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Theorem lflnegl 29888
Description: A functional plus its negative is the zero functional. (This is specialized for the purpose of proving ldualgrp 29958, 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 5555 . . . 4  |-  ( Base `  W )  e.  _V
31, 2eqeltri 2366 . . 3  |-  V  e. 
_V
43a1i 10 . 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 2296 . . . 4  |-  ( Base `  R )  =  (
Base `  R )
9 lflnegcl.f . . . 4  |-  F  =  (LFnl `  W )
107, 8, 1, 9lflf 29875 . . 3  |-  ( ( W  e.  LMod  /\  G  e.  F )  ->  G : V --> ( Base `  R
) )
115, 6, 10syl2anc 642 . 2  |-  ( ph  ->  G : V --> ( Base `  R ) )
12 lflnegl.o . . . 4  |-  .0.  =  ( 0g `  R )
13 fvex 5555 . . . 4  |-  ( 0g
`  R )  e. 
_V
1412, 13eqeltri 2366 . . 3  |-  .0.  e.  _V
1514a1i 10 . 2  |-  ( ph  ->  .0.  e.  _V )
16 lflnegcl.i . . . 4  |-  I  =  ( inv g `  R )
177lmodrng 15651 . . . . 5  |-  ( W  e.  LMod  ->  R  e. 
Ring )
18 rnggrp 15362 . . . . 5  |-  ( R  e.  Ring  ->  R  e. 
Grp )
195, 17, 183syl 18 . . . 4  |-  ( ph  ->  R  e.  Grp )
208, 16, 19grpinvf1o 14554 . . 3  |-  ( ph  ->  I : ( Base `  R ) -1-1-onto-> ( Base `  R
) )
21 f1of 5488 . . 3  |-  ( I : ( Base `  R
)
-1-1-onto-> ( Base `  R )  ->  I : ( Base `  R ) --> ( Base `  R ) )
2220, 21syl 15 . 2  |-  ( ph  ->  I : ( Base `  R ) --> ( Base `  R ) )
23 lflnegcl.n . . 3  |-  N  =  ( x  e.  V  |->  ( I `  ( G `  x )
) )
2423a1i 10 . 2  |-  ( ph  ->  N  =  ( x  e.  V  |->  ( I `
 ( G `  x ) ) ) )
25 lflnegl.p . . . 4  |-  .+  =  ( +g  `  R )
268, 25, 12, 16grplinv 14544 . . 3  |-  ( ( R  e.  Grp  /\  y  e.  ( Base `  R ) )  -> 
( ( I `  y )  .+  y
)  =  .0.  )
2719, 26sylan 457 . 2  |-  ( (
ph  /\  y  e.  ( Base `  R )
)  ->  ( (
I `  y )  .+  y )  =  .0.  )
284, 11, 15, 22, 24, 27caofinvl 6120 1  |-  ( ph  ->  ( N  o F 
.+  G )  =  ( V  X.  {  .0.  } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696   _Vcvv 2801   {csn 3653    e. cmpt 4093    X. cxp 4703   -->wf 5267   -1-1-onto->wf1o 5270   ` cfv 5271  (class class class)co 5874    o Fcof 6092   Basecbs 13164   +g cplusg 13224  Scalarcsca 13227   0gc0g 13416   Grpcgrp 14378   inv gcminusg 14379   Ringcrg 15353   LModclmod 15643  LFnlclfn 29869
This theorem is referenced by:  ldualgrplem  29957
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-riota 6320  df-map 6790  df-0g 13420  df-mnd 14383  df-grp 14505  df-minusg 14506  df-rng 15356  df-lmod 15645  df-lfl 29870
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