Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  lfladd0l Unicode version

Theorem lfladd0l 29886
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 5555 . . . 4  |-  ( Base `  W )  e.  _V
31, 2eqeltri 2366 . . 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 2296 . . . 4  |-  ( Base `  R )  =  (
Base `  R )
9 lfladd0l.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 lfladd0l.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 )
167lmodrng 15651 . . . 4  |-  ( W  e.  LMod  ->  R  e. 
Ring )
17 rnggrp 15362 . . . 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 14528 . . 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 6121 1  |-  ( ph  ->  ( ( V  X.  {  .0.  } )  o F  .+  G )  =  G )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696   _Vcvv 2801   {csn 3653    X. cxp 4703   -->wf 5267   ` cfv 5271  (class class class)co 5874    o Fcof 6092   Basecbs 13164   +g cplusg 13224  Scalarcsca 13227   0gc0g 13416   Grpcgrp 14378   Ringcrg 15353   LModclmod 15643  LFnlclfn 29869
This theorem is referenced by:  ldualgrplem  29957  ldual0v  29962
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-rng 15356  df-lmod 15645  df-lfl 29870
  Copyright terms: Public domain W3C validator