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Theorem lmhmvsca 16050
Description: The pointwise scalar product of a linear function and a constant is linear, over a commutative ring. (Contributed by Mario Carneiro, 22-Sep-2015.)
Hypotheses
Ref Expression
lmhmvsca.v  |-  V  =  ( Base `  M
)
lmhmvsca.s  |-  .x.  =  ( .s `  N )
lmhmvsca.j  |-  J  =  (Scalar `  N )
lmhmvsca.k  |-  K  =  ( Base `  J
)
Assertion
Ref Expression
lmhmvsca  |-  ( ( J  e.  CRing  /\  A  e.  K  /\  F  e.  ( M LMHom  N ) )  ->  ( ( V  X.  { A }
)  o F  .x.  F )  e.  ( M LMHom  N ) )

Proof of Theorem lmhmvsca
Dummy variables  v  u  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lmhmvsca.v . 2  |-  V  =  ( Base `  M
)
2 eqid 2389 . 2  |-  ( .s
`  M )  =  ( .s `  M
)
3 lmhmvsca.s . 2  |-  .x.  =  ( .s `  N )
4 eqid 2389 . 2  |-  (Scalar `  M )  =  (Scalar `  M )
5 lmhmvsca.j . 2  |-  J  =  (Scalar `  N )
6 eqid 2389 . 2  |-  ( Base `  (Scalar `  M )
)  =  ( Base `  (Scalar `  M )
)
7 lmhmlmod1 16038 . . 3  |-  ( F  e.  ( M LMHom  N
)  ->  M  e.  LMod )
873ad2ant3 980 . 2  |-  ( ( J  e.  CRing  /\  A  e.  K  /\  F  e.  ( M LMHom  N ) )  ->  M  e.  LMod )
9 lmhmlmod2 16037 . . 3  |-  ( F  e.  ( M LMHom  N
)  ->  N  e.  LMod )
1093ad2ant3 980 . 2  |-  ( ( J  e.  CRing  /\  A  e.  K  /\  F  e.  ( M LMHom  N ) )  ->  N  e.  LMod )
114, 5lmhmsca 16035 . . 3  |-  ( F  e.  ( M LMHom  N
)  ->  J  =  (Scalar `  M ) )
12113ad2ant3 980 . 2  |-  ( ( J  e.  CRing  /\  A  e.  K  /\  F  e.  ( M LMHom  N ) )  ->  J  =  (Scalar `  M ) )
13 fvex 5684 . . . . . . 7  |-  ( Base `  M )  e.  _V
141, 13eqeltri 2459 . . . . . 6  |-  V  e. 
_V
1514a1i 11 . . . . 5  |-  ( ( J  e.  CRing  /\  A  e.  K  /\  F  e.  ( M LMHom  N ) )  ->  V  e.  _V )
16 simpl2 961 . . . . 5  |-  ( ( ( J  e.  CRing  /\  A  e.  K  /\  F  e.  ( M LMHom  N ) )  /\  v  e.  V )  ->  A  e.  K )
17 eqid 2389 . . . . . . . 8  |-  ( Base `  N )  =  (
Base `  N )
181, 17lmhmf 16039 . . . . . . 7  |-  ( F  e.  ( M LMHom  N
)  ->  F : V
--> ( Base `  N
) )
19183ad2ant3 980 . . . . . 6  |-  ( ( J  e.  CRing  /\  A  e.  K  /\  F  e.  ( M LMHom  N ) )  ->  F : V
--> ( Base `  N
) )
2019ffvelrnda 5811 . . . . 5  |-  ( ( ( J  e.  CRing  /\  A  e.  K  /\  F  e.  ( M LMHom  N ) )  /\  v  e.  V )  ->  ( F `  v )  e.  ( Base `  N
) )
21 fconstmpt 4863 . . . . . 6  |-  ( V  X.  { A }
)  =  ( v  e.  V  |->  A )
2221a1i 11 . . . . 5  |-  ( ( J  e.  CRing  /\  A  e.  K  /\  F  e.  ( M LMHom  N ) )  ->  ( V  X.  { A } )  =  ( v  e.  V  |->  A ) )
2319feqmptd 5720 . . . . 5  |-  ( ( J  e.  CRing  /\  A  e.  K  /\  F  e.  ( M LMHom  N ) )  ->  F  =  ( v  e.  V  |->  ( F `  v
) ) )
