MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  lmhmvsca Structured version   Unicode version

Theorem lmhmvsca 17562
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 2467 . 2  |-  ( .s
`  M )  =  ( .s `  M
)
3 lmhmvsca.s . 2  |-  .x.  =  ( .s `  N )
4 eqid 2467 . 2  |-  (Scalar `  M )  =  (Scalar `  M )
5 lmhmvsca.j . 2  |-  J  =  (Scalar `  N )
6 eqid 2467 . 2  |-  ( Base `  (Scalar `  M )
)  =  ( Base `  (Scalar `  M )
)
7 lmhmlmod1 17550 . . 3  |-  ( F  e.  ( M LMHom  N
)  ->  M  e.  LMod )
873ad2ant3 1019 . 2  |-  ( ( J  e.  CRing  /\  A  e.  K  /\  F  e.  ( M LMHom  N ) )  ->  M  e.  LMod )
9 lmhmlmod2 17549 . . 3  |-  ( F  e.  ( M LMHom  N
)  ->  N  e.  LMod )
1093ad2ant3 1019 . 2  |-  ( ( J  e.  CRing  /\  A  e.  K  /\  F  e.  ( M LMHom  N ) )  ->  N  e.  LMod )
114, 5lmhmsca 17547 . . 3  |-  ( F  e.  ( M LMHom  N
)  ->  J  =  (Scalar `  M ) )
12113ad2ant3 1019 . 2  |-  ( ( J  e.  CRing  /\  A  e.  K  /\  F  e.  ( M LMHom  N ) )  ->  J  =  (Scalar `  M ) )
13 fvex 5882 . . . . . . 7  |-  ( Base `  M )  e.  _V
141, 13eqeltri 2551 . . . . . 6  |-  V  e. 
_V
1514a1i 11 . . . . 5  |-  ( ( J  e.  CRing  /\  A  e.  K  /\  F  e.  ( M LMHom  N ) )  ->  V  e.  _V )
16 simpl2 1000 . . . . 5  |-  ( ( ( J  e.  CRing  /\  A  e.  K  /\  F  e.  ( M LMHom  N ) )  /\  v  e.  V )  ->  A  e.  K )
17 eqid 2467 . . . . . . . 8  |-  ( Base `  N )  =  (
Base `  N )
181, 17lmhmf 17551 . . . . . . 7  |-  ( F  e.  ( M LMHom  N
)  ->  F : V
--> ( Base `  N
) )
19183ad2ant3 1019 . . . . . 6  |-  ( ( J  e.  CRing  /\  A  e.  K  /\  F  e.  ( M LMHom  N ) )  ->  F : V
--> ( Base `  N
) )
2019ffvelrnda 6032 . . . . 5  |-  ( ( ( J  e.  CRing  /\  A  e.  K  /\  F  e.  ( M LMHom  N ) )  /\  v  e.  V )  ->  ( F `  v )  e.  ( Base `  N
) )
21 fconstmpt 5049 . . . . . 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 5927 . . . . 5  |-  ( ( J  e.  CRing  /\  A  e.  K  /\  F  e.  ( M LMHom  N ) )  ->  F  =  ( v  e.  V  |->  ( F `  v
) ) )
2415, 16, 20, 22, 23offval2 6551 . . . 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 2468 . . . . 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 6303 . . . . 5  |-  ( u  =  ( F `  v )  ->  ( A  .x.  u )  =  ( A  .x.  ( F `  v )
) )
2720, 23, 25, 26fmptco 6065 . . . 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 2511 . . 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 997 . . . . 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 17442 . . . . 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 17548 . . . . 5  |-  ( F  e.  ( M LMHom  N
)  ->  F  e.  ( M  GrpHom  N ) )
34333ad2ant3 1019 . . . 4  |-  ( ( J  e.  CRing  /\  A  e.  K  /\  F  e.  ( M LMHom  N ) )  ->  F  e.  ( M  GrpHom  N ) )
35 ghmco 16158 . . . 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 2555 . 2  |-  ( ( J  e.  CRing  /\  A  e.  K  /\  F  e.  ( M LMHom  N ) )  ->  ( ( V  X.  { A }
)  oF  .x.  F )  e.  ( M  GrpHom  N ) )
38 simpl1 999 . . . . . 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 1000 . . . . . 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 5876 . . . . . . . . 9  |-  ( ( J  e.  CRing  /\  A  e.  K  /\  F  e.  ( M LMHom  N ) )  ->  ( Base `  J )  =  (
Base `  (Scalar `  M
) ) )
4230, 41syl5eq 2520 . . . . . . . 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 2558 . . . . . 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 2467 . . . . . . 7  |-  ( .r
`  J )  =  ( .r `  J
)
4630, 45crngcom 17085 . . . . . 6  |-  ( ( J  e.  CRing  /\  A  e.  K  /\  x  e.  K )  ->  ( A ( .r `  J ) x )  =  ( x ( .r `  J ) A ) )
4738, 39, 44, 46syl3anc 1228 . . . . 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 6310 . . . 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 6033 . . . . 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 17408 . . . . 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 1230 . . . 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 17408 . . . . 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 1230 . . . 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 2516 . . 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 17400 . . . . . 6  |-  ( ( M  e.  LMod  /\  x  e.  ( Base `  (Scalar `  M ) )  /\  y  e.  V )  ->  ( x ( .s
`  M ) y )  e.  V )
59583expb 1197 . . . . 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 5737 . . . . . . 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 17552 . . . . . . . 8  |-  ( ( F  e.  ( M LMHom 
N )  /\  x  e.  ( Base `  (Scalar `  M ) )  /\  y  e.  V )  ->  ( F `  (
x ( .s `  M ) y ) )  =  ( x 
.x.  ( F `  y ) ) )
66653expb 1197 . . . . . . 7  |-  ( ( F  e.  ( M LMHom 
N )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  V ) )  -> 
( F `  (
x ( .s `  M ) y ) )  =  ( x 
.x.  ( F `  y ) ) )
67663ad2antl3 1160 . . . . . 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 6558 . . . 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 2468 . . . . . 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 6558 . . . . 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 6311 . . 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 2518 . 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 17556 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 973    = wceq 1379    e. wcel 1767   _Vcvv 3118   {csn 4033    |-> cmpt 4511    X. cxp 5003    o. ccom 5009    Fn wfn 5589   -->wf 5590   ` cfv 5594  (class class class)co 6295    oFcof 6533   Basecbs 14507   .rcmulr 14573  Scalarcsca 14575   .scvsca 14576    GrpHom cghm 16136   CRingccrg 17071   LModclmod 17383   LMHom clmhm 17536
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-of 6535  df-om 6696  df-recs 7054  df-rdg 7088  df-er 7323  df-map 7434  df-en 7529  df-dom 7530  df-sdom 7531  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-2 10606  df-ndx 14510  df-slot 14511  df-base 14512  df-sets 14513  df-plusg 14585  df-0g 14714  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-mhm 15839  df-grp 15929  df-ghm 16137  df-cmn 16673  df-mgp 17014  df-cring 17073  df-lmod 17385  df-lmhm 17539
This theorem is referenced by:  mendlmod  31071
  Copyright terms: Public domain W3C validator