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Theorem isslmd 26216
Description: The predicate "is a semimodule". (Contributed by NM, 4-Nov-2013.) (Revised by Mario Carneiro, 19-Jun-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.)
Hypotheses
Ref Expression
isslmd.v  |-  V  =  ( Base `  W
)
isslmd.a  |-  .+  =  ( +g  `  W )
isslmd.s  |-  .x.  =  ( .s `  W )
isslmd.0  |-  .0.  =  ( 0g `  W )
isslmd.f  |-  F  =  (Scalar `  W )
isslmd.k  |-  K  =  ( Base `  F
)
isslmd.p  |-  .+^  =  ( +g  `  F )
isslmd.t  |-  .X.  =  ( .r `  F )
isslmd.u  |-  .1.  =  ( 1r `  F )
isslmd.o  |-  O  =  ( 0g `  F
)
Assertion
Ref Expression
isslmd  |-  ( W  e. SLMod 
<->  ( W  e. CMnd  /\  F  e. SRing  /\  A. q  e.  K  A. r  e.  K  A. x  e.  V  A. w  e.  V  ( (
( r  .x.  w
)  e.  V  /\  ( r  .x.  (
w  .+  x )
)  =  ( ( r  .x.  w ) 
.+  ( r  .x.  x ) )  /\  ( ( q  .+^  r )  .x.  w
)  =  ( ( q  .x.  w ) 
.+  ( r  .x.  w ) ) )  /\  ( ( ( q  .X.  r )  .x.  w )  =  ( q  .x.  ( r 
.x.  w ) )  /\  (  .1.  .x.  w )  =  w  /\  ( O  .x.  w )  =  .0.  ) ) ) )
Distinct variable groups:    r, q, w, x,  .X.    .+ , q,
r, w, x    .+^ , q, r, w, x    .1. , q,
r, w, x    .x. , q,
r, w, x    F, q, r, w, x    K, q, r, w, x    V, q, r, w, x    W, q, r, w, x    .0. , q, r, w, x    O, q, r, w, x

Proof of Theorem isslmd
Dummy variables  f 
a  g  k  p  s  t  v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fvex 5699 . . . . 5  |-  ( Base `  g )  e.  _V
2 fvex 5699 . . . . 5  |-  ( +g  `  g )  e.  _V
3 fvex 5699 . . . . . . 7  |-  ( .s
`  g )  e. 
_V
4 fvex 5699 . . . . . . 7  |-  (Scalar `  g )  e.  _V
5 fvex 5699 . . . . . . . . 9  |-  ( Base `  f )  e.  _V
6 fvex 5699 . . . . . . . . 9  |-  ( +g  `  f )  e.  _V
7 fvex 5699 . . . . . . . . 9  |-  ( .r
`  f )  e. 
_V
8 simp1 988 . . . . . . . . . . 11  |-  ( ( k  =  ( Base `  f )  /\  p  =  ( +g  `  f
)  /\  t  =  ( .r `  f ) )  ->  k  =  ( Base `  f )
)
9 simp2 989 . . . . . . . . . . . . . . . . . 18  |-  ( ( k  =  ( Base `  f )  /\  p  =  ( +g  `  f
)  /\  t  =  ( .r `  f ) )  ->  p  =  ( +g  `  f ) )
109oveqd 6106 . . . . . . . . . . . . . . . . 17  |-  ( ( k  =  ( Base `  f )  /\  p  =  ( +g  `  f
)  /\  t  =  ( .r `  f ) )  ->  ( q
p r )  =  ( q ( +g  `  f ) r ) )
1110oveq1d 6104 . . . . . . . . . . . . . . . 16  |-  ( ( k  =  ( Base `  f )  /\  p  =  ( +g  `  f
)  /\  t  =  ( .r `  f ) )  ->  ( (
q p r ) s w )  =  ( ( q ( +g  `  f ) r ) s w ) )
1211eqeq1d 2449 . . . . . . . . . . . . . . 15  |-  ( ( k  =  ( Base `  f )  /\  p  =  ( +g  `  f
)  /\  t  =  ( .r `  f ) )  ->  ( (
( q p r ) s w )  =  ( ( q s w ) a ( r s w ) )  <->  ( (
q ( +g  `  f
) r ) s w )  =  ( ( q s w ) a ( r s w ) ) ) )
13123anbi3d 1295 . . . . . . . . . . . . . 14  |-  ( ( k  =  ( Base `  f )  /\  p  =  ( +g  `  f
)  /\  t  =  ( .r `  f ) )  ->  ( (
( r s w )  e.  v  /\  ( r s ( w a x ) )  =  ( ( r s w ) a ( r s x ) )  /\  ( ( q p r ) s w )  =  ( ( q s w ) a ( r s w ) ) )  <-> 
( ( r s w )  e.  v  /\  ( r s ( w a x ) )  =  ( ( r s w ) a ( r s x ) )  /\  ( ( q ( +g  `  f
) r ) s w )  =  ( ( q s w ) a ( r s w ) ) ) ) )
14 simp3 990 . . . . . . . . . . . . . . . . . 18  |-  ( ( k  =  ( Base `  f )  /\  p  =  ( +g  `  f
)  /\  t  =  ( .r `  f ) )  ->  t  =  ( .r `  f ) )
1514oveqd 6106 . . . . . . . . . . . . . . . . 17  |-  ( ( k  =  ( Base `  f )  /\  p  =  ( +g  `  f
)  /\  t  =  ( .r `  f ) )  ->  ( q
t r )  =  ( q ( .r
`  f ) r ) )
1615oveq1d 6104 . . . . . . . . . . . . . . . 16  |-  ( ( k  =  ( Base `  f )  /\  p  =  ( +g  `  f
)  /\  t  =  ( .r `  f ) )  ->  ( (
q t r ) s w )  =  ( ( q ( .r `  f ) r ) s w ) )
1716eqeq1d 2449 . . . . . . . . . . . . . . 15  |-  ( ( k  =  ( Base `  f )  /\  p  =  ( +g  `  f
)  /\  t  =  ( .r `  f ) )  ->  ( (
( q t r ) s w )  =  ( q s ( r s w ) )  <->  ( (
q ( .r `  f ) r ) s w )  =  ( q s ( r s w ) ) ) )
18173anbi1d 1293 . . . . . . . . . . . . . 14  |-  ( ( k  =  ( Base `  f )  /\  p  =  ( +g  `  f
)  /\  t  =  ( .r `  f ) )  ->  ( (
( ( q t r ) s w )  =  ( q s ( r s w ) )  /\  ( ( 1r `  f ) s w )  =  w  /\  ( ( 0g `  f ) s w )  =  ( 0g
`  g ) )  <-> 
( ( ( q ( .r `  f
) r ) s w )  =  ( q s ( r s w ) )  /\  ( ( 1r
`  f ) s w )  =  w  /\  ( ( 0g
`  f ) s w )  =  ( 0g `  g ) ) ) )
1913, 18anbi12d 710 . . . . . . . . . . . . 13  |-  ( ( k  =  ( Base `  f )  /\  p  =  ( +g  `  f
)  /\  t  =  ( .r `  f ) )  ->  ( (
( ( r s w )  e.  v  /\  ( r s ( w a x ) )  =  ( ( r s w ) a ( r s x ) )  /\  ( ( q p r ) s w )  =  ( ( q s w ) a ( r s w ) ) )  /\  ( ( ( q t r ) s w )  =  ( q s ( r s w ) )  /\  (
( 1r `  f
) s w )  =  w  /\  (
( 0g `  f
) s w )  =  ( 0g `  g ) ) )  <-> 
( ( ( r s w )  e.  v  /\  ( r s ( w a x ) )  =  ( ( r s w ) a ( r s x ) )  /\  ( ( q ( +g  `  f
) r ) s w )  =  ( ( q s w ) a ( r s w ) ) )  /\  ( ( ( q ( .r
`  f ) r ) s w )  =  ( q s ( r s w ) )  /\  (
( 1r `  f
) s w )  =  w  /\  (
( 0g `  f
) s w )  =  ( 0g `  g ) ) ) ) )
20192ralbidv 2755 . . . . . . . . . . . 12  |-  ( ( k  =  ( Base `  f )  /\  p  =  ( +g  `  f
)  /\  t  =  ( .r `  f ) )  ->  ( A. x  e.  v  A. w  e.  v  (
( ( r s w )  e.  v  /\  ( r s ( w a x ) )  =  ( ( r s w ) a ( r s x ) )  /\  ( ( q p r ) s w )  =  ( ( q s w ) a ( r s w ) ) )  /\  ( ( ( q t r ) s w )  =  ( q s ( r s w ) )  /\  (
( 1r `  f
) s w )  =  w  /\  (
( 0g `  f
) s w )  =  ( 0g `  g ) ) )  <->  A. x  e.  v  A. w  e.  v 
( ( ( r s w )  e.  v  /\  ( r s ( w a x ) )  =  ( ( r s w ) a ( r s x ) )  /\  ( ( q ( +g  `  f
) r ) s w )  =  ( ( q s w ) a ( r s w ) ) )  /\  ( ( ( q ( .r
`  f ) r ) s w )  =  ( q s ( r s w ) )  /\  (
( 1r `  f
) s w )  =  w  /\  (
( 0g `  f
) s w )  =  ( 0g `  g ) ) ) ) )
218, 20raleqbidv 2929 . . . . . . . . . . 11  |-  ( ( k  =  ( Base `  f )  /\  p  =  ( +g  `  f
)  /\  t  =  ( .r `  f ) )  ->  ( A. r  e.  k  A. x  e.  v  A. w  e.  v  (
( ( r s w )  e.  v  /\  ( r s ( w a x ) )  =  ( ( r s w ) a ( r s x ) )  /\  ( ( q p r ) s w )  =  ( ( q s w ) a ( r s w ) ) )  /\  ( ( ( q t r ) s w )  =  ( q s ( r s w ) )  /\  (
( 1r `  f
) s w )  =  w  /\  (
( 0g `  f
) s w )  =  ( 0g `  g ) ) )  <->  A. r  e.  ( Base `  f ) A. x  e.  v  A. w  e.  v  (