2415, 16, 20, 22, 23offval2 6263 . . . 4  |-  ( ( J  e.  CRing  /\  A  e.  K  /\  F  e.  ( M LMHom  N ) )  ->  ( ( V  X.  { A }
)  o F  .x.  F )  =  ( v  e.  V  |->  ( A  .x.  ( F `
 v ) ) ) )
25 eqidd 2390 . . . . 5  |-  ( ( J  e.  CRing  /\  A  e.  K  /\  F  e.  ( M LMHom  N ) )  ->  ( u  e.  ( Base `  N
)  |->  ( A  .x.  u ) )  =  ( u  e.  (
Base `  N )  |->  ( A  .x.  u
) ) )
26 oveq2 6030 . . . . 5  |-  ( u  =  ( F `  v )  ->  ( A  .x.  u )  =  ( A  .x.  ( F `  v )
) )
2720, 23, 25, 26fmptco 5842 . . . 4  |-  ( ( J  e.  CRing  /\  A  e.  K  /\  F  e.  ( M LMHom  N ) )  ->  ( (
u  e.  ( Base `  N )  |->  ( A 
.x.  u ) )  o.  F )  =  ( v  e.  V  |->  ( A  .x.  ( F `  v )
) ) )
2824, 27eqtr4d 2424 . . 3  |-  ( ( J  e.  CRing  /\  A  e.  K  /\  F  e.  ( M LMHom  N ) )  ->  ( ( V  X.  { A }
)  o F  .x.  F )  =  ( ( u  e.  (
Base `  N )  |->  ( A  .x.  u
) )  o.  F
) )
29 simp2 958 . . . . 5  |-  ( ( J  e.  CRing  /\  A  e.  K  /\  F  e.  ( M LMHom  N ) )  ->  A  e.  K )
30 lmhmvsca.k . . . . . 6  |-  K  =  ( Base `  J
)
3117, 5, 3, 30lmodvsghm 15934 . . . . 5  |-  ( ( N  e.  LMod  /\  A  e.  K )  ->  (
u  e.  ( Base `  N )  |->  ( A 
.x.  u ) )  e.  ( N  GrpHom  N ) )
3210, 29, 31syl2anc 643 . . . 4  |-  ( ( J  e.  CRing  /\  A  e.  K  /\  F  e.  ( M LMHom  N ) )  ->  ( u  e.  ( Base `  N
)  |->  ( A  .x.  u ) )  e.  ( N  GrpHom  N ) )
33 lmghm 16036 . . . . 5  |-  ( F  e.  ( M LMHom  N
)  ->  F  e.  ( M  GrpHom  N ) )
34333ad2ant3 980 . . . 4  |-  ( ( J  e.  CRing  /\  A  e.  K  /\  F  e.  ( M LMHom  N ) )  ->  F  e.  ( M  GrpHom  N ) )
35 ghmco 14954 . . . 4  |-  ( ( ( u  e.  (
Base `  N )  |->  ( A  .x.  u
) )  e.  ( N  GrpHom  N )  /\  F  e.  ( M  GrpHom  N ) )  -> 
( ( u  e.  ( Base `  N
)  |->  ( A  .x.  u ) )  o.  F )  e.  ( M  GrpHom  N ) )
3632, 34, 35syl2anc 643 . . 3  |-  ( ( J  e.  CRing  /\  A  e.  K  /\  F  e.  ( M LMHom  N ) )  ->  ( (
u  e.  ( Base `  N )  |->  ( A 
.x.  u ) )  o.  F )  e.  ( M  GrpHom  N ) )
3728, 36eqeltrd 2463 . 2  |-  ( ( J  e.  CRing  /\  A  e.  K  /\  F  e.  ( M LMHom  N ) )  ->  ( ( V  X.  { A }
)  o F  .x.  F )  e.  ( M  GrpHom  N ) )
38 simpl1 960 . . . . . 6  |-  ( ( ( J  e.  CRing  /\  A  e.  K  /\  F  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  V ) )  ->  J  e.  CRing )
39 simpl2 961 . . . . . 6  |-  ( ( ( J  e.  CRing  /\  A  e.  K  /\  F  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  V ) )  ->  A  e.  K )
40 simprl 733 . . . . . . 7  |-  ( ( ( J  e.  CRing  /\  A  e.  K  /\  F  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  V ) )  ->  x  e.  ( Base `  (Scalar `  M )
) )
4112fveq2d 5674 . . . . . . . . 9  |-  ( ( J  e.  CRing  /\  A  e.  K  /\  F  e.  ( M LMHom  N ) )  ->  ( Base `  J )  =  (
