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Theorem lmhmvsca 17131
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 }
)  oF  .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 2443 . 2  |-  ( .s
`  M )  =  ( .s `  M
)
3 lmhmvsca.s . 2  |-  .x.  =  ( .s `  N )
4 eqid 2443 . 2  |-  (Scalar `  M )  =  (Scalar `  M )
5 lmhmvsca.j . 2  |-  J  =  (Scalar `  N )
6 eqid 2443 . 2  |-  ( Base `  (Scalar `  M )
)  =  ( Base `  (Scalar `  M )
)
7 lmhmlmod1 17119 . . 3  |-  ( F  e.  ( M LMHom  N
)  ->  M  e.  LMod )
873ad2ant3 1011 . 2  |-  ( ( J  e.  CRing  /\  A  e.  K  /\  F  e.  ( M LMHom  N ) )  ->  M  e.  LMod )
9 lmhmlmod2 17118 . . 3  |-  ( F  e.  ( M LMHom  N
)  ->  N  e.  LMod )
1093ad2ant3 1011 . 2  |-  ( ( J  e.  CRing  /\  A  e.  K  /\  F  e.  ( M LMHom  N ) )  ->  N  e.  LMod )
114, 5lmhmsca 17116 . . 3  |-  ( F  e.  ( M LMHom  N
)  ->  J  =  (Scalar `  M ) )
12113ad2ant3 1011 . 2  |-  ( ( J  e.  CRing  /\  A  e.  K  /\  F  e.  ( M LMHom  N ) )  ->  J  =  (Scalar `  M ) )
13 fvex 5706 . . . . . . 7  |-  ( Base `  M )  e.  _V
141, 13eqeltri 2513 . . . . . 6  |-  V  e. 
_V
1514a1i 11 . . . . 5  |-  ( ( J  e.  CRing  /\  A  e.  K  /\  F  e.  ( M LMHom  N ) )  ->  V  e.  _V )
16 simpl2 992 . . . . 5  |-  ( ( ( J  e.  CRing  /\  A  e.  K  /\  F  e.  ( M LMHom  N ) )  /\  v  e.  V )  ->  A  e.  K )
17 eqid 2443 . . . . . . . 8  |-  ( Base `  N )  =  (
Base `  N )
181, 17lmhmf 17120 . . . . . . 7  |-  ( F  e.  ( M LMHom  N
)  ->  F : V
--> ( Base `  N
) )
19183ad2ant3 1011 . . . . . 6  |-  ( ( J  e.  CRing  /\  A  e.  K  /\  F  e.  ( M LMHom  N ) )  ->  F : V
--> ( Base `  N
) )
2019ffvelrnda 5848 . . . . 5  |-  ( ( ( J  e.  CRing  /\  A  e.  K  /\  F  e.  ( M LMHom  N ) )  /\  v  e.  V )  ->  ( F `  v )  e.  ( Base `  N
) )
21 fconstmpt 4887 . . . . . 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 5749 . . . . 5  |-  ( ( J  e.  CRing  /\  A  e.  K  /\  F  e.  ( M LMHom  N ) )  ->  F  =  ( v  e.  V  |->  ( F `  v
) ) )
2415, 16, 20, 22, 23offval2 6341 . . . 4  |-  ( ( J  e.  CRing  /\  A  e.  K  /\  F  e.  ( M LMHom  N ) )  ->  ( ( V  X.  { A }
)  oF  .x.  F )  =  ( v  e.  V  |->  ( A  .x.  ( F `
 v ) ) ) )
25 eqidd 2444 . . . . 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 6104 . . . . 5  |-  ( u  =  ( F `  v )  ->  ( A  .x.  u )  =  ( A  .x.  ( F `  v )
) )
2720, 23, 25, 26fmptco 5881 . . . 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 2478 . . 3  |-  ( ( J  e.  CRing  /\  A  e.  K  /\  F  e.  ( M LMHom  N ) )  ->  ( ( V  X.  { A }
)  oF  .x.  F )  =  ( ( u  e.  (
Base `  N )  |->  ( A  .x.  u
) )  o.  F
) )
29 simp2 989 . . . . 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 17011 . . . . 5  |-  ( ( N  e.  LMod  /\  A  e.  K )  ->  (
u  e.  ( Base `  N )  |->  ( A 
.x.  u ) )  e.  ( N  GrpHom  N ) )
3210, 29, 31syl2anc 661 . . . 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 17117 . . . . 5  |-  ( F  e.  ( M LMHom  N
)  ->  F  e.  ( M  GrpHom  N ) )
34333ad2ant3 1011 . . . 4  |-  ( ( J  e.  CRing  /\  A  e.  K  /\  F  e.  ( M LMHom  N ) )  ->  F  e.  ( M  GrpHom  N ) )
35 ghmco 15771 . . . 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 661 . . 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 2517 . 2  |-  ( ( J  e.  CRing  /\  A  e.  K  /\  F  e.  ( M LMHom  N ) )  ->  ( ( V  X.  { A }
)  oF  .x.  F )  e.  ( M  GrpHom  N ) )
38 simpl1 991 . . . . . 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 992 . . . . . 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 755 . . . . . . 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 5700 . . . . . . . . 9  |-  ( ( J  e.  CRing  /\  A  e.  K  /\  F  e.  ( M LMHom  N ) )  ->  ( Base `  J )  =  (
Base `  (Scalar `  M
) ) )
4230, 41syl5eq 2487 . . . . . . . 8  |-  ( ( J  e.  CRing  /\  A  e.  K  /\  F  e.  ( M LMHom  N ) )  ->  K  =  ( Base `  (Scalar `  M
) ) )
4342adantr 465 . . . . . . 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 2520 . . . . . 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 2443 . . . . . . 7  |-  ( .r
`  J )  =  ( .r `  J
)
4630, 45crngcom 16664 . . . . . 6  |-  ( ( J  e.  CRing  /\  A  e.  K  /\  x  e.  K )  ->  ( A ( .r `  J ) x )  =  ( x ( .r `  J ) A ) )
