 4  |-  ( ph  ->  R  e.  K )
24 fconst6g 5430 . . . 4  |-  ( R  e.  K  ->  ( V  X.  { R }
) : V --> K )
2523, 24syl 15 . . 3  |-  ( ph  ->  ( V  X.  { R } ) : V --> K )
26 fvex 5539 . . . . 5  |-  ( Base `  W )  e.  _V
271, 26eqeltri 2353 . . . 4  |-  V  e. 
_V
2827a1i 10 . . 3  |-  ( ph  ->  V  e.  _V )
29 inidm 3378 . . 3  |-  ( V  i^i  V )  =  V
3019, 22, 25, 28, 28, 29off 6093 . 2  |-  ( ph  ->  ( G  o F 
.x.  ( V  X.  { R } ) ) : V --> K )
3114adantr 451 . . . . . 6  |-  ( (
ph  /\  ( r  e.  K  /\  x  e.  V  /\  y  e.  V ) )  ->  W  e.  LMod )
3220adantr 451 . . . . . 6  |-  ( (
ph  /\  ( r  e.  K  /\  x  e.  V  /\  y  e.  V ) )  ->  G  e.  F )
33 simpr1 961 . . . . . 6  |-  ( (
ph  /\  ( r  e.  K  /\  x  e.  V  /\  y  e.  V ) )  -> 
r  e.  K )
34 simpr2 962 . . . . . 6  |-  ( (
ph  /\  ( r  e.  K  /\  x  e.  V  /\  y  e.  V ) )  ->  x  e.  V )
35 simpr3 963 . . . . . 6  |-  ( (
ph  /\  ( r  e.  K  /\  x  e.  V  /\  y  e.  V ) )  -> 
y  e.  V )
36 eqid 2283 . . . . . . 7  |-  ( +g  `  W )  =  ( +g  `  W )
37 eqid 2283 . . . . . . 7  |-  ( .s
`  W )  =  ( .s `  W
)
38 eqid 2283 . . . . . . 7  |-  ( +g  `  D )  =  ( +g  `  D )
391, 36, 4, 37, 7, 38, 10, 12lfli 29251 . . . . . 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 1194 . . . . 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 5873 . . . 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 451 . . . . 5  |-  ( (
ph  /\  ( r  e.  K  /\  x  e.  V  /\  y  e.  V ) )  ->  D  e.  Ring )
434, 7, 1, 12lflcl 29254 . . . . . . 7  |-  ( ( W  e.  LMod  /\  G  e.  F  /\  x  e.  V )  ->  ( G `  x )  e.  K )
4431, 32, 34, 43syl3anc 1182 . . . . . 6  |-  ( (
ph  /\  ( r  e.  K  /\  x  e.  V  /\  y  e.  V ) )  -> 
( G `  x
)  e.  K )
457, 10rngcl 15354 . . . . . 6  |-  ( ( D  e.  Ring  /\  r  e.  K  /\  ( G `  x )  e.  K )  ->  (
r  .x.  ( G `  x ) )  e.  K )
4642, 33, 44, 45syl3anc 1182 . . . . 5  |-  ( (
ph  /\  ( r  e.  K  /\  x  e.  V  /\  y  e.  V ) )  -> 
( r  .x.  ( G `  x )
)  e.  K )
474, 7, 1, 12lflcl 29254 . . . . . 6  |-  ( ( W  e.  LMod  /\  G  e.  F  /\  y  e.  V )  ->  ( G `  y )  e.  K )
4831, 32, 35, 47syl3anc 1182 . . . . 5  |-  ( (
ph  /\  ( r  e.  K  /\  x  e.  V  /\  y  e.  V ) )  -> 
( G `  y
)  e.  K )
4923adantr 451 . . . . 5  |-  ( (
ph  /\  ( r  e.  K  /\  x  e.  V  /\  y  e.  V ) )  ->  R  e.  K )
507, 38, 10rngdir 15360 . . . . 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 1184 . . . 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, 10rngass 15357 . . . . . 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 1184 . . . . 5  |-  ( (
ph  /\  ( r  e.  K  /\  x  e.  V  /\  y  e.  V ) )  -> 
( ( r  .x.  ( G `  x ) )  .x.  R )  =  ( r  .x.  ( ( G `  x )  .x.  R
) ) )
5453oveq1d 5873 . . . 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 2319 . . 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 15644 . . . . . 6  |-  ( ( W  e.  LMod  /\  r  e.  K  /\  x  e.  V )  ->  (
r ( .s `  W ) x )  e.  V )
5731, 33, 34, 56syl3anc 1182 . . . . 5  |-  ( (
ph  /\  ( r  e.  K  /\  x  e.  V  /\  y  e.  V ) )  -> 
( r ( .s
`  W ) x )  e.  V )