Base `  (Scalar `  M
) ) )
4230, 41syl5eq 2433 . . . . . . . 8  |-  ( ( J  e.  CRing  /\  A  e.  K  /\  F  e.  ( M LMHom  N ) )  ->  K  =  ( Base `  (Scalar `  M
) ) )
4342adantr 452 . . . . . . 7  |-  ( ( ( J  e.  CRing  /\  A  e.  K  /\  F  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  V ) )  ->  K  =  ( Base `  (Scalar `  M )
) )
4440, 43eleqtrrd 2466 . . . . . 6  |-  ( ( ( J  e.  CRing  /\  A  e.  K  /\  F  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  V ) )  ->  x  e.  K )
45 eqid 2389 . . . . . . 7  |-  ( .r
`  J )  =  ( .r `  J
)
4630, 45crngcom 15607 . . . . . 6  |-  ( ( J  e.  CRing  /\  A  e.  K  /\  x  e.  K )  ->  ( A ( .r `  J ) x )  =  ( x ( .r `  J ) A ) )
4738, 39, 44, 46syl3anc 1184 . . . . 5  |-  ( ( ( J  e.  CRing  /\  A  e.  K  /\  F  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  V ) )  -> 
( A ( .r
`  J ) x )  =  ( x ( .r `  J
) A ) )
4847oveq1d 6037 . . . 4  |-  ( ( ( J  e.  CRing  /\  A  e.  K  /\  F  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  V ) )  -> 
( ( A ( .r `  J ) x )  .x.  ( F `  y )
)  =  ( ( x ( .r `  J ) A ) 
.x.  ( F `  y ) ) )
4910adantr 452 . . . . 5  |-  ( ( ( J  e.  CRing  /\  A  e.  K  /\  F  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  V ) )  ->  N  e.  LMod )
5019adantr 452 . . . . . 6  |-  ( ( ( J  e.  CRing  /\  A  e.  K  /\  F  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  V ) )  ->  F : V --> ( Base `  N ) )
51 simprr 734 . . . . . 6  |-  ( ( ( J  e.  CRing  /\  A  e.  K  /\  F  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  V ) )  -> 
y  e.  V )
5250, 51ffvelrnd 5812 . . . . 5  |-  ( ( ( J  e.  CRing  /\  A  e.  K  /\  F  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  V ) )  -> 
( F `  y
)  e.  ( Base `  N ) )
5317, 5, 3, 30, 45lmodvsass 15904 . . . . 5  |-  ( ( N  e.  LMod  /\  ( A  e.  K  /\  x  e.  K  /\  ( F `  y )  e.  ( Base `  N
) ) )  -> 
( ( A ( .r `  J ) x )  .x.  ( F `  y )
)  =  ( A 
.x.  ( x  .x.  ( F `  y ) ) ) )
5449, 39, 44, 52, 53syl13anc 1186 . . . 4  |-  ( ( ( J  e.  CRing  /\  A  e.  K  /\  F  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  V ) )  -> 
( ( A ( .r `  J ) x )  .x.  ( F `  y )
)  =  ( A 
.x.  ( x  .x.  ( F `  y ) ) ) )
5517, 5, 3, 30, 45lmodvsass 15904 . . . . 5  |-  ( ( N  e.  LMod  /\  (
x  e.  K  /\  A  e.  K  /\  ( F `  y )  e.  ( Base `  N
) ) )  -> 
( ( x ( .r `  J ) A )  .x.  ( F `  y )
)  =  ( x 
.x.  ( A  .x.  ( F `  y ) ) ) )
5649, 44, 39, 52, 55syl13anc 1186 . . . 4  |-  ( ( ( J  e.  CRing  /\  A  e.  K  /\  F  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  V ) )  -> 
