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Theorem lmhmvsca 15818
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 2296 . 2  |-  ( .s
`  M )  =  ( .s `  M
)
3 lmhmvsca.s . 2  |-  .x.  =  ( .s `  N )
4 eqid 2296 . 2  |-  (Scalar `  M )  =  (Scalar `  M )
5 lmhmvsca.j . 2  |-  J  =  (Scalar `  N )
6 eqid 2296 . 2  |-  ( Base `  (Scalar `  M )
)  =  ( Base `  (Scalar `  M )
)
7 lmhmlmod1 15806 . . 3  |-  ( F  e.  ( M LMHom  N
)  ->  M  e.  LMod )
873ad2ant3 978 . 2  |-  ( ( J  e.  CRing  /\  A  e.  K  /\  F  e.  ( M LMHom  N ) )  ->  M  e.  LMod )
9 lmhmlmod2 15805 . . 3  |-  ( F  e.  ( M LMHom  N
)  ->  N  e.  LMod )
1093ad2ant3 978 . 2  |-  ( ( J  e.  CRing  /\  A  e.  K  /\  F  e.  ( M LMHom  N ) )  ->  N  e.  LMod )
114, 5lmhmsca 15803 . . 3  |-  ( F  e.  ( M LMHom  N
)  ->  J  =  (Scalar `  M ) )
12113ad2ant3 978 . 2  |-  ( ( J  e.  CRing  /\  A  e.  K  /\  F  e.  ( M LMHom  N ) )  ->  J  =  (Scalar `  M ) )
13 fvex 5555 . . . . . . 7  |-  ( Base `  M )  e.  _V
141, 13eqeltri 2366 . . . . . 6  |-  V  e. 
_V
1514a1i 10 . . . . 5  |-  ( ( J  e.  CRing  /\  A  e.  K  /\  F  e.  ( M LMHom  N ) )  ->  V  e.  _V )
16 simpl2 959 . . . . 5  |-  ( ( ( J  e.  CRing  /\  A  e.  K  /\  F  e.  ( M LMHom  N ) )  /\  v  e.  V )  ->  A  e.  K )
17 eqid 2296 . . . . . . . 8  |-  ( Base `  N )  =  (
Base `  N )
181, 17lmhmf 15807 . . . . . . 7  |-  ( F  e.  ( M LMHom  N
)  ->  F : V
--> ( Base `  N
) )
19183ad2ant3 978 . . . . . 6  |-  ( ( J  e.  CRing  /\  A  e.  K  /\  F  e.  ( M LMHom  N ) )  ->  F : V
--> ( Base `  N
) )
20 ffvelrn 5679 . . . . . 6  |-  ( ( F : V --> ( Base `  N )  /\  v  e.  V )  ->  ( F `  v )  e.  ( Base `  N
) )
2119, 20sylan 457 . . . . 5  |-  ( ( ( J  e.  CRing  /\  A  e.  K  /\  F  e.  ( M LMHom  N ) )  /\  v  e.  V )  ->  ( F `  v )  e.  ( Base `  N
) )
22 fconstmpt 4748 . . . . . 6  |-  ( V  X.  { A }
)  =  ( v  e.  V  |->  A )
2322a1i 10 . . . . 5  |-  ( ( J  e.  CRing  /\  A  e.  K  /\  F  e.  ( M LMHom  N ) )  ->  ( V  X.  { A } )  =  ( v  e.  V  |->  A ) )
2419feqmptd 5591 . . . . 5  |-  ( ( J  e.  CRing  /\  A  e.  K  /\  F  e.  ( M LMHom  N ) )  ->  F  =  ( v  e.  V  |->  ( F `  v
) ) )
2515, 16, 21, 23, 24offval2 6111 . . . 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 ) ) ) )
26 eqidd 2297 . . . . 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
) ) )
27 oveq2 5882 . . . . 5  |-  ( u  =  ( F `  v )  ->  ( A  .x.  u )  =  ( A  .x.  ( F `  v )
) )
2821, 24, 26, 27fmptco 5707 . . . 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 )
) ) )
2925, 28eqtr4d 2331 . . 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
) )
30 simp2 956 . . . . 5  |-  ( ( J  e.  CRing  /\  A  e.  K  /\  F  e.  ( M LMHom  N ) )  ->  A  e.  K )
31 lmhmvsca.k . . . . . 6  |-  K  =  ( Base `  J
)
3217, 5, 3, 31lmodvsghm 15702 . . . . 5  |-  ( ( N  e.  LMod  /\  A  e.  K )  ->  (
u  e.  ( Base `  N )  |->  ( A 
.x.  u ) )  e.  ( N  GrpHom  N ) )