( ( r s w )  e.  v  /\  ( r s ( w a x ) )  =  ( ( r s w ) a ( r s x ) )  /\  ( ( q ( +g  `  f
) r ) s w )  =  ( ( q s w ) a ( r s w ) ) )  /\  ( ( ( q ( .r
`  f ) r ) s w )  =  ( q s ( r s w ) )  /\  (
( 1r `  f
) s w )  =  w  /\  (
( 0g `  f
) s w )  =  ( 0g `  g ) ) ) ) )
228, 21raleqbidv 2929 . . . . . . . . . 10  |-  ( ( k  =  ( Base `  f )  /\  p  =  ( +g  `  f
)  /\  t  =  ( .r `  f ) )  ->  ( A. q  e.  k  A. r  e.  k  A. x  e.  v  A. w  e.  v  (
( ( r s w )  e.  v  /\  ( r s ( w a x ) )  =  ( ( r s w ) a ( r s x ) )  /\  ( ( q p r ) s w )  =  ( ( q s w ) a ( r s w ) ) )  /\  ( ( ( q t r ) s w )  =  ( q s ( r s w ) )  /\  (
( 1r `  f
) s w )  =  w  /\  (
( 0g `  f
) s w )  =  ( 0g `  g ) ) )  <->  A. q  e.  ( Base `  f ) A. r  e.  ( Base `  f ) A. x  e.  v  A. w  e.  v  ( (
( r s w )  e.  v  /\  ( r s ( w a x ) )  =  ( ( r s w ) a ( r s x ) )  /\  ( ( q ( +g  `  f ) r ) s w )  =  ( ( q s w ) a ( r s w ) ) )  /\  ( ( ( q ( .r `  f ) r ) s w )  =  ( q s ( r s w ) )  /\  ( ( 1r `  f ) s w )  =  w  /\  ( ( 0g `  f ) s w )  =  ( 0g `  g
) ) ) ) )
2322anbi2d 703 . . . . . . . . 9  |-  ( ( k  =  ( Base `  f )  /\  p  =  ( +g  `  f
)  /\  t  =  ( .r `  f ) )  ->  ( (
f  e. SRing  /\  A. q  e.  k  A. r  e.  k  A. x  e.  v  A. w  e.  v  ( (
( r s w )  e.  v  /\  ( r s ( w a x ) )  =  ( ( r s w ) a ( r s x ) )  /\  ( ( q p r ) s w )  =  ( ( q s w ) a ( r s w ) ) )  /\  ( ( ( q t r ) s w )  =  ( q s ( r s w ) )  /\  ( ( 1r `  f ) s w )  =  w  /\  ( ( 0g `  f ) s w )  =  ( 0g `  g
) ) ) )  <-> 
( f  e. SRing  /\  A. q  e.  ( Base `  f ) A. r  e.  ( Base `  f
) A. x  e.  v  A. w  e.  v  ( ( ( r s w )  e.  v  /\  (
r s ( w a x ) )  =  ( ( r s w ) a ( r s x ) )  /\  (
( q ( +g  `  f ) r ) s w )  =  ( ( q s w ) a ( r s w ) ) )  /\  (
( ( q ( .r `  f ) r ) s w )  =  ( q s ( r s w ) )  /\  ( ( 1r `  f ) s w )  =  w  /\  ( ( 0g `  f ) s w )  =  ( 0g
`  g ) ) ) ) ) )
245, 6, 7, 23sbc3ie 3262 . . . . . . . 8  |-  ( [. ( Base `  f )  /  k ]. [. ( +g  `  f )  /  p ]. [. ( .r
`  f )  / 
t ]. ( f  e. SRing  /\  A. q  e.  k 
A. r  e.  k 
A. x  e.  v 
A. w  e.  v  ( ( ( r s w )  e.  v  /\  ( r s ( w a x ) )  =  ( ( r s w ) a ( r s x ) )  /\  ( ( q p r ) s w )  =  ( ( q s w ) a ( r s w ) ) )  /\  (
( ( q t r ) s w )  =  ( q s ( r s w ) )  /\  ( ( 1r `  f ) s w )  =  w  /\  ( ( 0g `  f ) s w )  =  ( 0g
`  g ) ) ) )  <->  ( f  e. SRing  /\  A. q  e.  ( Base `  f
) A. r  e.  ( Base `  f
) A. x  e.  v  A. w  e.  v  ( ( ( r s w )  e.  v  /\  (
r s ( w a x ) )  =  ( ( r s w ) a ( r s x ) )  /\  (
( q ( +g  `  f ) r ) s w )  =  ( ( q s w ) a ( r s w ) ) )  /\  (
( ( q ( .r `  f ) r ) s w )  =  ( q s ( r s w ) )  /\  ( ( 1r `  f ) s w )  =  w  /\  ( ( 0g `  f ) s w )  =  ( 0g
`  g ) ) ) ) )
25 simpr 461 . . . . . . . . . 10  |-  ( ( s  =  ( .s
`  g )  /\  f  =  (Scalar `  g
) )  ->  f  =  (Scalar `  g )
)
2625eleq1d 2507 . . . . . . . . 9  |-  ( ( s  =  ( .s
`  g )  /\  f  =  (Scalar `  g
) )  ->  (
f  e. SRing  <->  (Scalar `  g )  e. SRing ) )
2725fveq2d 5693 . . . . . . . . . 10  |-  ( ( s  =  ( .s
`  g )  /\  f  =  (Scalar `  g
) )  ->  ( Base `  f )  =  ( Base `  (Scalar `  g ) ) )
28 simpl 457 . . . . . . . . . . . . . . . 16  |-  ( ( s  =  ( .s
`  g )  /\  f  =  (Scalar `  g
) )  ->  s  =  ( .s `  g ) )
2928oveqd 6106 . . . . . . . . . . . . . . 15  |-  ( ( s  =  ( .s
`  g )  /\  f  =  (Scalar `  g
) )  ->  (
r s w )  =  ( r ( .s `  g ) w ) )
3029eleq1d 2507 . . . . . . . . . . . . . 14  |-  ( ( s  =  ( .s
`  g )  /\  f  =  (Scalar `  g
) )  ->  (
( r s w )  e.  v  <->  ( r
( .s `  g
) w )  e.  v ) )
3128oveqd 6106 . . . . . . . . . . . . . . 15  |-  ( ( s  =  ( .s
`  g )  /\  f  =  (Scalar `  g
) )  ->  (
r s ( w a x ) )  =  ( r ( .s `  g ) ( w a x ) ) )
3228oveqd 6106 . . . . . . . . . . . . . . . 16  |-  ( ( s  =  ( .s
`  g )  /\  f  =  (Scalar `  g
) )  ->  (
r s x )  =  ( r ( .s `  g ) x ) )
3329, 32oveq12d 6107 . . . . . . . . . . . . . . 15  |-  ( ( s  =  ( .s
`  g )  /\  f  =  (Scalar `  g
) )  ->  (
( r s w ) a ( r s x ) )  =  ( ( r ( .s `  g
) w ) a ( r ( .s
`  g ) x ) ) )
3431, 33eqeq12d 2455 . . . . . . . . . . . . . 14  |-  ( ( s  =  ( .s
`  g )  /\  f  =  (Scalar `  g
) )  ->  (
( r s ( w a x ) )  =  ( ( r s w ) a ( r s x ) )  <->  ( r
( .s `  g
) ( w a x ) )  =  ( ( r ( .s `  g ) w ) a ( r ( .s `  g ) x ) ) ) )
3525fveq2d 5693 . . . . . . . . . . . . . . . . 17  |-  ( ( s  =  ( .s
`  g )  /\  f  =  (Scalar `  g
) )  ->  ( +g  `  f )  =  ( +g  `  (Scalar `  g ) ) )
3635oveqd 6106 . . . . . . . . . . . . . . . 16  |-  ( ( s  =  ( .s
`  g )  /\  f  =  (Scalar `  g
) )  ->  (
q ( +g  `  f
) r )  =  ( q ( +g  `  (Scalar `  g )
) r ) )
37 eqidd 2442 . . . . . . . . . . . . . . . 16  |-  ( ( s  =  ( .s
`  g )  /\  f  =  (Scalar `  g
) )  ->  w  =  w )
3828, 36, 37oveq123d 6110 . . . . . . . . . . . . . . 15  |-  ( ( s  =  ( .s
`  g )  /\  f  =  (Scalar `  g
) )  ->  (
( q ( +g  `  f ) r ) s w )  =  ( ( q ( +g  `  (Scalar `  g ) ) r ) ( .s `  g ) w ) )
3928oveqd 6106 . . . . . . . . . . . . . . . 16  |-  ( ( s  =  ( .s
`  g )  /\  f  =  (Scalar `  g
) )  ->  (
q s w )  =  ( q ( .s `  g ) w ) )
4039, 29oveq12d 6107 . . . . . . . . . . . . . . 15  |-  ( ( s  =  ( .s
`  g )  /\  f  =  (Scalar `  g
) )  ->  (
( q s w ) a ( r s w ) )  =  ( ( q ( .s `  g
) w ) a ( r ( .s
`  g ) w ) ) )
4138, 40eqeq12d 2455 . . . . . . . . . . . . . 14  |-  ( ( s  =  ( .s
`  g )  /\  f  =  (Scalar `  g
) )  ->  (
( ( q ( +g  `  f ) r ) s w )  =  ( ( q s w ) a ( r s w ) )  <->  ( (
q ( +g  `  (Scalar `  g ) ) r ) ( .s `  g ) w )  =  ( ( q ( .s `  g
) w ) a ( r ( .s
`  g ) w ) ) ) )
4230, 34, 413anbi123d 1289 . . . . . . . . . . . . 13  |-  ( ( s  =  ( .s
`  g )  /\  f  =  (Scalar `  g
) )  ->  (
( ( r s w )  e.  v  /\  ( r s ( w a x ) )  =  ( ( r s w ) a ( r s x ) )  /\  ( ( q ( +g  `  f
) r ) s w )  =  ( ( q s w ) a ( r s w ) ) )  <->  ( ( r ( .s `  g
) w )  e.  v  /\  ( r ( .s `  g
) ( w a x ) )  =  ( ( r ( .s `  g ) w ) a ( r ( .s `  g ) x ) )  /\  ( ( q ( +g  `  (Scalar `  g ) ) r ) ( .s `  g ) w )  =  ( ( q ( .s `  g