( ( x ( .r `  J ) A )  .x.  ( F `  y )
)  =  ( x 
.x.  ( A  .x.  ( F `  y ) ) ) )
5748, 54, 563eqtr3d 2429 . . 3  |-  ( ( ( J  e.  CRing  /\  A  e.  K  /\  F  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  V ) )  -> 
( A  .x.  (
x  .x.  ( F `  y ) ) )  =  ( x  .x.  ( A  .x.  ( F `
 y ) ) ) )
581, 4, 2, 6lmodvscl 15896 . . . . . 6  |-  ( ( M  e.  LMod  /\  x  e.  ( Base `  (Scalar `  M ) )  /\  y  e.  V )  ->  ( x ( .s
`  M ) y )  e.  V )
59583expb 1154 . . . . 5  |-  ( ( M  e.  LMod  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  V ) )  -> 
( x ( .s
`  M ) y )  e.  V )
608, 59sylan 458 . . . 4  |-  ( ( ( J  e.  CRing  /\  A  e.  K  /\  F  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  V ) )  -> 
( x ( .s
`  M ) y )  e.  V )
6114a1i 11 . . . . 5  |-  ( ( ( J  e.  CRing  /\  A  e.  K  /\  F  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  V ) )  ->  V  e.  _V )
62 ffn 5533 . . . . . . 7  |-  ( F : V --> ( Base `  N )  ->  F  Fn  V )
6319, 62syl 16 . . . . . 6  |-  ( ( J  e.  CRing  /\  A  e.  K  /\  F  e.  ( M LMHom  N ) )  ->  F  Fn  V )
6463adantr 452 . . . . 5  |-  ( ( ( J  e.  CRing  /\  A  e.  K  /\  F  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  V ) )  ->  F  Fn  V )
654, 6, 1, 2, 3lmhmlin 16040 . . . . . . . 8  |-  ( ( F  e.  ( M LMHom 
N )  /\  x  e.  ( Base `  (Scalar `  M ) )  /\  y  e.  V )  ->  ( F `  (
x ( .s `  M ) y ) )  =  ( x 
.x.  ( F `  y ) ) )
66653expb 1154 . . . . . . 7  |-  ( ( F  e.  ( M LMHom 
N )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  V ) )  -> 
( F `  (
x ( .s `  M ) y ) )  =  ( x 
.x.  ( F `  y ) ) )
67663ad2antl3 1121 . . . . . 6  |-  ( ( ( J  e.  CRing  /\  A  e.  K  /\  F  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  V ) )  -> 
( F `  (
x ( .s `  M ) y ) )  =  ( x 
.x.  ( F `  y ) ) )
6867adantr 452 . . . . 5  |-  ( ( ( ( J  e. 
CRing  /\  A  e.  K  /\  F  e.  ( M LMHom  N ) )  /\  ( x  e.  ( Base `  (Scalar `  M
) )  /\  y  e.  V ) )  /\  ( x ( .s
`  M ) y )  e.  V )  ->  ( F `  ( x ( .s
`  M ) y ) )  =  ( x  .x.  ( F `
 y ) ) )
6961, 39, 64, 68ofc1 6268 . . . 4  |-  ( ( ( ( J  e. 
CRing  /\  A  e.  K  /\  F  e.  ( M LMHom  N ) )  /\  ( x  e.  ( Base `  (Scalar `  M
) )  /\  y  e.  V ) )  /\  ( x ( .s
`  M ) y )  e.  V )  ->  ( ( ( V  X.  { A } )  o F 
.x.  F ) `  ( x ( .s
`  M ) y ) )  =  ( A  .x.  ( x 
.x.  ( F `  y ) ) ) )
7060, 69mpdan 650 . . 3  |-  ( ( ( J  e.  CRing  /\  A  e.  K  /\  F  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  V ) )  -> 
( ( ( V  X.  { A }
)  o F  .x.  F ) `  (
x ( .s `  M ) y ) )  =  ( A 
.x.  ( x  .x.  ( F `  y ) ) ) )
71 eqidd 2390 . . . . . 6  |-  ( ( ( ( J  e. 