3310, 30, 32syl2anc 642 . . . 4  |-  ( ( J  e.  CRing  /\  A  e.  K  /\  F  e.  ( M LMHom  N ) )  ->  ( u  e.  ( Base `  N
)  |->  ( A  .x.  u ) )  e.  ( N  GrpHom  N ) )
34 lmghm 15804 . . . . 5  |-  ( F  e.  ( M LMHom  N
)  ->  F  e.  ( M  GrpHom  N ) )
35343ad2ant3 978 . . . 4  |-  ( ( J  e.  CRing  /\  A  e.  K  /\  F  e.  ( M LMHom  N ) )  ->  F  e.  ( M  GrpHom  N ) )
36 ghmco 14718 . . . 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 ) )
3733, 35, 36syl2anc 642 . . 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 ) )
3829, 37eqeltrd 2370 . 2  |-  ( ( J  e.  CRing  /\  A  e.  K  /\  F  e.  ( M LMHom  N ) )  ->  ( ( V  X.  { A }
)  o F  .x.  F )  e.  ( M  GrpHom  N ) )
39 simpl1 958 . . . . . 6  |-  ( ( ( J  e.  CRing  /\  A  e.  K  /\  F  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  V ) )  ->  J  e.  CRing )
40 simpl2 959 . . . . . 6  |-  ( ( ( J  e.  CRing  /\  A  e.  K  /\  F  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  V ) )  ->  A  e.  K )
41 simprl 732 . . . . . . 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 )
) )
4212fveq2d 5545 . . . . . . . . 9  |-  ( ( J  e.  CRing  /\  A  e.  K  /\  F  e.  ( M LMHom  N ) )  ->  ( Base `  J )  =  (
Base `  (Scalar `  M
) ) )
4331, 42syl5eq 2340 . . . . . . . 8  |-  ( ( J  e.  CRing  /\  A  e.  K  /\  F  e.  ( M LMHom  N ) )  ->  K  =  ( Base `  (Scalar `  M
) ) )
4443adantr 451 . . . . . . 7  |-  ( ( ( J  e.  CRing  /\  A  e.  K  /\  F  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  V ) )  ->  K  =  ( Base `  (Scalar `  M )
) )
4541, 44eleqtrrd 2373 . . . . . 6  |-  ( ( ( J  e.  CRing  /\  A  e.  K  /\  F  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  V ) )  ->  x  e.  K )
46 eqid 2296 . . . . . . 7  |-  ( .r
`  J )  =  ( .r `  J
)
4731, 46crngcom 15371 . . . . . 6  |-  ( ( J  e.  CRing  /\  A  e.  K  /\  x  e.  K )  ->  ( A ( .r `  J ) x )  =  ( x ( .r `  J ) A ) )
4839, 40, 45, 47syl3anc 1182 . . . . 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 ) )
4948oveq1d 5889 . . . 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 ) ) )
5010adantr 451 . . . . 5  |-  ( ( ( J  e.  CRing  /\  A  e.  K  /\  F  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  V ) )  ->  N  e.  LMod )
5119adantr 451 . . . . . 6  |-  ( ( ( J  e.  CRing  /\  A  e.  K  /\  F  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  V ) )  ->  F : V --> ( Base `  N ) )
52 simprr 733 . . . . . 6  |-  ( ( ( J  e.  CRing  /\  A  e.  K  /\  F  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  V ) )  -> 
y  e.  V )
53 ffvelrn 5679 . . . . . 6  |-  ( ( F : V --> ( Base `  N )  /\  y  e.  V )  ->  ( F `  y )  e.  ( Base `  N
) )
5451, 52, 53syl2anc 642 . . . . 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 ) )
5517, 5, 3, 31, 46lmodvsass 15670 . . . . 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 ) ) ) )
5650, 40, 45, 54, 55syl13anc 1184 . . . 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 ) ) ) )
5717, 5, 3, 31, 46lmodvsass 15670 . . . . 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 ) ) ) )