) w ) a ( r ( .s
`  g ) w ) ) ) ) )
4325fveq2d 5693 . . . . . . . . . . . . . . . . 17  |-  ( ( s  =  ( .s
`  g )  /\  f  =  (Scalar `  g
) )  ->  ( .r `  f )  =  ( .r `  (Scalar `  g ) ) )
4443oveqd 6106 . . . . . . . . . . . . . . . 16  |-  ( ( s  =  ( .s
`  g )  /\  f  =  (Scalar `  g
) )  ->  (
q ( .r `  f ) r )  =  ( q ( .r `  (Scalar `  g ) ) r ) )
4528, 44, 37oveq123d 6110 . . . . . . . . . . . . . . 15  |-  ( ( s  =  ( .s
`  g )  /\  f  =  (Scalar `  g
) )  ->  (
( q ( .r
`  f ) r ) s w )  =  ( ( q ( .r `  (Scalar `  g ) ) r ) ( .s `  g ) w ) )
46 eqidd 2442 . . . . . . . . . . . . . . . 16  |-  ( ( s  =  ( .s
`  g )  /\  f  =  (Scalar `  g
) )  ->  q  =  q )
4728, 46, 29oveq123d 6110 . . . . . . . . . . . . . . 15  |-  ( ( s  =  ( .s
`  g )  /\  f  =  (Scalar `  g
) )  ->  (
q s ( r s w ) )  =  ( q ( .s `  g ) ( r ( .s
`  g ) w ) ) )
4845, 47eqeq12d 2455 . . . . . . . . . . . . . 14  |-  ( ( s  =  ( .s
`  g )  /\  f  =  (Scalar `  g
) )  ->  (
( ( q ( .r `  f ) r ) s w )  =  ( q s ( r s w ) )  <->  ( (
q ( .r `  (Scalar `  g ) ) r ) ( .s
`  g ) w )  =  ( q ( .s `  g
) ( r ( .s `  g ) w ) ) ) )
4925fveq2d 5693 . . . . . . . . . . . . . . . 16  |-  ( ( s  =  ( .s
`  g )  /\  f  =  (Scalar `  g
) )  ->  ( 1r `  f )  =  ( 1r `  (Scalar `  g ) ) )
5028, 49, 37oveq123d 6110 . . . . . . . . . . . . . . 15  |-  ( ( s  =  ( .s
`  g )  /\  f  =  (Scalar `  g
) )  ->  (
( 1r `  f
) s w )  =  ( ( 1r
`  (Scalar `  g )
) ( .s `  g ) w ) )
5150eqeq1d 2449 . . . . . . . . . . . . . 14  |-  ( ( s  =  ( .s
`  g )  /\  f  =  (Scalar `  g
) )  ->  (
( ( 1r `  f ) s w )  =  w  <->  ( ( 1r `  (Scalar `  g
) ) ( .s
`  g ) w )  =  w ) )
5225fveq2d 5693 . . . . . . . . . . . . . . . 16  |-  ( ( s  =  ( .s
`  g )  /\  f  =  (Scalar `  g
) )  ->  ( 0g `  f )  =  ( 0g `  (Scalar `  g ) ) )
5328, 52, 37oveq123d 6110 . . . . . . . . . . . . . . 15  |-  ( ( s  =  ( .s
`  g )  /\  f  =  (Scalar `  g
) )  ->  (
( 0g `  f
) s w )  =  ( ( 0g
`  (Scalar `  g )
) ( .s `  g ) w ) )
5453eqeq1d 2449 . . . . . . . . . . . . . 14  |-  ( ( s  =  ( .s
`  g )  /\  f  =  (Scalar `  g
) )  ->  (
( ( 0g `  f ) s w )  =  ( 0g
`  g )  <->  ( ( 0g `  (Scalar `  g
) ) ( .s
`  g ) w )  =  ( 0g
`  g ) ) )
5548, 51, 543anbi123d 1289 . . . . . . . . . . . . 13  |-  ( ( s  =  ( .s
`  g )  /\  f  =  (Scalar `  g
) )  ->  (
( ( ( q ( .r `  f
) r ) s w )  =  ( q s ( r s w ) )  /\  ( ( 1r
`  f ) s w )  =  w  /\  ( ( 0g
`  f ) s w )  =  ( 0g `  g ) )  <->  ( ( ( q ( .r `  (Scalar `  g ) ) r ) ( .s
`  g ) w )  =  ( q ( .s `  g
) ( r ( .s `  g ) w ) )  /\  ( ( 1r `  (Scalar `  g ) ) ( .s `  g
) w )  =  w  /\  ( ( 0g `  (Scalar `  g ) ) ( .s `  g ) w )  =  ( 0g `  g ) ) ) )
5642, 55anbi12d 710 . . . . . . . . . . . 12  |-  ( ( s  =  ( .s
`  g )  /\  f  =  (Scalar `  g
) )  ->  (
( ( ( r s w )  e.  v  /\  ( r s ( w a x ) )  =  ( ( r s w ) a ( r s x ) )  /\  ( ( q ( +g  `  f
) r ) s w )  =  ( ( q s w ) a ( r s w ) ) )  /\  ( ( ( q ( .r
`  f ) r ) s w )  =  ( q s ( r s w ) )  /\  (
( 1r `  f
) s w )  =  w  /\  (
( 0g `  f
) s w )  =  ( 0g `  g ) ) )  <-> 
( ( ( r ( .s `  g
) w )  e.  v  /\  ( r ( .s `  g
) ( w a x ) )  =  ( ( r ( .s `  g ) w ) a ( r ( .s `  g ) x ) )  /\  ( ( q ( +g  `  (Scalar `  g ) ) r ) ( .s `  g ) w )  =  ( ( q ( .s `  g
) w ) a ( r ( .s
`  g ) w ) ) )  /\  ( ( ( q ( .r `  (Scalar `  g ) ) r ) ( .s `  g ) w )  =  ( q ( .s `  g ) ( r ( .s
`  g ) w ) )  /\  (
( 1r `  (Scalar `  g ) ) ( .s `  g ) w )  =  w  /\  ( ( 0g
`  (Scalar `  g )
) ( .s `  g ) w )  =  ( 0g `  g ) ) ) ) )
57562ralbidv 2755 . . . . . . . . . . 11  |-  ( ( s  =  ( .s
`  g )  /\  f  =  (Scalar `  g
) )  ->  ( A. x  e.  v  A. w  e.  v 
( ( ( r s w )  e.  v  /\  ( r s ( w a x ) )  =  ( ( r s w ) a ( r s x ) )  /\  ( ( q ( +g  `  f
) r ) s w )  =  ( ( q s w ) a ( r s w ) ) )  /\  ( ( ( q ( .r
`  f ) r ) s w )  =  ( q s ( r s w ) )  /\  (
( 1r `  f
) s w )  =  w  /\  (
( 0g `  f
) s w )  =  ( 0g `  g ) ) )  <->  A. x  e.  v  A. w  e.  v 
( ( ( r ( .s `  g
) w )  e.  v  /\  ( r ( .s `  g
) ( w a x ) )  =  ( ( r ( .s `  g ) w ) a ( r ( .s `  g ) x ) )  /\  ( ( q ( +g  `  (Scalar `  g ) ) r ) ( .s `  g ) w )  =  ( ( q ( .s `  g
) w ) a ( r ( .s
`  g ) w ) ) )  /\  ( ( ( q ( .r `  (Scalar `  g ) ) r ) ( .s `  g ) w )  =  ( q ( .s `  g ) ( r ( .s
`  g ) w ) )  /\  (
( 1r `  (Scalar `  g ) ) ( .s `  g ) w )  =  w  /\  ( ( 0g
`  (Scalar `  g )
) ( .s `  g ) w )  =  ( 0g `  g ) ) ) ) )
5827, 57raleqbidv 2929 . . . . . . . . . 10  |-  ( ( s  =  ( .s
`  g )  /\  f  =  (Scalar `  g
) )  ->  ( A. r  e.  ( Base `  f ) A. x  e.  v  A. w  e.  v  (
( ( r s w )  e.  v  /\  ( r s ( w a x ) )  =  ( ( r s w ) a ( r s x ) )  /\  ( ( q ( +g  `  f
) r ) s w )  =  ( ( q s w ) a ( r s w ) ) )  /\  ( ( ( q ( .r
`  f ) r ) s w )  =  ( q s ( r s w ) )  /\  (
( 1r `  f
) s w )  =  w  /\  (
( 0g `  f
) s w )  =  ( 0g `  g ) ) )  <->  A. r  e.  ( Base `  (Scalar `  g
) ) A. x  e.  v  A. w  e.  v  ( (
( r ( .s
`  g ) w )  e.  v  /\  ( r ( .s
`  g ) ( w a x ) )  =  ( ( r ( .s `  g ) w ) a ( r ( .s `  g ) x ) )  /\  ( ( q ( +g  `  (Scalar `  g ) ) r ) ( .s `  g ) w )  =  ( ( q ( .s `  g
) w ) a ( r ( .s
`  g ) w ) ) )  /\  ( ( ( q ( .r `  (Scalar `  g ) ) r ) ( .s `  g ) w )  =  ( q ( .s `  g ) ( r ( .s
`  g ) w ) )  /\  (
( 1r `  (Scalar `  g ) ) ( .s `  g ) w )  =  w  /\  ( ( 0g
`  (Scalar `  g )
) ( .s `  g ) w )  =  ( 0g `  g ) ) ) ) )
5927, 58raleqbidv 2929 . . . . . . . . 9  |-  ( ( s  =  ( .s
`  g )  /\  f  =  (Scalar `  g
) )  ->  ( A. q  e.  ( Base `  f ) A. r  e.  ( Base `  f ) A. x  e.  v  A. w  e.  v  ( (
( r s w )  e.  v  /\  ( r s ( w a x ) )  =  ( ( r s w ) a ( r s x ) )  /\  ( ( q ( +g  `  f ) r ) s w )  =  ( ( q s w ) a ( r s w ) ) )  /\  ( ( ( q ( .r `  f ) r ) s w )  =  ( q s ( r s w ) )  /\  ( ( 1r `  f ) s w )  =  w  /\  ( ( 0g `  f ) s w )  =  ( 0g `  g
) ) )  <->  A. q  e.  ( Base `  (Scalar `  g ) ) A. r  e.  ( Base `  (Scalar `  g )
) A. x  e.  v  A. w  e.  v  ( ( ( r ( .s `  g ) w )  e.  v  /\  (
r ( .s `  g ) ( w a x ) )  =  ( ( r ( .s `  g
) w ) a ( r ( .s
`  g ) x ) )  /\  (
