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Theorem lflnegl 29811
Description: A functional plus its negative is the zero functional. (This is specialized for the purpose of proving ldualgrp 29881, and we do not define a general operation here.) (Contributed by NM, 22-Oct-2014.)
Hypotheses
Ref Expression
lflnegcl.v  |-  V  =  ( Base `  W
)
lflnegcl.r  |-  R  =  (Scalar `  W )
lflnegcl.i  |-  I  =  ( inv g `  R )
lflnegcl.n  |-  N  =  ( x  e.  V  |->  ( I `  ( G `  x )
) )
lflnegcl.f  |-  F  =  (LFnl `  W )
lflnegcl.w  |-  ( ph  ->  W  e.  LMod )
lflnegcl.g  |-  ( ph  ->  G  e.  F )
lflnegl.p  |-  .+  =  ( +g  `  R )
lflnegl.o  |-  .0.  =  ( 0g `  R )
Assertion
Ref Expression
lflnegl  |-  ( ph  ->  ( N  o F 
.+  G )  =  ( V  X.  {  .0.  } ) )
Distinct variable groups:    x, G    x, I    x, R    x, V    x, W    ph, x
Allowed substitution hints:    .+ ( x)    F( x)    N( x)    .0. ( x)

Proof of Theorem lflnegl
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 lflnegcl.v . . . 4  |-  V  =  ( Base `  W
)
2 fvex 5734 . . . 4  |-  ( Base `  W )  e.  _V
31, 2eqeltri 2505 . . 3  |-  V  e. 
_V
43a1i 11 . 2  |-  ( ph  ->  V  e.  _V )
5 lflnegcl.w . . 3  |-  ( ph  ->  W  e.  LMod )
6 lflnegcl.g . . 3  |-  ( ph  ->  G  e.  F )
7 lflnegcl.r . . . 4  |-  R  =  (Scalar `  W )
8 eqid 2435 . . . 4  |-  ( Base `  R )  =  (
Base `  R )
9 lflnegcl.f . . . 4  |-  F  =  (LFnl `  W )
107, 8, 1, 9lflf 29798 . . 3  |-  ( ( W  e.  LMod  /\  G  e.  F )  ->  G : V --> ( Base `  R
) )
115, 6, 10syl2anc 643 . 2  |-  ( ph  ->  G : V --> ( Base `  R ) )
12 lflnegl.o . . . 4  |-  .0.  =  ( 0g `  R )
13 fvex 5734 . . . 4  |-  ( 0g
`  R )  e. 
_V
1412, 13eqeltri 2505 . . 3  |-  .0.  e.  _V
1514a1i 11 . 2  |-  ( ph  ->  .0.  e.  _V )
16 lflnegcl.i . . . 4  |-  I  =  ( inv g `  R )
177lmodrng 15950 . . . . 5  |-  ( W  e.  LMod  ->  R  e. 
Ring )
18 rnggrp 15661 . . . . 5  |-  ( R  e.  Ring  ->  R  e. 
Grp )
195, 17, 183syl 19 . . . 4  |-  ( ph  ->  R  e.  Grp )
208, 16, 19grpinvf1o 14853 . . 3  |-  ( ph  ->  I : ( Base `  R ) -1-1-onto-> ( Base `  R
) )
21 f1of 5666 . . 3  |-  ( I : ( Base `  R
)
-1-1-onto-> ( Base `  R )  ->  I : ( Base `  R ) --> ( Base `  R ) )
2220, 21syl 16 . 2  |-  ( ph  ->  I : ( Base `  R ) --> ( Base `  R ) )
23 lflnegcl.n . . 3  |-  N  =  ( x  e.  V  |->  ( I `  ( G `  x )
) )
2423a1i 11 . 2  |-  ( ph  ->  N  =  ( x  e.  V  |->  ( I `
 ( G `  x ) ) ) )
25 lflnegl.p . . . 4  |-  .+  =  ( +g  `  R )
268, 25, 12, 16grplinv 14843 . . 3  |-  ( ( R  e.  Grp  /\  y  e.  ( Base `  R ) )  -> 
( ( I `  y )  .+  y
)  =  .0.  )
2719, 26sylan 458 . 2  |-  ( (
ph  /\  y  e.  ( Base `  R )
)  ->  ( (
I `  y )  .+  y )  =  .0.  )
284, 11, 15, 22, 24, 27caofinvl 6323 1  |-  ( ph  ->  ( N  o F 
.+  G )  =  ( V  X.  {  .0.  } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   _Vcvv 2948   {csn 3806    e. cmpt 4258    X. cxp 4868   -->wf 5442   -1-1-onto->wf1o 5445   ` cfv 5446  (class class class)co 6073    o Fcof 6295   Basecbs 13461   +g cplusg 13521  Scalarcsca 13524   0gc0g 13715   Grpcgrp 14677   inv gcminusg 14678   Ringcrg 15652   LModclmod 15942  LFnlclfn 29792
This theorem is referenced by:  ldualgrplem  29880
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-of 6297  df-riota 6541  df-map 7012  df-0g 13719  df-mnd 14682  df-grp 14804  df-minusg 14805  df-rng 15655  df-lmod 15944  df-lfl 29793
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