5850, 45, 40, 54, 57syl13anc 1184 . . . 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 ) ) ) )
5949, 56, 583eqtr3d 2336 . . 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 ) ) ) )
601, 4, 2, 6lmodvscl 15660 . . . . . 6  |-  ( ( M  e.  LMod  /\  x  e.  ( Base `  (Scalar `  M ) )  /\  y  e.  V )  ->  ( x ( .s
`  M ) y )  e.  V )
61603expb 1152 . . . . 5  |-  ( ( M  e.  LMod  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  V ) )  -> 
( x ( .s
`  M ) y )  e.  V )
628, 61sylan 457 . . . 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 )
6314a1i 10 . . . . 5  |-  ( ( ( J  e.  CRing  /\  A  e.  K  /\  F  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  V ) )  ->  V  e.  _V )
64 ffn 5405 . . . . . . 7  |-  ( F : V --> ( Base `  N )  ->  F  Fn  V )
6519, 64syl 15 . . . . . 6  |-  ( ( J  e.  CRing  /\  A  e.  K  /\  F  e.  ( M LMHom  N ) )  ->  F  Fn  V )
6665adantr 451 . . . . 5  |-  ( ( ( J  e.  CRing  /\  A  e.  K  /\  F  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  V ) )  ->  F  Fn  V )
674, 6, 1, 2, 3lmhmlin 15808 . . . . . . . 8  |-  ( ( F  e.  ( M LMHom 
N )  /\  x  e.  ( Base `  (Scalar `  M ) )  /\  y  e.  V )  ->  ( F `  (
x ( .s `  M ) y ) )  =  ( x 
.x.  ( F `  y ) ) )
68673expb 1152 . . . . . . 7  |-  ( ( F  e.  ( M LMHom 
N )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  V ) )  -> 
( F `  (
x ( .s `  M ) y ) )  =  ( x 
.x.  ( F `  y ) ) )
69683ad2antl3 1119 . . . . . 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 ) ) )
7069adantr 451 . . . . 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 ) ) )
7163, 40, 66, 70ofc1 6116 . . . 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 ) ) ) )
7262, 71mpdan 649 . . 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 ) ) ) )
73 eqidd 2297 . . . . . 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 ) )
7463, 40, 66, 73ofc1 6116 . . . . 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 ) ) )
7552, 74mpdan 649 . . . 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 ) ) )
7675oveq2d 5890 . . 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 ) ) ) )
7759, 72, 763eqtr4d 2338 . 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 ) ) )
781, 2, 3, 4, 5, 6, 8, 10, 12, 38, 77islmhmd 15812 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 358    /\ w3a 934    = wceq 1632    e. wcel 1696   _Vcvv 2801   {csn 3653    e. cmpt 4093    X. cxp 4703    o. ccom 4709    Fn wfn 5266   -->wf 5267   ` cfv 5271  (class class class)co 5874    o Fcof 6092   Basecbs 13164   .rcmulr 13225  Scalarcsca 13227   .scvsca 13228    GrpHom cghm 14696   CRingccrg 15354   LModclmod 15643   LMHom clmhm 15792
This theorem is referenced by:  mendlmod  27604
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-plusg 13237  df-0g 13420  df-mnd 14383  df-mhm 14431  df-grp 14505  df-ghm 14697  df-cmn 15107  df-mgp 15342  df-cring 15357  df-lmod 15645  df-lmhm 15795
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