( q ( +g  `  (Scalar `  g )
) r ) ( .s `  g ) w )  =  ( ( q ( .s
`  g ) w ) a ( r ( .s `  g
) w ) ) )  /\  ( ( ( q ( .r
`  (Scalar `  g )
) r ) ( .s `  g ) w )  =  ( q ( .s `  g ) ( r ( .s `  g
) w ) )  /\  ( ( 1r
`  (Scalar `  g )
) ( .s `  g ) w )  =  w  /\  (
( 0g `  (Scalar `  g ) ) ( .s `  g ) w )  =  ( 0g `  g ) ) ) ) )
6026, 59anbi12d 710 . . . . . . . 8  |-  ( ( s  =  ( .s
`  g )  /\  f  =  (Scalar `  g
) )  ->  (
( f  e. SRing  /\  A. q  e.  ( Base `  f ) A. r  e.  ( Base `  f
) A. x  e.  v  A. w  e.  v  ( ( ( r s w )  e.  v  /\  (
r s ( w a x ) )  =  ( ( r s w ) a ( r s x ) )  /\  (
( q ( +g  `  f ) r ) s w )  =  ( ( q s w ) a ( r s w ) ) )  /\  (
( ( q ( .r `  f ) r ) s w )  =  ( q s ( r s w ) )  /\  ( ( 1r `  f ) s w )  =  w  /\  ( ( 0g `  f ) s w )  =  ( 0g
`  g ) ) ) )  <->  ( (Scalar `  g )  e. SRing  /\  A. q  e.  ( Base `  (Scalar `  g )
) A. r  e.  ( Base `  (Scalar `  g ) ) A. x  e.  v  A. w  e.  v  (
( ( r ( .s `  g ) w )  e.  v  /\  ( r ( .s `  g ) ( w a x ) )  =  ( ( r ( .s
`  g ) w ) a ( r ( .s `  g
) x ) )  /\  ( ( q ( +g  `  (Scalar `  g ) ) r ) ( .s `  g ) w )  =  ( ( q ( .s `  g
) w ) a ( r ( .s
`  g ) w ) ) )  /\  ( ( ( q ( .r `  (Scalar `  g ) ) r ) ( .s `  g ) w )  =  ( q ( .s `  g ) ( r ( .s
`  g ) w ) )  /\  (
( 1r `  (Scalar `  g ) ) ( .s `  g ) w )  =  w  /\  ( ( 0g
`  (Scalar `  g )
) ( .s `  g ) w )  =  ( 0g `  g ) ) ) ) ) )
6124, 60syl5bb 257 . . . . . . 7  |-  ( ( s  =  ( .s
`  g )  /\  f  =  (Scalar `  g
) )  ->  ( [. ( Base `  f
)  /  k ]. [. ( +g  `  f
)  /  p ]. [. ( .r `  f
)  /  t ]. ( f  e. SRing  /\  A. q  e.  k  A. r  e.  k  A. x  e.  v  A. w  e.  v  (
( ( r s w )  e.  v  /\  ( r s ( w a x ) )  =  ( ( r s w ) a ( r s x ) )  /\  ( ( q p r ) s w )  =  ( ( q s w ) a ( r s w ) ) )  /\  ( ( ( q t r ) s w )  =  ( q s ( r s w ) )  /\  (
( 1r `  f
) s w )  =  w  /\  (
( 0g `  f
) s w )  =  ( 0g `  g ) ) ) )  <->  ( (Scalar `  g )  e. SRing  /\  A. q  e.  ( Base `  (Scalar `  g )
) A. r  e.  ( Base `  (Scalar `  g ) ) A. x  e.  v  A. w  e.  v  (
( ( r ( .s `  g ) w )  e.  v  /\  ( r ( .s `  g ) ( w a x ) )  =  ( ( r ( .s
`  g ) w ) a ( r ( .s `  g
) x ) )  /\  ( ( q ( +g  `  (Scalar `  g ) ) r ) ( .s `  g ) w )  =  ( ( q ( .s `  g
) w ) a ( r ( .s
`  g ) w ) ) )  /\  ( ( ( q ( .r `  (Scalar `  g ) ) r ) ( .s `  g ) w )  =  ( q ( .s `  g ) ( r ( .s
`  g ) w ) )  /\  (
( 1r `  (Scalar `  g ) ) ( .s `  g ) w )  =  w  /\  ( ( 0g
`  (Scalar `  g )
) ( .s `  g ) w )  =  ( 0g `  g ) ) ) ) ) )
623, 4, 61sbc2ie 3260 . . . . . 6  |-  ( [. ( .s `  g )  /  s ]. [. (Scalar `  g )  /  f ]. [. ( Base `  f
)  /  k ]. [. ( +g  `  f
)  /  p ]. [. ( .r `  f
)  /  t ]. ( f  e. SRing  /\  A. q  e.  k  A. r  e.  k  A. x  e.  v  A. w  e.  v  (
( ( r s w )  e.  v  /\  ( r s ( w a x ) )  =  ( ( r s w ) a ( r s x ) )  /\  ( ( q p r ) s w )  =  ( ( q s w ) a ( r s w ) ) )  /\  ( ( ( q t r ) s w )  =  ( q s ( r s w ) )  /\  (
( 1r `  f
) s w )  =  w  /\  (
( 0g `  f
) s w )  =  ( 0g `  g ) ) ) )  <->  ( (Scalar `  g )  e. SRing  /\  A. q  e.  ( Base `  (Scalar `  g )
) A. r  e.  ( Base `  (Scalar `  g ) ) A. x  e.  v  A. w  e.  v  (
( ( r ( .s `  g ) w )  e.  v  /\  ( r ( .s `  g ) ( w a x ) )  =  ( ( r ( .s
`  g ) w ) a ( r ( .s `  g
) x ) )  /\  ( ( q ( +g  `  (Scalar `  g ) ) r ) ( .s `  g ) w )  =  ( ( q ( .s `  g
) w ) a ( r ( .s
`  g ) w ) ) )  /\  ( ( ( q ( .r `  (Scalar `  g ) ) r ) ( .s `  g ) w )  =  ( q ( .s `  g ) ( r ( .s
`  g ) w ) )  /\  (
( 1r `  (Scalar `  g ) ) ( .s `  g ) w )  =  w  /\  ( ( 0g
`  (Scalar `  g )
) ( .s `  g ) w )  =  ( 0g `  g ) ) ) ) )
63 simpl 457 . . . . . . . . 9  |-  ( ( v  =  ( Base `  g )  /\  a  =  ( +g  `  g
) )  ->  v  =  ( Base `  g
) )
6463eleq2d 2508 . . . . . . . . . . . 12  |-  ( ( v  =  ( Base `  g )  /\  a  =  ( +g  `  g
) )  ->  (
( r ( .s
`  g ) w )  e.  v  <->  ( r
( .s `  g
) w )  e.  ( Base `  g
) ) )
65 simpr 461 . . . . . . . . . . . . . . 15  |-  ( ( v  =  ( Base `  g )  /\  a  =  ( +g  `  g
) )  ->  a  =  ( +g  `  g
) )
6665oveqd 6106 . . . . . . . . . . . . . 14  |-  ( ( v  =  ( Base `  g )  /\  a  =  ( +g  `  g
) )  ->  (
w a x )  =  ( w ( +g  `  g ) x ) )
6766oveq2d 6105 . . . . . . . . . . . . 13  |-  ( ( v  =  ( Base `  g )  /\  a  =  ( +g  `  g
) )  ->  (
r ( .s `  g ) ( w a x ) )  =  ( r ( .s `  g ) ( w ( +g  `  g ) x ) ) )
6865oveqd 6106 . . . . . . . . . . . . 13  |-  ( ( v  =  ( Base `  g )  /\  a  =  ( +g  `  g
) )  ->  (
( r ( .s
`  g ) w ) a ( r ( .s `  g
) x ) )  =  ( ( r ( .s `  g
) w ) ( +g  `  g ) ( r ( .s
`  g ) x ) ) )
6967, 68eqeq12d 2455 . . . . . . . . . . . 12  |-  ( ( v  =  ( Base `  g )  /\  a  =  ( +g  `  g
) )  ->  (
( r ( .s
`  g ) ( w a x ) )  =  ( ( r ( .s `  g ) w ) a ( r ( .s `  g ) x ) )  <->  ( r
( .s `  g
) ( w ( +g  `  g ) x ) )  =  ( ( r ( .s `  g ) w ) ( +g  `  g ) ( r ( .s `  g
) x ) ) ) )
7065oveqd 6106 . . . . . . . . . . . . 13  |-  ( ( v  =  ( Base `  g )  /\  a  =  ( +g  `  g
) )  ->  (
( q ( .s
`  g ) w ) a ( r ( .s `  g
) w ) )  =  ( ( q ( .s `  g
) w ) ( +g  `  g ) ( r ( .s
`  g ) w ) ) )
7170eqeq2d 2452 . . . . . . . . . . . 12  |-  ( ( v  =  ( Base `  g )  /\  a  =  ( +g  `  g
) )  ->  (
( ( q ( +g  `  (Scalar `  g ) ) r ) ( .s `  g ) w )  =  ( ( q ( .s `  g
) w ) a ( r ( .s
`  g ) w ) )  <->  ( (
q ( +g  `  (Scalar `  g ) ) r ) ( .s `  g ) w )  =  ( ( q ( .s `  g
) w ) ( +g  `  g ) ( r ( .s
`  g ) w ) ) ) )
7264, 69, 713anbi123d 1289 . . . . . . . . . . 11  |-  ( ( v  =  ( Base `  g )  /\  a  =  ( +g  `  g
) )  ->  (
( ( r ( .s `  g ) w )  e.  v  /\  ( r ( .s `  g ) ( w a x ) )  =  ( ( r ( .s
`  g ) w ) a ( r ( .s `  g
) x ) )  /\  ( ( q ( +g  `  (Scalar `  g ) ) r ) ( .s `  g ) w )  =  ( ( q ( .s `  g
) w ) a ( r ( .s
`  g ) w ) ) )  <->  ( (
r ( .s `  g ) w )  e.  ( Base `  g
)  /\  ( r
( .s `  g
) ( w ( +g  `  g ) x ) )  =  ( ( r ( .s `  g ) w ) ( +g  `  g ) ( r ( .s `  g
) x ) )  /\  ( ( q ( +g  `  (Scalar `  g ) ) r ) ( .s `  g ) w )  =  ( ( q ( .s `  g
) w ) ( +g  `  g ) ( r ( .s
`  g ) w ) ) ) ) )
7372anbi1d 704 . . . . . . . . . 10  |-  ( ( v  =  ( Base `  g )  /\  a  =  ( +g  `  g