CRing  /\  A  e.  K  /\  F  e.  ( M LMHom  N ) )  /\  ( x  e.  ( Base `  (Scalar `  M
) )  /\  y  e.  V ) )  /\  y  e.  V )  ->  ( F `  y
)  =  ( F `
 y ) )
7261, 39, 64, 71ofc1 6268 . . . . 5  |-  ( ( ( ( J  e. 
CRing  /\  A  e.  K  /\  F  e.  ( M LMHom  N ) )  /\  ( x  e.  ( Base `  (Scalar `  M
) )  /\  y  e.  V ) )  /\  y  e.  V )  ->  ( ( ( V  X.  { A }
)  o F  .x.  F ) `  y
)  =  ( A 
.x.  ( F `  y ) ) )
7351, 72mpdan 650 . . . 4  |-  ( ( ( J  e.  CRing  /\  A  e.  K  /\  F  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  V ) )  -> 
( ( ( V  X.  { A }
)  o F  .x.  F ) `  y
)  =  ( A 
.x.  ( F `  y ) ) )
7473oveq2d 6038 . . 3  |-  ( ( ( J  e.  CRing  /\  A  e.  K  /\  F  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  V ) )  -> 
( x  .x.  (
( ( V  X.  { A } )  o F  .x.  F ) `
 y ) )  =  ( x  .x.  ( A  .x.  ( F `
 y ) ) ) )
7557, 70, 743eqtr4d 2431 . 2  |-  ( ( ( J  e.  CRing  /\  A  e.  K  /\  F  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  V ) )  -> 
( ( ( V  X.  { A }
)  o F  .x.  F ) `  (
x ( .s `  M ) y ) )  =  ( x 
.x.  ( ( ( V  X.  { A } )  o F 
.x.  F ) `  y ) ) )
761, 2, 3, 4, 5, 6, 8, 10, 12, 37, 75islmhmd 16044 1  |-  ( ( J  e.  CRing  /\  A  e.  K  /\  F  e.  ( M LMHom  N ) )  ->  ( ( V  X.  { A }
)  o F  .x.  F )  e.  ( M LMHom  N ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   _Vcvv 2901   {csn 3759    e. cmpt 4209    X. cxp 4818    o. ccom 4824    Fn wfn 5391   -->wf 5392   ` cfv 5396  (class class class)co 6022    o Fcof 6244   Basecbs 13398   .rcmulr 13459  Scalarcsca 13461   .scvsca 13462    GrpHom cghm 14932   CRingccrg 15590   LModclmod 15879   LMHom clmhm 16024
This theorem is referenced by:  mendlmod  27172
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  ax-cnex 8981  ax-resscn 8982  ax-1cn 8983  ax-icn 8984  ax-addcl 8985  ax-addrcl 8986  ax-mulcl 8987  ax-mulrcl 8988  ax-mulcom 8989  ax-addass 8990  ax-mulass 8991  ax-distr 8992  ax-i2m1 8993  ax-1ne0 8994  ax-1rid 8995  ax-rnegex 8996  ax-rrecex 8997  ax-cnre 8998  ax-pre-lttri 8999  ax-pre-lttrn 9000  ax-pre-ltadd 9001  ax-pre-mulgt0 9002
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  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-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rmo 2659  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-pss 3281  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-tp 3767  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-tr 4246  df-eprel 4437  df-id 4441  df-po 4446  df-so 4447  df-fr 4484  df-we 4486  df-ord 4527  df-on 4528  df-lim 4529  df-suc 4530  df-om 4788  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-riota 6487  df-recs 6571  df-rdg 6606  df-er 6843  df-map 6958  df-en 7048  df-dom 7049  df-sdom 7050  df-pnf 9057  df-mnf 9058  df-xr 9059  df-ltxr 9060  df-le 9061  df-sub 9227  df-neg 9228  df-nn 9935  df-2 9992  df-ndx 13401  df-slot 13402  df-base 13403  df-sets 13404  df-plusg 13471  df-0g 13656  df-mnd 14619  df-mhm 14667  df-grp 14741  df-ghm 14933  df-cmn 15343  df-mgp 15578  df-cring 15593  df-lmod 15881  df-lmhm 16027
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