581, 36lmodvacl 15641 . . . . 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 1182 . . . 4  |-  ( (
ph  /\  ( r  e.  K  /\  x  e.  V  /\  y  e.  V ) )  -> 
( ( r ( .s `  W ) x ) ( +g  `  W ) y )  e.  V )
60 ffn 5389 . . . . . 6  |-  ( G : V --> K  ->  G  Fn  V )
6122, 60syl 15 . . . . 5  |-  ( ph  ->  G  Fn  V )
62 eqidd 2284 . . . . 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 6101 . . . 4  |-  ( (
ph  /\  ( (
r ( .s `  W ) x ) ( +g  `  W
) y )  e.  V )  ->  (
( G  o F 
.x.  ( V  X.  { R } ) ) `
 ( ( r ( .s `  W
) x ) ( +g  `  W ) y ) )  =  ( ( G `  ( ( r ( .s `  W ) x ) ( +g  `  W ) y ) )  .x.  R ) )
6459, 63syldan 456 . . 3  |-  ( (
ph  /\  ( r  e.  K  /\  x  e.  V  /\  y  e.  V ) )  -> 
( ( G  o F  .x.  ( V  X.  { R } ) ) `
 ( ( r ( .s `  W
) x ) ( +g  `  W ) y ) )  =  ( ( G `  ( ( r ( .s `  W ) x ) ( +g  `  W ) y ) )  .x.  R ) )
65 eqidd 2284 . . . . . . 7  |-  ( (
ph  /\  x  e.  V )  ->  ( G `  x )  =  ( G `  x ) )
6628, 23, 61, 65ofc2 6101 . . . . . 6  |-  ( (
ph  /\  x  e.  V )  ->  (
( G  o F 
.x.  ( V  X.  { R } ) ) `
 x )  =  ( ( G `  x )  .x.  R
) )
6734, 66syldan 456 . . . . 5  |-  ( (
ph  /\  ( r  e.  K  /\  x  e.  V  /\  y  e.  V ) )  -> 
( ( G  o F  .x.  ( V  X.  { R } ) ) `
 x )  =  ( ( G `  x )  .x.  R
) )
6867oveq2d 5874 . . . 4  |-  ( (
ph  /\  ( r  e.  K  /\  x  e.  V  /\  y  e.  V ) )  -> 
( r  .x.  (
( G  o F 
.x.  ( V  X.  { R } ) ) `
 x ) )  =  ( r  .x.  ( ( G `  x )  .x.  R
) ) )
69 eqidd 2284 . . . . . 6  |-  ( (
ph  /\  y  e.  V )  ->  ( G `  y )  =  ( G `  y ) )
7028, 23, 61, 69ofc2 6101 . . . . 5  |-  ( (
ph  /\  y  e.  V )  ->  (
( G  o F 
.x.  ( V  X.  { R } ) ) `
 y )  =  ( ( G `  y )  .x.  R
) )
7135, 70syldan 456 . . . 4  |-  ( (
ph  /\  ( r  e.  K  /\  x  e.  V  /\  y  e.  V ) )  -> 
( ( G  o F  .x.  ( V  X.  { R } ) ) `
 y )  =  ( ( G `  y )  .x.  R
) )
7268, 71oveq12d 5876 . . 3  |-  ( (
ph  /\  ( r  e.  K  /\  x  e.  V  /\  y  e.  V ) )  -> 
( ( r  .x.  ( ( G  o F  .x.  ( V  X.  { R } ) ) `
 x ) ) ( +g  `  D
) ( ( G  o F  .x.  ( V  X.  { R }
) ) `  y
) )  =  ( ( r  .x.  (
( G `  x
)  .x.  R )
) ( +g  `  D
) ( ( G `
 y )  .x.  R ) ) )
7355, 64, 723eqtr4d 2325 . 2  |-  ( (
ph  /\  ( r  e.  K  /\  x  e.  V  /\  y  e.  V ) )  -> 
( ( G  o F  .x.  ( V  X.  { R } ) ) `
 ( ( r ( .s `  W
) x ) ( +g  `  W ) y ) )  =  ( ( r  .x.  ( ( G  o F  .x.  ( V  X.  { R } ) ) `
 x ) ) ( +g  `  D
) ( ( G  o F  .x.  ( V  X.  { R }
) ) `  y
) ) )
742, 3, 5, 6, 8, 9, 11, 13, 30, 73, 14islfld 29252 1  |-  ( ph  ->  ( G  o F 
.x.  ( V  X.  { R } ) )  e.  F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   _Vcvv 2788   {csn 3640    X. cxp 4687    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858    o Fcof 6076   Basecbs 13148   +g cplusg 13208   .rcmulr 13209  Scalarcsca 13211   .scvsca 13212   Ringcrg 15337   LModclmod 15627  LFnlclfn 29247
This theorem is referenced by:  lkrsc  29287  lfl1dim  29311  ldualvscl  29329  ldualvsass  29331
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-plusg 13221  df-mnd 14367  df-grp 14489  df-mgp 15326  df-rng 15340  df-lmod 15629  df-lfl 29248
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