) )  ->  (
( ( ( r ( .s `  g
) w )  e.  v  /\  ( r ( .s `  g
) ( w a x ) )  =  ( ( r ( .s `  g ) w ) a ( r ( .s `  g ) x ) )  /\  ( ( q ( +g  `  (Scalar `  g ) ) r ) ( .s `  g ) w )  =  ( ( q ( .s `  g
) w ) a ( r ( .s
`  g ) w ) ) )  /\  ( ( ( q ( .r `  (Scalar `  g ) ) r ) ( .s `  g ) w )  =  ( q ( .s `  g ) ( r ( .s
`  g ) w ) )  /\  (
( 1r `  (Scalar `  g ) ) ( .s `  g ) w )  =  w  /\  ( ( 0g
`  (Scalar `  g )
) ( .s `  g ) w )  =  ( 0g `  g ) ) )  <-> 
( ( ( r ( .s `  g
) w )  e.  ( Base `  g
)  /\  ( r
( .s `  g
) ( w ( +g  `  g ) x ) )  =  ( ( r ( .s `  g ) w ) ( +g  `  g ) ( r ( .s `  g
) x ) )  /\  ( ( q ( +g  `  (Scalar `  g ) ) r ) ( .s `  g ) w )  =  ( ( q ( .s `  g
) w ) ( +g  `  g ) ( r ( .s
`  g ) w ) ) )  /\  ( ( ( q ( .r `  (Scalar `  g ) ) r ) ( .s `  g ) w )  =  ( q ( .s `  g ) ( r ( .s
`  g ) w ) )  /\  (
( 1r `  (Scalar `  g ) ) ( .s `  g ) w )  =  w  /\  ( ( 0g
`  (Scalar `  g )
) ( .s `  g ) w )  =  ( 0g `  g ) ) ) ) )
7463, 73raleqbidv 2929 . . . . . . . . 9  |-  ( ( v  =  ( Base `  g )  /\  a  =  ( +g  `  g
) )  ->  ( A. w  e.  v 
( ( ( r ( .s `  g
) w )  e.  v  /\  ( r ( .s `  g
) ( w a x ) )  =  ( ( r ( .s `  g ) w ) a ( r ( .s `  g ) x ) )  /\  ( ( q ( +g  `  (Scalar `  g ) ) r ) ( .s `  g ) w )  =  ( ( q ( .s `  g
) w ) a ( r ( .s
`  g ) w ) ) )  /\  ( ( ( q ( .r `  (Scalar `  g ) ) r ) ( .s `  g ) w )  =  ( q ( .s `  g ) ( r ( .s
`  g ) w ) )  /\  (
( 1r `  (Scalar `  g ) ) ( .s `  g ) w )  =  w  /\  ( ( 0g
`  (Scalar `  g )
) ( .s `  g ) w )  =  ( 0g `  g ) ) )  <->  A. w  e.  ( Base `  g ) ( ( ( r ( .s `  g ) w )  e.  (
Base `  g )  /\  ( r ( .s
`  g ) ( w ( +g  `  g
) x ) )  =  ( ( r ( .s `  g
) w ) ( +g  `  g ) ( r ( .s
`  g ) x ) )  /\  (
( q ( +g  `  (Scalar `  g )
) r ) ( .s `  g ) w )  =  ( ( q ( .s
`  g ) w ) ( +g  `  g
) ( r ( .s `  g ) w ) ) )  /\  ( ( ( q ( .r `  (Scalar `  g ) ) r ) ( .s
`  g ) w )  =  ( q ( .s `  g
) ( r ( .s `  g ) w ) )  /\  ( ( 1r `  (Scalar `  g ) ) ( .s `  g
) w )  =  w  /\  ( ( 0g `  (Scalar `  g ) ) ( .s `  g ) w )  =  ( 0g `  g ) ) ) ) )
7563, 74raleqbidv 2929 . . . . . . . 8  |-  ( ( v  =  ( Base `  g )  /\  a  =  ( +g  `  g
) )  ->  ( A. x  e.  v  A. w  e.  v 
( ( ( r ( .s `  g
) w )  e.  v  /\  ( r ( .s `  g
) ( w a x ) )  =  ( ( r ( .s `  g ) w ) a ( r ( .s `  g ) x ) )  /\  ( ( q ( +g  `  (Scalar `  g ) ) r ) ( .s `  g ) w )  =  ( ( q ( .s `  g
) w ) a ( r ( .s
`  g ) w ) ) )  /\  ( ( ( q ( .r `  (Scalar `  g ) ) r ) ( .s `  g ) w )  =  ( q ( .s `  g ) ( r ( .s
`  g ) w ) )  /\  (
( 1r `  (Scalar `  g ) ) ( .s `  g ) w )  =  w  /\  ( ( 0g
`  (Scalar `  g )
) ( .s `  g ) w )  =  ( 0g `  g ) ) )  <->  A. x  e.  ( Base `  g ) A. w  e.  ( Base `  g ) ( ( ( r ( .s
`  g ) w )  e.  ( Base `  g )  /\  (
r ( .s `  g ) ( w ( +g  `  g
) x ) )  =  ( ( r ( .s `  g
) w ) ( +g  `  g ) ( r ( .s
`  g ) x ) )  /\  (
( q ( +g  `  (Scalar `  g )
) r ) ( .s `  g ) w )  =  ( ( q ( .s
`  g ) w ) ( +g  `  g
) ( r ( .s `  g ) w ) ) )  /\  ( ( ( q ( .r `  (Scalar `  g ) ) r ) ( .s
`  g ) w )  =  ( q ( .s `  g
) ( r ( .s `  g ) w ) )  /\  ( ( 1r `  (Scalar `  g ) ) ( .s `  g
) w )  =  w  /\  ( ( 0g `  (Scalar `  g ) ) ( .s `  g ) w )  =  ( 0g `  g ) ) ) ) )
76752ralbidv 2755 . . . . . . 7  |-  ( ( v  =  ( Base `  g )  /\  a  =  ( +g  `  g
) )  ->  ( A. q  e.  ( Base `  (Scalar `  g
) ) A. r  e.  ( Base `  (Scalar `  g ) ) A. x  e.  v  A. w  e.  v  (
( ( r ( .s `  g ) w )  e.  v  /\  ( r ( .s `  g ) ( w a x ) )  =  ( ( r ( .s
`  g ) w ) a ( r ( .s `  g
) x ) )  /\  ( ( q ( +g  `  (Scalar `  g ) ) r ) ( .s `  g ) w )  =  ( ( q ( .s `  g
) w ) a ( r ( .s
`  g ) w ) ) )  /\  ( ( ( q ( .r `  (Scalar `  g ) ) r ) ( .s `  g ) w )  =  ( q ( .s `  g ) ( r ( .s
`  g ) w ) )  /\  (
( 1r `  (Scalar `  g ) ) ( .s `  g ) w )  =  w  /\  ( ( 0g
`  (Scalar `  g )
) ( .s `  g ) w )  =  ( 0g `  g ) ) )  <->  A. q  e.  ( Base `  (Scalar `  g
) ) A. r  e.  ( Base `  (Scalar `  g ) ) A. x  e.  ( Base `  g ) A. w  e.  ( Base `  g
) ( ( ( r ( .s `  g ) w )  e.  ( Base `  g
)  /\  ( r
( .s `  g
) ( w ( +g  `  g ) x ) )  =  ( ( r ( .s `  g ) w ) ( +g  `  g ) ( r ( .s `  g
) x ) )  /\  ( ( q ( +g  `  (Scalar `  g ) ) r ) ( .s `  g ) w )  =  ( ( q ( .s `  g
) w ) ( +g  `  g ) ( r ( .s
`  g ) w ) ) )  /\  ( ( ( q ( .r `  (Scalar `  g ) ) r ) ( .s `  g ) w )  =  ( q ( .s `  g ) ( r ( .s
`  g ) w ) )  /\  (
( 1r `  (Scalar `  g ) ) ( .s `  g ) w )  =  w  /\  ( ( 0g
`  (Scalar `  g )
) ( .s `  g ) w )  =  ( 0g `  g ) ) ) ) )
7776anbi2d 703 . . . . . 6  |-  ( ( v  =  ( Base `  g )  /\  a  =  ( +g  `  g
) )  ->  (
( (Scalar `  g
)  e. SRing  /\  A. q  e.  ( Base `  (Scalar `  g ) ) A. r  e.  ( Base `  (Scalar `  g )
) A. x  e.  v  A. w  e.  v  ( ( ( r ( .s `  g ) w )  e.  v  /\  (
r ( .s `  g ) ( w a x ) )  =  ( ( r ( .s `  g
) w ) a ( r ( .s
`  g ) x ) )  /\  (
( q ( +g  `  (Scalar `  g )
) r ) ( .s `  g ) w )  =  ( ( q ( .s
`  g ) w ) a ( r ( .s `  g
) w ) ) )  /\  ( ( ( q ( .r
`  (Scalar `  g )
) r ) ( .s `  g ) w )  =  ( q ( .s `  g ) ( r ( .s `  g
) w ) )  /\  ( ( 1r
`  (Scalar `  g )
) ( .s `  g ) w )  =  w  /\  (
( 0g `  (Scalar `  g ) ) ( .s `  g ) w )  =  ( 0g `  g ) ) ) )  <->  ( (Scalar `  g )  e. SRing  /\  A. q  e.  ( Base `  (Scalar `  g )
) A. r  e.  ( Base `  (Scalar `  g ) ) A. x  e.  ( Base `  g ) A. w  e.  ( Base `  g
) ( ( ( r ( .s `  g ) w )  e.  ( Base `  g
)  /\  ( r
( .s `  g
) ( w ( +g  `  g ) x ) )  =  ( ( r ( .s `  g ) w ) ( +g  `  g ) ( r ( .s `  g
) x ) )  /\  ( ( q ( +g  `  (Scalar `  g ) ) r ) ( .s `  g ) w )  =  ( ( q ( .s `  g
) w ) ( +g  `  g ) ( r ( .s
`  g ) w ) ) )  /\  ( ( ( q ( .r `  (Scalar `  g ) ) r ) ( .s `  g ) w )  =  ( q ( .s `  g ) ( r ( .s
`  g ) w ) )  /\  (
( 1r `  (Scalar `  g ) ) ( .s `  g ) w )  =  w  /\  ( ( 0g
`  (Scalar `  g )
) ( .s `  g ) w )  =  ( 0g `  g ) ) ) ) ) )
7862, 77syl5bb 257 . . . . 5  |-  ( ( v  =  ( Base `  g )  /\  a  =  ( +g  `  g
) )  ->  ( [. ( .s `  g
)  /  s ]. [. (Scalar `  g )  /  f ]. [. ( Base `  f )  / 
k ]. [. ( +g  `  f )  /  p ]. [. ( .r `  f )  /  t ]. ( f  e. SRing  /\  A. q  e.  k  A. r  e.  k  A. x  e.  v  A. w  e.  v  (