4738, 39, 44, 46syl3anc 1218 . . . . 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 6111 . . . 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 465 . . . . 5  |-  ( ( ( J  e.  CRing  /\  A  e.  K  /\  F  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  V ) )  ->  N  e.  LMod )
5019adantr 465 . . . . . 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 756 . . . . . 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 5849 . . . . 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 16978 . . . . 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 1220 . . . 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 16978 . . . . 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 1220 . . . 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 2483 . . 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 16970 . . . . . 6  |-  ( ( M  e.  LMod  /\  x  e.  ( Base `  (Scalar `  M ) )  /\  y  e.  V )  ->  ( x ( .s
`  M ) y )  e.  V )
59583expb 1188 . . . . 5  |-  ( ( M  e.  LMod  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  V ) )  -> 
( x ( .s
`  M ) y )  e.  V )
608, 59sylan 471 . . . 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 5564 . . . . . . 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 465 . . . . 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 17121 . . . . . . . 8  |-  ( ( F  e.  ( M LMHom 
N )  /\  x  e.  ( Base `  (Scalar `  M ) )  /\  y  e.  V )  ->  ( F `  (
x ( .s `  M ) y ) )  =  ( x 
.x.  ( F `  y ) ) )
66653expb 1188 . . . . . . 7  |-  ( ( F  e.  ( M LMHom 
N )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  V ) )  -> 
( F `  (
x ( .s `  M ) y ) )  =  ( x 
.x.  ( F `  y ) ) )
67663ad2antl3 1152 . . . . . 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 465 . . . . 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 6348 . . . 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 } )  oF  .x.  F ) `  ( x ( .s
`  M ) y ) )  =  ( A  .x.  ( x 
.x.  ( F `  y ) ) ) )
7060, 69mpdan 668 . . 3  |-  ( ( ( J  e.  CRing  /\  A  e.  K  /\  F  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  V ) )  -> 
( ( ( V  X.  { A }
)  oF  .x.  F ) `  (
x ( .s `  M ) y ) )  =  ( A 
.x.  ( x  .x.  ( F `  y ) ) ) )
71 eqidd 2444 . . . . . 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 6348 . . . . 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 }
)  oF  .x.  F ) `  y
)  =  ( A 
.x.  ( F `  y ) ) )
7351, 72mpdan 668 . . . 4  |-  ( ( ( J  e.  CRing  /\  A  e.  K  /\  F  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  V ) )  -> 
( ( ( V  X.  { A }
)  oF  .x.  F ) `  y
)  =  ( A 
.x.  ( F `  y ) ) )
7473oveq2d 6112 . . 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 } )  oF  .x.  F ) `
 y ) )  =  ( x  .x.  ( A  .x.  ( F `
 y ) ) ) )
7557, 70, 743eqtr4d 2485 . 2  |-  ( ( ( J  e.  CRing  /\  A  e.  K  /\  F  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  V ) )  -> 
( ( ( V  X.  { A }
)  oF  .x.  F ) `  (
x ( .s `  M ) y ) )  =  ( x 
.x.  ( ( ( V  X.  { A } )  oF  .x.  F ) `  y ) ) )
761, 2, 3, 4, 5, 6, 8, 10, 12, 37, 75islmhmd 17125 1  |-  ( ( J  e.  CRing  /\  A  e.  K  /\  F  e.  ( M LMHom  N ) )  ->  ( ( V  X.  { A }
)  oF  .x.  F )  e.  ( M LMHom  N ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   _Vcvv 2977   {csn 3882    e. cmpt 4355    X. cxp 4843    o. ccom 4849    Fn wfn 5418   -->wf 5419   ` cfv 5423  (class class class)co 6096    oFcof 6323   Basecbs 14179   .rcmulr 14244  Scalarcsca 14246   .scvsca 14247    GrpHom cghm 15749   CRingccrg 16651   LModclmod 16953   LMHom clmhm 17105
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-of 6325  df-om 6482  df-recs 6837  df-rdg 6871  df-er 7106  df-map 7221  df-en 7316  df-dom 7317  df-sdom 7318  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-nn 10328  df-2 10385  df-ndx 14182  df-slot 14183  df-base 14184  df-sets 14185  df-plusg 14256  df-0g 14385  df-mnd 15420  df-mhm 15469  df-grp 15550  df-ghm 15750  df-cmn 16284  df-mgp 16597  df-cring 16653  df-lmod 16955  df-lmhm 17108
This theorem is referenced by:  mendlmod  29555
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