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Theorem lfladdass 29190
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 5684 . . 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 2389 . . . 4  |-  ( Base `  R )  =  (
Base `  R )
7 eqid 2389 . . . 4  |-  ( Base `  W )  =  (
Base `  W )
8 lfladdcl.f . . . 4  |-  F  =  (LFnl `  W )
95, 6, 7, 8lflf 29180 . . 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 29180 . . 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 29180 . . 3  |-  ( ( W  e.  LMod  /\  I  e.  F )  ->  I : ( Base `  W
) --> ( Base `  R
) )
163, 14, 15syl2anc 643 . 2  |-  ( ph  ->  I : ( Base `  W ) --> ( Base `  R ) )
175lmodrng 15887 . . . 4  |-  ( W  e.  LMod  ->  R  e. 
Ring )
18 rnggrp 15598 . . . 4  |-  ( R  e.  Ring  ->  R  e. 
Grp )
193, 17, 183syl 19 . . 3  |-  ( ph  ->  R  e.  Grp )
20 lfladdcl.p . . . 4  |-  .+  =  ( +g  `  R )
216, 20grpass 14748 . . 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 6279 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 1649    e. wcel 1717   _Vcvv 2901   -->wf 5392   ` cfv 5396  (class class class)co 6022    o Fcof 6244   Basecbs 13398   +g cplusg 13458  Scalarcsca 13461   Grpcgrp 14614   Ringcrg 15589   LModclmod 15879  LFnlclfn 29174
This theorem is referenced by:  ldualgrplem  29262
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-reu 2658  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-of 6246  df-map 6958  df-mnd 14619  df-grp 14741  df-rng 15592  df-lmod 15881  df-lfl 29175
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