( ( r s w )  e.  v  /\  ( r s ( w a x ) )  =  ( ( r s w ) a ( r s x ) )  /\  ( ( q p r ) s w )  =  ( ( q s w ) a ( r s w ) ) )  /\  ( ( ( q t r ) s w )  =  ( q s ( r s w ) )  /\  (
( 1r `  f
) s w )  =  w  /\  (
( 0g `  f
) s w )  =  ( 0g `  g ) ) ) )  <->  ( (Scalar `  g )  e. SRing  /\  A. q  e.  ( Base `  (Scalar `  g )
) A. r  e.  ( Base `  (Scalar `  g ) ) A. x  e.  ( Base `  g ) A. w  e.  ( Base `  g
) ( ( ( r ( .s `  g ) w )  e.  ( Base `  g
)  /\  ( r
( .s `  g
) ( w ( +g  `  g ) x ) )  =  ( ( r ( .s `  g ) w ) ( +g  `  g ) ( r ( .s `  g
) x ) )  /\  ( ( q ( +g  `  (Scalar `  g ) ) r ) ( .s `  g ) w )  =  ( ( q ( .s `  g
) w ) ( +g  `  g ) ( r ( .s
`  g ) w ) ) )  /\  ( ( ( q ( .r `  (Scalar `  g ) ) r ) ( .s `  g ) w )  =  ( q ( .s `  g ) ( r ( .s
`  g ) w ) )  /\  (
( 1r `  (Scalar `  g ) ) ( .s `  g ) w )  =  w  /\  ( ( 0g
`  (Scalar `  g )
) ( .s `  g ) w )  =  ( 0g `  g ) ) ) ) ) )
791, 2, 78sbc2ie 3260 . . . 4  |-  ( [. ( Base `  g )  /  v ]. [. ( +g  `  g )  / 
a ]. [. ( .s
`  g )  / 
s ]. [. (Scalar `  g )  /  f ]. [. ( Base `  f
)  /  k ]. [. ( +g  `  f
)  /  p ]. [. ( .r `  f
)  /  t ]. ( f  e. SRing  /\  A. q  e.  k  A. r  e.  k  A. x  e.  v  A. w  e.  v  (
( ( r s w )  e.  v  /\  ( r s ( w a x ) )  =  ( ( r s w ) a ( r s x ) )  /\  ( ( q p r ) s w )  =  ( ( q s w ) a ( r s w ) ) )  /\  ( ( ( q t r ) s w )  =  ( q s ( r s w ) )  /\  (
( 1r `  f
) s w )  =  w  /\  (
( 0g `  f
) s w )  =  ( 0g `  g ) ) ) )  <->  ( (Scalar `  g )  e. SRing  /\  A. q  e.  ( Base `  (Scalar `  g )
) A. r  e.  ( Base `  (Scalar `  g ) ) A. x  e.  ( Base `  g ) A. w  e.  ( Base `  g
) ( ( ( r ( .s `  g ) w )  e.  ( Base `  g
)  /\  ( r
( .s `  g
) ( w ( +g  `  g ) x ) )  =  ( ( r ( .s `  g ) w ) ( +g  `  g ) ( r ( .s `  g
) x ) )  /\  ( ( q ( +g  `  (Scalar `  g ) ) r ) ( .s `  g ) w )  =  ( ( q ( .s `  g
) w ) ( +g  `  g ) ( r ( .s
`  g ) w ) ) )  /\  ( ( ( q ( .r `  (Scalar `  g ) ) r ) ( .s `  g ) w )  =  ( q ( .s `  g ) ( r ( .s
`  g ) w ) )  /\  (
( 1r `  (Scalar `  g ) ) ( .s `  g ) w )  =  w  /\  ( ( 0g
`  (Scalar `  g )
) ( .s `  g ) w )  =  ( 0g `  g ) ) ) ) )
80 fveq2 5689 . . . . . . 7  |-  ( g  =  W  ->  (Scalar `  g )  =  (Scalar `  W ) )
81 isslmd.f . . . . . . 7  |-  F  =  (Scalar `  W )
8280, 81syl6eqr 2491 . . . . . 6  |-  ( g  =  W  ->  (Scalar `  g )  =  F )
8382eleq1d 2507 . . . . 5  |-  ( g  =  W  ->  (
(Scalar `  g )  e. SRing  <-> 
F  e. SRing ) )
8482fveq2d 5693 . . . . . . 7  |-  ( g  =  W  ->  ( Base `  (Scalar `  g
) )  =  (
Base `  F )
)
85 isslmd.k . . . . . . 7  |-  K  =  ( Base `  F
)
8684, 85syl6eqr 2491 . . . . . 6  |-  ( g  =  W  ->  ( Base `  (Scalar `  g
) )  =  K )
87 fveq2 5689 . . . . . . . . 9  |-  ( g  =  W  ->  ( Base `  g )  =  ( Base `  W
) )
88 isslmd.v . . . . . . . . 9  |-  V  =  ( Base `  W
)
8987, 88syl6eqr 2491 . . . . . . . 8  |-  ( g  =  W  ->  ( Base `  g )  =  V )
90 fveq2 5689 . . . . . . . . . . . . . 14  |-  ( g  =  W  ->  ( .s `  g )  =  ( .s `  W
) )
91 isslmd.s . . . . . . . . . . . . . 14  |-  .x.  =  ( .s `  W )
9290, 91syl6eqr 2491 . . . . . . . . . . . . 13  |-  ( g  =  W  ->  ( .s `  g )  = 
.x.  )
9392oveqd 6106 . . . . . . . . . . . 12  |-  ( g  =  W  ->  (
r ( .s `  g ) w )  =  ( r  .x.  w ) )
9493, 89eleq12d 2509 . . . . . . . . . . 11  |-  ( g  =  W  ->  (
( r ( .s
`  g ) w )  e.  ( Base `  g )  <->  ( r  .x.  w )  e.  V
) )
95 eqidd 2442 . . . . . . . . . . . . 13  |-  ( g  =  W  ->  r  =  r )
96 fveq2 5689 . . . . . . . . . . . . . . 15  |-  ( g  =  W  ->  ( +g  `  g )  =  ( +g  `  W
) )
97 isslmd.a . . . . . . . . . . . . . . 15  |-  .+  =  ( +g  `  W )
9896, 97syl6eqr 2491 . . . . . . . . . . . . . 14  |-  ( g  =  W  ->  ( +g  `  g )  = 
.+  )
9998oveqd 6106 . . . . . . . . . . . . 13  |-  ( g  =  W  ->  (
w ( +g  `  g
) x )  =  ( w  .+  x
) )
10092, 95, 99oveq123d 6110 . . . . . . . . . . . 12  |-  ( g  =  W  ->  (
r ( .s `  g ) ( w ( +g  `  g
) x ) )  =  ( r  .x.  ( w  .+  x ) ) )
10192oveqd 6106 . . . . . . . . . . . . 13  |-  ( g  =  W  ->  (
r ( .s `  g ) x )  =  ( r  .x.  x ) )
10298, 93, 101oveq123d 6110 . . . . . . . . . . . 12  |-  ( g  =  W  ->  (
( r ( .s
`  g ) w ) ( +g  `  g
) ( r ( .s `  g ) x ) )  =  ( ( r  .x.  w )  .+  (
r  .x.  x )
) )
103100, 102eqeq12d 2455 . . . . . . . . . . 11  |-  ( g  =  W  ->  (
( r ( .s
`  g ) ( w ( +g  `  g
) x ) )  =  ( ( r ( .s `  g
) w ) ( +g  `  g ) ( r ( .s
`  g ) x ) )  <->  ( r  .x.  ( w  .+  x
) )  =  ( ( r  .x.  w
)  .+  ( r  .x.  x ) ) ) )
10482fveq2d 5693 . . . . . . . . . . . . . . 15  |-  ( g  =  W  ->  ( +g  `  (Scalar `  g
) )  =  ( +g  `  F ) )
105 isslmd.p . . . . . . . . . . . . . . 15  |-  .+^  =  ( +g  `  F )
106104, 105syl6eqr 2491 . . . . . . . . . . . . . 14  |-  ( g  =  W  ->  ( +g  `  (Scalar `  g
) )  =  .+^  )
107106oveqd 6106 . . . . . . . . . . . . 13  |-  ( g  =  W  ->  (
q ( +g  `  (Scalar `  g ) ) r )  =  ( q 
.+^  r ) )
108 eqidd 2442 . . . . . . . . . . . . 13  |-  ( g  =  W  ->  w  =  w )
10992, 107, 108oveq123d 6110 . . . . . . . . . . . 12  |-  ( g  =  W  ->  (
( q ( +g  `  (Scalar `  g )
) r ) ( .s `  g ) w )  =  ( ( q  .+^  r ) 
.x.  w ) )
11092oveqd 6106 . . . . . . . . . . . . 13  |-  ( g  =  W  ->  (
q ( .s `  g ) w )  =  ( q  .x.  w ) )
11198, 110, 93oveq123d 6110 . . . . . . . . . . . 12  |-  ( g  =  W  ->  (
( q ( .s
`  g ) w ) ( +g  `  g
) ( r ( .s `  g ) w ) )  =  ( ( q  .x.  w )  .+  (
r  .x.  w )
) )
112109, 111eqeq12d 2455 . . . . . . . . . . 11  |-  ( g  =  W  ->  (
( ( q ( +g  `  (Scalar `  g ) ) r ) ( .s `  g ) w )  =  ( ( q ( .s `  g
) w ) ( +g  `  g ) ( r ( .s
`  g ) w ) )  <->  ( (
q  .+^  r )  .x.  w )  =  ( ( q  .x.  w
)  .+  ( r  .x.  w ) ) ) )
11394, 103, 1123anbi123d 1289 . . . . . . . . . 10  |-  ( g  =  W  ->  (
( ( r ( .s `  g ) w )  e.  (
Base `  g )  /\  ( r ( .s
`  g ) ( w ( +g  `  g
) x ) )  =  ( ( r ( .s `  g
) w ) ( +g  `  g ) ( r ( .s
`  g ) x ) )  /\  (
( q ( +g  `  (Scalar `  g )
) r ) ( .s `  g ) w )  =  ( ( q ( .s
`  g ) w ) ( +g  `  g
) ( r ( .s `  g ) w ) ) )  <-> 
( ( r  .x.  w )  e.  V  /\  ( r  .x.  (
