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Theorem lflvscl 33080
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 2455 . 2  |-  ( ph  ->  ( +g  `  W
)  =  ( +g  `  W ) )
4 lflsccl.d . . 3  |-  D  =  (Scalar `  W )
54a1i 11 . 2  |-  ( ph  ->  D  =  (Scalar `  W ) )
6 eqidd 2455 . 2  |-  ( ph  ->  ( .s `  W
)  =  ( .s
`  W ) )
7 lflsccl.k . . 3  |-  K  =  ( Base `  D
)
87a1i 11 . 2  |-  ( ph  ->  K  =  ( Base `  D ) )
9 eqidd 2455 . 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 )
154lmodrng 17082 . . . . 5  |-  ( W  e.  LMod  ->  D  e. 
Ring )
1614, 15syl 16 . . . 4  |-  ( ph  ->  D  e.  Ring )
177, 10rngcl 16784 . . . . 5  |-  ( ( D  e.  Ring  /\  x  e.  K  /\  y  e.  K )  ->  (
x  .x.  y )  e.  K )
18173expb 1189 . . . 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 33066 . . . 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 5710 . . . 4  |-  ( R  e.  K  ->  ( V  X.  { R }
) : V --> K )
2523, 24syl 16 . . 3  |-  ( ph  ->  ( V  X.  { R } ) : V --> K )
26 fvex 5812 . . . . 5  |-  ( Base `  W )  e.  _V
271, 26eqeltri 2538 . . . 4  |-  V  e. 
_V
2827a1i 11 . . 3  |-  ( ph  ->  V  e.  _V )
29 inidm 3670 . . 3  |-  ( V  i^i  V )  =  V
3019, 22, 25, 28, 28, 29off 6447 . 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 994 . . . . . 6  |-  ( (
ph  /\  ( r  e.  K  /\  x  e.  V  /\  y  e.  V ) )  -> 
r  e.  K )
34 simpr2 995 . . . . . 6  |-  ( (
ph  /\  ( r  e.  K  /\  x  e.  V  /\  y  e.  V ) )  ->  x  e.  V )
35 simpr3 996 . . . . . 6  |-  ( (
ph  /\  ( r  e.  K  /\  x  e.  V  /\  y  e.  V ) )  -> 
y  e.  V )
36 eqid 2454 . . . . . . 7  |-  ( +g  `  W )  =  ( +g  `  W )
37 eqid 2454 . . . . . . 7  |-  ( .s
`  W )  =  ( .s `  W
)
38 eqid 2454 . . . . . . 7  |-  ( +g  `  D )  =  ( +g  `  D )
391, 36, 4, 37, 7, 38, 10, 12lfli 33064 . . . . . 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 1231 . . . . 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 6218 . . . 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 33067 . . . . . . 7  |-  ( ( W  e.  LMod  /\  G  e.  F  /\  x  e.  V )  ->  ( G `  x )  e.  K )
4431, 32, 34, 43syl3anc 1219 . . . . . 6  |-  ( (
ph  /\  ( r  e.  K  /\  x  e.  V  /\  y  e.  V ) )  -> 
( G `  x
)  e.  K )
457, 10rngcl 16784 . . . . . 6  |-  ( ( D  e.  Ring  /\  r  e.  K  /\  ( G `  x )  e.  K )  ->  (
r  .x.  ( G `  x ) )  e.  K )
4642, 33, 44, 45syl3anc 1219 . . . . 5  |-  ( (
ph  /\  ( r  e.  K  /\  x  e.  V  /\  y  e.  V ) )  -> 
( r  .x.  ( G `  x )
)  e.  K )
474, 7, 1, 12lflcl 33067 . . . . . 6  |-  ( ( W  e.  LMod  /\  G  e.  F  /\  y  e.  V )  ->  ( G `  y )  e.  K )
4831, 32, 35, 47syl3anc 1219 . . . . 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, 10rngdir 16790 . . . . 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 1221 . . . 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 16787 . . . . . 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 1221 . . . . 5  |-  ( (
ph  /\  ( r  e.  K  /\  x  e.  V  /\  y  e.  V ) )  -> 
( ( r  .x.  ( G `  x ) )  .x.  R )  =  ( r  .x.  ( ( G `  x )  .x.  R
) ) )
5453oveq1d 6218 . . . 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 2499 . . 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 17091 . . . . . 6  |-  ( ( W  e.  LMod  /\  r  e.  K  /\  x  e.  V )  ->  (
r ( .s `  W ) x )  e.  V )
5731, 33, 34, 56syl3anc 1219 . . . . 5  |-  ( (
ph  /\  ( r  e.  K  /\  x  e.  V  /\  y  e.  V ) )  -> 
( r ( .s
`  W ) x )  e.  V )
581, 36lmodvacl 17088 . . . . 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 1219 . . . 4  |-  ( (
ph  /\  ( r  e.  K  /\  x  e.  V  /\  y  e.  V ) )  -> 
( ( r ( .s `  W ) x ) ( +g  `  W ) y )  e.  V )
60 ffn 5670 . . . . . 6  |-  ( G : V --> K  ->  G  Fn  V )
6122, 60syl 16 . . . . 5  |-  ( ph  ->  G  Fn  V )
62 eqidd 2455 . . . . 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 6457 . . . 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 2455 . . . . . . 7  |-  ( (
ph  /\  x  e.  V )  ->  ( G `  x )  =  ( G `  x ) )
6628, 23, 61, 65ofc2 6457 . . . . . 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 6219 . . . 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 2455 . . . . . 6  |-  ( (
ph  /\  y  e.  V )  ->  ( G `  y )  =  ( G `  y ) )
7028, 23, 61, 69ofc2 6457 . . . . 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 6221 . . 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 2505 . 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 33065 1  |-  ( ph  ->  ( G  oF  .x.  ( V  X.  { R } ) )  e.  F )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   _Vcvv 3078   {csn 3988    X. cxp 4949    Fn wfn 5524   -->wf 5525   ` cfv 5529  (class class class)co 6203    oFcof 6431   Basecbs 14295   +g cplusg 14360   .rcmulr 14361  Scalarcsca 14363   .scvsca 14364   Ringcrg 16771   LModclmod 17074  LFnlclfn 33060
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 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9452  ax-resscn 9453  ax-1cn 9454  ax-icn 9455  ax-addcl 9456  ax-addrcl 9457  ax-mulcl 9458  ax-mulrcl 9459  ax-mulcom 9460  ax-addass 9461  ax-mulass 9462  ax-distr 9463  ax-i2m1 9464  ax-1ne0 9465  ax-1rid 9466  ax-rnegex 9467  ax-rrecex 9468  ax-cnre 9469  ax-pre-lttri 9470  ax-pre-lttrn 9471  ax-pre-ltadd 9472  ax-pre-mulgt0 9473
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-of 6433  df-om 6590  df-recs 6945  df-rdg 6979  df-er 7214  df-map 7329  df-en 7424  df-dom 7425  df-sdom 7426  df-pnf 9534  df-mnf 9535  df-xr 9536  df-ltxr 9537  df-le 9538  df-sub 9711  df-neg 9712  df-nn 10437  df-2 10494  df-ndx 14298  df-slot 14299  df-base 14300  df-sets 14301  df-plusg 14373  df-mnd 15537  df-grp 15667  df-mgp 16717  df-rng 16773  df-lmod 17076  df-lfl 33061
This theorem is referenced by:  lkrsc  33100  lfl1dim  33124  ldualvscl  33142  ldualvsass  33144
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