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

Theorem lfladdass 29263
Description: Associativity of functional addition. (Contributed by NM, 19-Oct-2014.)
Hypotheses
Ref Expression
lfladdcl.r  |-  R  =  (Scalar `  W )
lfladdcl.p  |-  .+  =  ( +g  `  R )
lfladdcl.f  |-  F  =  (LFnl `  W )
lfladdcl.w  |-  ( ph  ->  W  e.  LMod )
lfladdcl.g  |-  ( ph  ->  G  e.  F )
lfladdcl.h  |-  ( ph  ->  H  e.  F )
lfladdass.i  |-  ( ph  ->  I  e.  F )
Assertion
Ref Expression
lfladdass  |-  ( ph  ->  ( ( G  o F  .+  H )  o F  .+  I )  =  ( G  o F  .+  ( H  o F  .+  I ) ) )

Proof of Theorem lfladdass
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fvex 5539 . . 3  |-  ( Base `  W )  e.  _V
21a1i 10 . 2  |-  ( ph  ->  ( Base `  W
)  e.  _V )
3 lfladdcl.w . . 3  |-  ( ph  ->  W  e.  LMod )
4 lfladdcl.g . . 3  |-  ( ph  ->  G  e.  F )
5 lfladdcl.r . . . 4  |-  R  =  (Scalar `  W )
6 eqid 2283 . . . 4  |-  ( Base `  R )  =  (
Base `  R )
7 eqid 2283 . . . 4  |-  ( Base `  W )  =  (
Base `  W )
8 lfladdcl.f . . . 4  |-  F  =  (LFnl `  W )
95, 6, 7, 8lflf 29253 . . 3  |-  ( ( W  e.  LMod  /\  G  e.  F )  ->  G : ( Base `  W
) --> ( Base `  R
) )
103, 4, 9syl2anc 642 . 2  |-  ( ph  ->  G : ( Base `  W ) --> ( Base `  R ) )
11 lfladdcl.h . . 3  |-  ( ph  ->  H  e.  F )
125, 6, 7, 8lflf 29253 . . 3  |-  ( ( W  e.  LMod  /\  H  e.  F )  ->  H : ( Base `  W
) --> ( Base `  R
) )
133, 11, 12syl2anc 642 . 2  |-  ( ph  ->  H : ( Base `  W ) --> ( Base `  R ) )
14 lfladdass.i . . 3  |-  ( ph  ->  I  e.  F )
155, 6, 7, 8lflf 29253 . . 3  |-  ( ( W  e.  LMod  /\  I  e.  F )  ->  I : ( Base `  W
) --> ( Base `  R
) )
163, 14, 15syl2anc 642 . 2  |-  ( ph  ->  I : ( Base `  W ) --> ( Base `  R ) )
175lmodrng 15635 . . . 4  |-  ( W  e.  LMod  ->  R  e. 
Ring )
18 rnggrp 15346 . . . 4  |-  ( R  e.  Ring  ->  R  e. 
Grp )
193, 17, 183syl 18 . . 3  |-  ( ph  ->  R  e.  Grp )
20 lfladdcl.p . . . 4  |-  .+  =  ( +g  `  R )
216, 20grpass 14496 . . 3  |-  ( ( R  e.  Grp  /\  ( x  e.  ( Base `  R )  /\  y  e.  ( Base `  R )  /\  z  e.  ( Base `  R
) ) )  -> 
( ( x  .+  y )  .+  z
)  =  ( x 
.+  ( y  .+  z ) ) )
2219, 21sylan 457 . 2  |-  ( (
ph  /\  ( x  e.  ( Base `  R
)  /\  y  e.  ( Base `  R )  /\  z  e.  ( Base `  R ) ) )  ->  ( (
x  .+  y )  .+  z )  =  ( x  .+  ( y 
.+  z ) ) )
232, 10, 13, 16, 22caofass 6111 1  |-  ( ph  ->  ( ( G  o F  .+  H )  o F  .+  I )  =  ( G  o F  .+  ( H  o F  .+  I ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934    = wceq 1623    e. wcel 1684   _Vcvv 2788   -->wf 5251   ` cfv 5255  (class class class)co 5858    o Fcof 6076   Basecbs 13148   +g cplusg 13208  Scalarcsca 13211   Grpcgrp 14362   Ringcrg 15337   LModclmod 15627  LFnlclfn 29247
This theorem is referenced by:  ldualgrplem  29335
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-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-map 6774  df-mnd 14367  df-grp 14489  df-rng 15340  df-lmod 15629  df-lfl 29248
  Copyright terms: Public domain W3C validator