Metamath Proof Explorer

Theorem lfladdass

Description: Associativity of functional addition. (Contributed by NM, 19-Oct-2014)

Ref Expression
Hypotheses 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 lfladdass 
|- ( ph -> ( ( G oF .+ H ) oF .+ I ) = ( G oF .+ ( H oF .+ I ) ) )

Proof

Step Hyp Ref Expression
1 lfladdcl.r 
 |-  R = ( Scalar ` W )
2 lfladdcl.p 
 |-  .+ = ( +g ` R )
3 lfladdcl.f 
 |-  F = ( LFnl ` W )
4 lfladdcl.w 
 |-  ( ph -> W e. LMod )
5 lfladdcl.g 
 |-  ( ph -> G e. F )
6 lfladdcl.h 
 |-  ( ph -> H e. F )
7 lfladdass.i 
 |-  ( ph -> I e. F )
8 fvexd 
 |-  ( ph -> ( Base ` W ) e. _V )
9 eqid 
 |-  ( Base ` R ) = ( Base ` R )
10 eqid 
 |-  ( Base ` W ) = ( Base ` W )
11 1 9 10 3 lflf 
 |-  ( ( W e. LMod /\ G e. F ) -> G : ( Base ` W ) --> ( Base ` R ) )
12 4 5 11 syl2anc 
 |-  ( ph -> G : ( Base ` W ) --> ( Base ` R ) )
13 1 9 10 3 lflf 
 |-  ( ( W e. LMod /\ H e. F ) -> H : ( Base ` W ) --> ( Base ` R ) )
14 4 6 13 syl2anc 
 |-  ( ph -> H : ( Base ` W ) --> ( Base ` R ) )
15 1 9 10 3 lflf 
 |-  ( ( W e. LMod /\ I e. F ) -> I : ( Base ` W ) --> ( Base ` R ) )
16 4 7 15 syl2anc 
 |-  ( ph -> I : ( Base ` W ) --> ( Base ` R ) )
17 1 lmodring 
 |-  ( W e. LMod -> R e. Ring )
18 ringgrp 
 |-  ( R e. Ring -> R e. Grp )
19 4 17 18 3syl 
 |-  ( ph -> R e. Grp )
20 9 2 grpass 
 |-  ( ( R e. Grp /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) /\ z e. ( Base ` R ) ) ) -> ( ( x .+ y ) .+ z ) = ( x .+ ( y .+ z ) ) )
21 19 20 sylan 
 |-  ( ( ph /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) /\ z e. ( Base ` R ) ) ) -> ( ( x .+ y ) .+ z ) = ( x .+ ( y .+ z ) ) )
22 8 12 14 16 21 caofass 
 |-  ( ph -> ( ( G oF .+ H ) oF .+ I ) = ( G oF .+ ( H oF .+ I ) ) )