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Theorem lfladd0l 29264
Description: Functional addition with the zero functional. (Contributed by NM, 21-Oct-2014.)
Hypotheses
Ref Expression
lfladd0l.v  |-  V  =  ( Base `  W
)
lfladd0l.r  |-  R  =  (Scalar `  W )
lfladd0l.p  |-  .+  =  ( +g  `  R )
lfladd0l.o  |-  .0.  =  ( 0g `  R )
lfladd0l.f  |-  F  =  (LFnl `  W )
lfladd0l.w  |-  ( ph  ->  W  e.  LMod )
lfladd0l.g  |-  ( ph  ->  G  e.  F )
Assertion
Ref Expression
lfladd0l  |-  ( ph  ->  ( ( V  X.  {  .0.  } )  o F  .+  G )  =  G )

Proof of Theorem lfladd0l
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 lfladd0l.v . . . 4  |-  V  =  ( Base `  W
)
2 fvex 5539 . . . 4  |-  ( Base `  W )  e.  _V
31, 2eqeltri 2353 . . 3  |-  V  e. 
_V
43a1i 10 . 2  |-  ( ph  ->  V  e.  _V )
5 lfladd0l.w . . 3  |-  ( ph  ->  W  e.  LMod )
6 lfladd0l.g . . 3  |-  ( ph  ->  G  e.  F )
7 lfladd0l.r . . . 4  |-  R  =  (Scalar `  W )
8 eqid 2283 . . . 4  |-  ( Base `  R )  =  (
Base `  R )
9 lfladd0l.f . . . 4  |-  F  =  (LFnl `  W )
107, 8, 1, 9lflf 29253 . . 3  |-  ( ( W  e.  LMod  /\  G  e.  F )  ->  G : V --> ( Base `  R
) )
115, 6, 10syl2anc 642 . 2  |-  ( ph  ->  G : V --> ( Base `  R ) )
12 lfladd0l.o . . . 4  |-  .0.  =  ( 0g `  R )
13 fvex 5539 . . . 4  |-  ( 0g
`  R )  e. 
_V
1412, 13eqeltri 2353 . . 3  |-  .0.  e.  _V
1514a1i 10 . 2  |-  ( ph  ->  .0.  e.  _V )
167lmodrng 15635 . . . 4  |-  ( W  e.  LMod  ->  R  e. 
Ring )
17 rnggrp 15346 . . . 4  |-  ( R  e.  Ring  ->  R  e. 
Grp )
185, 16, 173syl 18 . . 3  |-  ( ph  ->  R  e.  Grp )
19 lfladd0l.p . . . 4  |-  .+  =  ( +g  `  R )
208, 19, 12grplid 14512 . . 3  |-  ( ( R  e.  Grp  /\  k  e.  ( Base `  R ) )  -> 
(  .0.  .+  k
)  =  k )
2118, 20sylan 457 . 2  |-  ( (
ph  /\  k  e.  ( Base `  R )
)  ->  (  .0.  .+  k )  =  k )
224, 11, 15, 21caofid0l 6105 1  |-  ( ph  ->  ( ( V  X.  {  .0.  } )  o F  .+  G )  =  G )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   _Vcvv 2788   {csn 3640    X. cxp 4687   -->wf 5251   ` cfv 5255  (class class class)co 5858    o Fcof 6076   Basecbs 13148   +g cplusg 13208  Scalarcsca 13211   0gc0g 13400   Grpcgrp 14362   Ringcrg 15337   LModclmod 15627  LFnlclfn 29247
This theorem is referenced by:  ldualgrplem  29335  ldual0v  29340
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
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  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-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  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-map 6774  df-0g 13404  df-mnd 14367  df-grp 14489  df-rng 15340  df-lmod 15629  df-lfl 29248
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