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Theorem lfladdass 29772
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 5734 . . 3  |-  ( Base `  W )  e.  _V
21a1i 11 . 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 2435 . . . 4  |-  ( Base `  R )  =  (
Base `  R )
7 eqid 2435 . . . 4  |-  ( Base `  W )  =  (
Base `  W )
8 lfladdcl.f . . . 4  |-  F  =  (LFnl `  W )
95, 6, 7, 8lflf 29762 . . 3  |-  ( ( W  e.  LMod  /\  G  e.  F )  ->  G : ( Base `  W
) --> ( Base `  R
) )
103, 4, 9syl2anc 643 . 2  |-  ( ph  ->  G : ( Base `  W ) --> ( Base `  R ) )
11 lfladdcl.h . . 3  |-  ( ph  ->  H  e.  F )
125, 6, 7, 8lflf 29762 . . 3  |-  ( ( W  e.  LMod  /\  H  e.  F )  ->  H : ( Base `  W
) --> ( Base `  R
) )
133, 11, 12syl2anc 643 . 2  |-  ( ph  ->  H : ( Base `  W ) --> ( Base `  R ) )
14 lfladdass.i . . 3  |-  ( ph  ->  I  e.  F )
155, 6, 7, 8lflf 29762 . . 3  |-  ( ( W  e.  LMod  /\  I  e.  F )  ->  I : ( Base `  W
) --> ( Base `  R
) )
163, 14, 15syl2anc 643 . 2  |-  ( ph  ->  I : ( Base `  W ) --> ( Base `  R ) )
175lmodrng 15948 . . . 4  |-  ( W  e.  LMod  ->  R  e. 
Ring )
18 rnggrp 15659 . . . 4  |-  ( R  e.  Ring  ->  R  e. 
Grp )
193, 17, 183syl 19 . . 3  |-  ( ph  ->  R  e.  Grp )
20 lfladdcl.p . . . 4  |-  .+  =  ( +g  `  R )
216, 20grpass 14809 . . 3  |-  ( ( R  e.  Grp  /\  ( x  e.  ( Base `  R )  /\  y  e.  ( Base `  R )  /\  z  e.  ( Base `  R
) ) )  -> 
( ( x  .+  y )  .+  z
)  =  ( x 
.+  ( y  .+  z ) ) )
2219, 21sylan 458 . 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 6330 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 936    = wceq 1652    e. wcel 1725   _Vcvv 2948   -->wf 5442   ` cfv 5446  (class class class)co 6073    o Fcof 6295   Basecbs 13459   +g cplusg 13519  Scalarcsca 13522   Grpcgrp 14675   Ringcrg 15650   LModclmod 15940  LFnlclfn 29756
This theorem is referenced by:  ldualgrplem  29844
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-of 6297  df-map 7012  df-mnd 14680  df-grp 14802  df-rng 15653  df-lmod 15942  df-lfl 29757
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