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Theorem lflnegl 29242
Description: A functional plus its negative is the zero functional. (This is specialized for the purpose of proving ldualgrp 29312, 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 5675 . . . 4  |-  ( Base `  W )  e.  _V
31, 2eqeltri 2450 . . 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 2380 . . . 4  |-  ( Base `  R )  =  (
Base `  R )
9 lflnegcl.f . . . 4  |-  F  =  (LFnl `  W )
107, 8, 1, 9lflf 29229 . . 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 5675 . . . 4  |-  ( 0g
`  R )  e. 
_V
1412, 13eqeltri 2450 . . 3  |-  .0.  e.  _V
1514a1i 11 . 2  |-  ( ph  ->  .0.  e.  _V )
16 lflnegcl.i . . . 4  |-  I  =  ( inv g `  R )
177lmodrng 15878 . . . . 5  |-  ( W  e.  LMod  ->  R  e. 
Ring )
18 rnggrp 15589 . . . . 5  |-  ( R  e.  Ring  ->  R  e. 
Grp )
195, 17, 183syl 19 . . . 4  |-  ( ph  ->  R  e.  Grp )
208, 16, 19grpinvf1o 14781 . . 3  |-  ( ph  ->  I : ( Base `  R ) -1-1-onto-> ( Base `  R
) )
21 f1of 5607 . . 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 14771 . . 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 6263 1  |-  ( ph  ->  ( N  o F 
.+  G )  =  ( V  X.  {  .0.  } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1717   _Vcvv 2892   {csn 3750    e. cmpt 4200    X. cxp 4809   -->wf 5383   -1-1-onto->wf1o 5386   ` cfv 5387  (class class class)co 6013    o Fcof 6235   Basecbs 13389   +g cplusg 13449  Scalarcsca 13452   0gc0g 13643   Grpcgrp 14605   inv gcminusg 14606   Ringcrg 15580   LModclmod 15870  LFnlclfn 29223
This theorem is referenced by:  ldualgrplem  29311
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-of 6237  df-riota 6478  df-map 6949  df-0g 13647  df-mnd 14610  df-grp 14732  df-minusg 14733  df-rng 15583  df-lmod 15872  df-lfl 29224
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