w  .+  x )
)  =  ( ( r  .x.  w ) 
.+  ( r  .x.  x ) )  /\  ( ( q  .+^  r )  .x.  w
)  =  ( ( q  .x.  w ) 
.+  ( r  .x.  w ) ) ) ) )
11482fveq2d 5693 . . . . . . . . . . . . . . 15  |-  ( g  =  W  ->  ( .r `  (Scalar `  g
) )  =  ( .r `  F ) )
115 isslmd.t . . . . . . . . . . . . . . 15  |-  .X.  =  ( .r `  F )
116114, 115syl6eqr 2491 . . . . . . . . . . . . . 14  |-  ( g  =  W  ->  ( .r `  (Scalar `  g
) )  =  .X.  )
117116oveqd 6106 . . . . . . . . . . . . 13  |-  ( g  =  W  ->  (
q ( .r `  (Scalar `  g ) ) r )  =  ( q  .X.  r )
)
11892, 117, 108oveq123d 6110 . . . . . . . . . . . 12  |-  ( g  =  W  ->  (
( q ( .r
`  (Scalar `  g )
) r ) ( .s `  g ) w )  =  ( ( q  .X.  r
)  .x.  w )
)
119 eqidd 2442 . . . . . . . . . . . . 13  |-  ( g  =  W  ->  q  =  q )
12092, 119, 93oveq123d 6110 . . . . . . . . . . . 12  |-  ( g  =  W  ->  (
q ( .s `  g ) ( r ( .s `  g
) w ) )  =  ( q  .x.  ( r  .x.  w
) ) )
121118, 120eqeq12d 2455 . . . . . . . . . . 11  |-  ( g  =  W  ->  (
( ( q ( .r `  (Scalar `  g ) ) r ) ( .s `  g ) w )  =  ( q ( .s `  g ) ( r ( .s
`  g ) w ) )  <->  ( (
q  .X.  r )  .x.  w )  =  ( q  .x.  ( r 
.x.  w ) ) ) )
12282fveq2d 5693 . . . . . . . . . . . . . 14  |-  ( g  =  W  ->  ( 1r `  (Scalar `  g
) )  =  ( 1r `  F ) )
123 isslmd.u . . . . . . . . . . . . . 14  |-  .1.  =  ( 1r `  F )
124122, 123syl6eqr 2491 . . . . . . . . . . . . 13  |-  ( g  =  W  ->  ( 1r `  (Scalar `  g
) )  =  .1.  )
12592, 124, 108oveq123d 6110 . . . . . . . . . . . 12  |-  ( g  =  W  ->  (
( 1r `  (Scalar `  g ) ) ( .s `  g ) w )  =  (  .1.  .x.  w )
)
126125eqeq1d 2449 . . . . . . . . . . 11  |-  ( g  =  W  ->  (
( ( 1r `  (Scalar `  g ) ) ( .s `  g
) w )  =  w  <->  (  .1.  .x.  w )  =  w ) )
12782fveq2d 5693 . . . . . . . . . . . . . 14  |-  ( g  =  W  ->  ( 0g `  (Scalar `  g
) )  =  ( 0g `  F ) )
128 isslmd.o . . . . . . . . . . . . . 14  |-  O  =  ( 0g `  F
)
129127, 128syl6eqr 2491 . . . . . . . . . . . . 13  |-  ( g  =  W  ->  ( 0g `  (Scalar `  g
) )  =  O )
13092, 129, 108oveq123d 6110 . . . . . . . . . . . 12  |-  ( g  =  W  ->  (
( 0g `  (Scalar `  g ) ) ( .s `  g ) w )  =  ( O  .x.  w ) )
131 fveq2 5689 . . . . . . . . . . . . 13  |-  ( g  =  W  ->  ( 0g `  g )  =  ( 0g `  W
) )
132 isslmd.0 . . . . . . . . . . . . 13  |-  .0.  =  ( 0g `  W )
133131, 132syl6eqr 2491 . . . . . . . . . . . 12  |-  ( g  =  W  ->  ( 0g `  g )  =  .0.  )
134130, 133eqeq12d 2455 . . . . . . . . . . 11  |-  ( g  =  W  ->  (
( ( 0g `  (Scalar `  g ) ) ( .s `  g
) w )  =  ( 0g `  g
)  <->  ( O  .x.  w )  =  .0.  ) )
135121, 126, 1343anbi123d 1289 . . . . . . . . . 10  |-  ( g  =  W  ->  (
( ( ( q ( .r `  (Scalar `  g ) ) r ) ( .s `  g ) w )  =  ( q ( .s `  g ) ( r ( .s
`  g ) w ) )  /\  (
( 1r `  (Scalar `  g ) ) ( .s `  g ) w )  =  w  /\  ( ( 0g
`  (Scalar `  g )
) ( .s `  g ) w )  =  ( 0g `  g ) )  <->  ( (
( q  .X.  r
)  .x.  w )  =  ( q  .x.  ( r  .x.  w
) )  /\  (  .1.  .x.  w )  =  w  /\  ( O 
.x.  w )  =  .0.  ) ) )
136113, 135anbi12d 710 . . . . . . . . 9  |-  ( g  =  W  ->  (
( ( ( r ( .s `  g
) w )  e.  ( Base `  g
)  /\  ( r
( .s `  g
) ( w ( +g  `  g ) x ) )  =  ( ( r ( .s `  g ) w ) ( +g  `  g ) ( r ( .s `  g
) x ) )  /\  ( ( q ( +g  `  (Scalar `  g ) ) r ) ( .s `  g ) w )  =  ( ( q ( .s `  g
) w ) ( +g  `  g ) ( r ( .s
`  g ) w ) ) )  /\  ( ( ( q ( .r `  (Scalar `  g ) ) r ) ( .s `  g ) w )  =  ( q ( .s `  g ) ( r ( .s
`  g ) w ) )  /\  (
( 1r `  (Scalar `  g ) ) ( .s `  g ) w )  =  w  /\  ( ( 0g
`  (Scalar `  g )
) ( .s `  g ) w )  =  ( 0g `  g ) ) )  <-> 
( ( ( r 
.x.  w )  e.  V  /\  ( r 
.x.  ( w  .+  x ) )  =  ( ( r  .x.  w )  .+  (
r  .x.  x )
)  /\  ( (
q  .+^  r )  .x.  w )  =  ( ( q  .x.  w
)  .+  ( r  .x.  w ) ) )  /\  ( ( ( q  .X.  r )  .x.  w )  =  ( q  .x.  ( r 
.x.  w ) )  /\  (  .1.  .x.  w )  =  w  /\  ( O  .x.  w )  =  .0.  ) ) ) )
13789, 136raleqbidv 2929 . . . . . . . 8  |-  ( g  =  W  ->  ( A. w  e.  ( Base `  g ) ( ( ( r ( .s `  g ) w )  e.  (
Base `  g )  /\  ( r ( .s
`  g ) ( w ( +g  `  g
) x ) )  =  ( ( r ( .s `  g
) w ) ( +g  `  g ) ( r ( .s
`  g ) x ) )  /\  (
( q ( +g  `  (Scalar `  g )
) r ) ( .s `  g ) w )  =  ( ( q ( .s
`  g ) w ) ( +g  `  g
) ( r ( .s `  g ) w ) ) )  /\  ( ( ( q ( .r `  (Scalar `  g ) ) r ) ( .s
`  g ) w )  =  ( q ( .s `  g
) ( r ( .s `  g ) w ) )  /\  ( ( 1r `  (Scalar `  g ) ) ( .s `  g
) w )  =  w  /\  ( ( 0g `  (Scalar `  g ) ) ( .s `  g ) w )  =  ( 0g `  g ) ) )  <->  A. w  e.  V  ( (
( r  .x.  w
)  e.  V  /\  ( r  .x.  (
w  .+  x )
)  =  ( ( r  .x.  w ) 
.+  ( r  .x.  x ) )  /\  ( ( q  .+^  r )  .x.  w
)  =  ( ( q  .x.  w ) 
.+  ( r  .x.  w ) ) )  /\  ( ( ( q  .X.  r )  .x.  w )  =  ( q  .x.  ( r 
.x.  w ) )  /\  (  .1.  .x.  w )  =  w  /\  ( O  .x.  w )  =  .0.  ) ) ) )
13889, 137raleqbidv 2929 . . . . . . 7  |-  ( g  =  W  ->  ( A. x  e.  ( Base `  g ) A. w  e.  ( Base `  g ) ( ( ( r ( .s
`  g ) w )  e.  ( Base `  g )  /\  (
r ( .s `  g ) ( w ( +g  `  g
) x ) )  =  ( ( r ( .s `  g
) w ) ( +g  `  g ) ( r ( .s
`  g ) x ) )  /\  (
( q ( +g  `  (Scalar `  g )
) r ) ( .s `  g ) w )  =  ( ( q ( .s
`  g ) w ) ( +g  `  g
) ( r ( .s `  g ) w ) ) )  /\  ( ( ( q ( .r `  (Scalar `  g ) ) r ) ( .s
`  g ) w )  =  ( q ( .s `  g
) ( r ( .s `  g ) w ) )  /\  ( ( 1r `  (Scalar `  g ) ) ( .s `  g
) w )  =  w  /\  ( ( 0g `  (Scalar `  g ) ) ( .s `  g ) w )  =  ( 0g `  g ) ) )  <->  A. x  e.  V  A. w  e.  V  ( (
( r  .x.  w
)  e.  V  /\  ( r  .x.  (
w  .+  x )
)  =  ( ( r  .x.  w ) 
.+  ( r  .x.  x ) )  /\  ( ( q  .+^  r )  .x.  w
)  =  ( ( q  .x.  w ) 
.+  ( r  .x.  w ) ) )  /\  ( ( ( q  .X.  r )  .x.  w )  =  ( q  .x.  ( r 
.x.  w ) )  /\  (  .1.  .x.  w )  =  w  /\  ( O  .x.  w )  =  .0.  ) ) ) )
13986, 138raleqbidv 2929 . . . . . 6  |-  ( g  =  W  ->  ( A. r  e.  ( Base `  (Scalar `  g
) ) A. x  e.  ( Base `  g
) A. w  e.  ( Base `  g
) ( ( ( r ( .s `  g ) w )  e.  ( Base `  g
)  /\  ( r
( .s `  g
) ( w ( +g  `  g ) x ) )  =  ( ( r ( .s `  g ) w ) ( +g  `  g ) ( r ( .s `  g
) x ) )  /\  ( ( q ( +g  `  (Scalar `  g ) ) r ) ( .s `  g ) w )  =  ( ( q ( .s `  g
