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Theorem lmhmvsca 15802
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 2283 . 2  |-  ( .s
`  M )  =  ( .s `  M
)
3 lmhmvsca.s . 2  |-  .x.  =  ( .s `  N )
4 eqid 2283 . 2  |-  (Scalar `  M )  =  (Scalar `  M )
5 lmhmvsca.j . 2  |-  J  =  (Scalar `  N )
6 eqid 2283 . 2  |-  ( Base `  (Scalar `  M )
)  =  ( Base `  (Scalar `  M )
)
7 lmhmlmod1 15790 . . 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 15789 . . 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 15787 . . 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 5539 . . . . . . 7  |-  ( Base `  M )  e.  _V
141, 13eqeltri 2353 . . . . . 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 2283 . . . . . . . 8  |-  ( Base `  N )  =  (
Base `  N )
181, 17lmhmf 15791 . . . . . . 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 5663 . . . . . 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 4732 . . . . . 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 5575 . . . . 5  |-  ( ( J  e.  CRing  /\  A  e.  K  /\  F  e.  ( M LMHom  N ) )  ->  F  =  ( v  e.  V  |->  ( F `  v
) ) )
2515, 16, 21, 23, 24offval2 6095 . . . 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 2284 . . . . 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 5866 . . . . 5  |-  ( u  =  ( F `  v )  ->  ( A  .x.  u )  =  ( A  .x.  ( F `  v )
) )
2821, 24, 26, 27fmptco 5691 . . . 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 2318 . . 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 15686 . . . . 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 15788 . . . . 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 14702 . . . 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 2357 . 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 5529 . . . . . . . . 9  |-  ( ( J  e.  CRing  /\  A  e.  K  /\  F  e.  ( M LMHom  N ) )  ->  ( Base `  J )  =  (
Base `  (Scalar `  M
) ) )
4331, 42syl5eq 2327 . . . . . . . 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 2360 . . . . . 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 2283 . . . . . . 7  |-  ( .r
`  J )  =  ( .r `  J
)
4731, 46crngcom 15355 . . . . . 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 5873 . . . 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 5663 . . . . . 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 15654 . . . . 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 15654 . . . . 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 2323 . . 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 15644 . . . . . 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 5389 . . . . . . 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 15792 . . . . . . . 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 6100 . . . 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 2284 . . . . . 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 6100 . . . . 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 5874 . . 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 2325 . 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 15796 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 1623    e. wcel 1684   _Vcvv 2788   {csn 3640    e. cmpt 4077    X. cxp 4687    o. ccom 4693    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858    o Fcof 6076   Basecbs 13148   .rcmulr 13209  Scalarcsca 13211   .scvsca 13212    GrpHom cghm 14680   CRingccrg 15338   LModclmod 15627   LMHom clmhm 15776
This theorem is referenced by:  mendlmod  27501
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-plusg 13221  df-0g 13404  df-mnd 14367  df-mhm 14415  df-grp 14489  df-ghm 14681  df-cmn 15091  df-mgp 15326  df-cring 15341  df-lmod 15629  df-lmhm 15779
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