) w ) ( +g  `  g ) ( r ( .s
`  g ) w ) ) )  /\  ( ( ( q ( .r `  (Scalar `  g ) ) r ) ( .s `  g ) w )  =  ( q ( .s `  g ) ( r ( .s
`  g ) w ) )  /\  (
( 1r `  (Scalar `  g ) ) ( .s `  g ) w )  =  w  /\  ( ( 0g
`  (Scalar `  g )
) ( .s `  g ) w )  =  ( 0g `  g ) ) )  <->  A. r  e.  K  A. x  e.  V  A. w  e.  V  ( ( ( r 
.x.  w )  e.  V  /\  ( r 
.x.  ( w  .+  x ) )  =  ( ( r  .x.  w )  .+  (
r  .x.  x )
)  /\  ( (
q  .+^  r )  .x.  w )  =  ( ( q  .x.  w
)  .+  ( r  .x.  w ) ) )  /\  ( ( ( q  .X.  r )  .x.  w )  =  ( q  .x.  ( r 
.x.  w ) )  /\  (  .1.  .x.  w )  =  w  /\  ( O  .x.  w )  =  .0.  ) ) ) )
14086, 139raleqbidv 2929 . . . . 5  |-  ( g  =  W  ->  ( A. q  e.  ( Base `  (Scalar `  g
) ) A. r  e.  ( Base `  (Scalar `  g ) ) A. x  e.  ( Base `  g ) A. w  e.  ( Base `  g
) ( ( ( r ( .s `  g ) w )  e.  ( Base `  g
)  /\  ( r
( .s `  g
) ( w ( +g  `  g ) x ) )  =  ( ( r ( .s `  g ) w ) ( +g  `  g ) ( r ( .s `  g
) x ) )  /\  ( ( q ( +g  `  (Scalar `  g ) ) r ) ( .s `  g ) w )  =  ( ( q ( .s `  g
) w ) ( +g  `  g ) ( r ( .s
`  g ) w ) ) )  /\  ( ( ( q ( .r `  (Scalar `  g ) ) r ) ( .s `  g ) w )  =  ( q ( .s `  g ) ( r ( .s
`  g ) w ) )  /\  (
( 1r `  (Scalar `  g ) ) ( .s `  g ) w )  =  w  /\  ( ( 0g
`  (Scalar `  g )
) ( .s `  g ) w )  =  ( 0g `  g ) ) )  <->  A. q  e.  K  A. r  e.  K  A. x  e.  V  A. w  e.  V  ( ( ( r 
.x.  w )  e.  V  /\  ( r 
.x.  ( w  .+  x ) )  =  ( ( r  .x.  w )  .+  (
r  .x.  x )
)  /\  ( (
q  .+^  r )  .x.  w )  =  ( ( q  .x.  w
)  .+  ( r  .x.  w ) ) )  /\  ( ( ( q  .X.  r )  .x.  w )  =  ( q  .x.  ( r 
.x.  w ) )  /\  (  .1.  .x.  w )  =  w  /\  ( O  .x.  w )  =  .0.  ) ) ) )
14183, 140anbi12d 710 . . . 4  |-  ( g  =  W  ->  (
( (Scalar `  g
)  e. SRing  /\  A. q  e.  ( Base `  (Scalar `  g ) ) A. r  e.  ( Base `  (Scalar `  g )
) A. x  e.  ( Base `  g
) A. w  e.  ( Base `  g
) ( ( ( r ( .s `  g ) w )  e.  ( Base `  g
)  /\  ( r
( .s `  g
) ( w ( +g  `  g ) x ) )  =  ( ( r ( .s `  g ) w ) ( +g  `  g ) ( r ( .s `  g
) x ) )  /\  ( ( q ( +g  `  (Scalar `  g ) ) r ) ( .s `  g ) w )  =  ( ( q ( .s `  g
) w ) ( +g  `  g ) ( r ( .s
`  g ) w ) ) )  /\  ( ( ( q ( .r `  (Scalar `  g ) ) r ) ( .s `  g ) w )  =  ( q ( .s `  g ) ( r ( .s
`  g ) w ) )  /\  (
( 1r `  (Scalar `  g ) ) ( .s `  g ) w )  =  w  /\  ( ( 0g
`  (Scalar `  g )
) ( .s `  g ) w )  =  ( 0g `  g ) ) ) )  <->  ( F  e. SRing  /\  A. q  e.  K  A. r  e.  K  A. x  e.  V  A. w  e.  V  ( ( ( r 
.x.  w )  e.  V  /\  ( r 
.x.  ( w  .+  x ) )  =  ( ( r  .x.  w )  .+  (
r  .x.  x )
)  /\  ( (
q  .+^  r )  .x.  w )  =  ( ( q  .x.  w
)  .+  ( r  .x.  w ) ) )  /\  ( ( ( q  .X.  r )  .x.  w )  =  ( q  .x.  ( r 
.x.  w ) )  /\  (  .1.  .x.  w )  =  w  /\  ( O  .x.  w )  =  .0.  ) ) ) ) )
14279, 141syl5bb 257 . . 3  |-  ( g  =  W  ->  ( [. ( Base `  g
)  /  v ]. [. ( +g  `  g
)  /  a ]. [. ( .s `  g
)  /  s ]. [. (Scalar `  g )  /  f ]. [. ( Base `  f )  / 
k ]. [. ( +g  `  f )  /  p ]. [. ( .r `  f )  /  t ]. ( f  e. SRing  /\  A. q  e.  k  A. r  e.  k  A. x  e.  v  A. w  e.  v  (
( ( r s w )  e.  v  /\  ( r s ( w a x ) )  =  ( ( r s w ) a ( r s x ) )  /\  ( ( q p r ) s w )  =  ( ( q s w ) a ( r s w ) ) )  /\  ( ( ( q t r ) s w )  =  ( q s ( r s w ) )  /\  (
( 1r `  f
) s w )  =  w  /\  (
( 0g `  f
) s w )  =  ( 0g `  g ) ) ) )  <->  ( F  e. SRing  /\  A. q  e.  K  A. r  e.  K  A. x  e.  V  A. w  e.  V  ( ( ( r 
.x.  w )  e.  V  /\  ( r 
.x.  ( w  .+  x ) )  =  ( ( r  .x.  w )  .+  (
r  .x.  x )
)  /\  ( (
q  .+^  r )  .x.  w )  =  ( ( q  .x.  w
)  .+  ( r  .x.  w ) ) )  /\  ( ( ( q  .X.  r )  .x.  w )  =  ( q  .x.  ( r 
.x.  w ) )  /\  (  .1.  .x.  w )  =  w  /\  ( O  .x.  w )  =  .0.  ) ) ) ) )
143 df-slmd 26215 . . 3  |- SLMod  =  {
g  e. CMnd  |  [. ( Base `  g )  / 
v ]. [. ( +g  `  g )  /  a ]. [. ( .s `  g )  /  s ]. [. (Scalar `  g
)  /  f ]. [. ( Base `  f
)  /  k ]. [. ( +g  `  f
)  /  p ]. [. ( .r `  f
)  /  t ]. ( f  e. SRing  /\  A. q  e.  k  A. r  e.  k  A. x  e.  v  A. w  e.  v  (
( ( r s w )  e.  v  /\  ( r s ( w a x ) )  =  ( ( r s w ) a ( r s x ) )  /\  ( ( q p r ) s w )  =  ( ( q s w ) a ( r s w ) ) )  /\  ( ( ( q t r ) s w )  =  ( q s ( r s w ) )  /\  (
( 1r `  f
) s w )  =  w  /\  (
( 0g `  f
) s w )  =  ( 0g `  g ) ) ) ) }
144142, 143elrab2 3117 . 2  |-  ( W  e. SLMod 
<->  ( W  e. CMnd  /\  ( F  e. SRing  /\  A. q  e.  K  A. r  e.  K  A. x  e.  V  A. w  e.  V  (
( ( r  .x.  w )  e.  V  /\  ( r  .x.  (
w  .+  x )
)  =  ( ( r  .x.  w ) 
.+  ( r  .x.  x ) )  /\  ( ( q  .+^  r )  .x.  w
)  =  ( ( q  .x.  w ) 
.+  ( r  .x.  w ) ) )  /\  ( ( ( q  .X.  r )  .x.  w )  =  ( q  .x.  ( r 
.x.  w ) )  /\  (  .1.  .x.  w )  =  w  /\  ( O  .x.  w )  =  .0.  ) ) ) ) )
145 3anass 969 . 2  |-  ( ( W  e. CMnd  /\  F  e. SRing  /\  A. q  e.  K  A. r  e.  K  A. x  e.  V  A. w  e.  V  ( ( ( r  .x.  w )  e.  V  /\  (
r  .x.  ( w  .+  x ) )  =  ( ( r  .x.  w )  .+  (
r  .x.  x )
)  /\  ( (
q  .+^  r )  .x.  w )  =  ( ( q  .x.  w
)  .+  ( r  .x.  w ) ) )  /\  ( ( ( q  .X.  r )  .x.  w )  =  ( q  .x.  ( r 
.x.  w ) )  /\  (  .1.  .x.  w )  =  w  /\  ( O  .x.  w )  =  .0.  ) ) )  <->  ( W  e. CMnd  /\  ( F  e. SRing  /\  A. q  e.  K  A. r  e.  K  A. x  e.  V  A. w  e.  V  ( ( ( r 
.x.  w )  e.  V  /\  ( r 
.x.  ( w  .+  x ) )  =  ( ( r  .x.  w )  .+  (
r  .x.  x )
)  /\  ( (
q  .+^  r )  .x.  w )  =  ( ( q  .x.  w
)  .+  ( r  .x.  w ) ) )  /\  ( ( ( q  .X.  r )  .x.  w )  =  ( q  .x.  ( r 
.x.  w ) )  /\  (  .1.  .x.  w )  =  w  /\  ( O  .x.  w )  =  .0.  ) ) ) ) )
146144, 145bitr4i 252 1  |-  ( W  e. SLMod 
<->  ( W  e. CMnd  /\  F  e. SRing  /\  A. q  e.  K  A. r  e.  K  A. x  e.  V  A. w  e.  V  ( (
( r  .x.  w
)  e.  V  /\  ( r  .x.  (
w  .+  x )
)  =  ( ( r  .x.  w ) 
.+  ( r  .x.  x ) )  /\  ( ( q  .+^  r )  .x.  w
)  =  ( ( q  .x.  w ) 
.+  ( r  .x.  w ) ) )  /\  ( ( ( q  .X.  r )  .x.  w )  =  ( q  .x.  ( r 
.x.  w ) )  /\  (  .1.  .x.  w )  =  w  /\  ( O  .x.  w )  =  .0.  ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   A.wral 2713   [.wsbc 3184   ` cfv 5416  (class class class)co 6089   Basecbs 14172   +g cplusg 14236   .rcmulr 14237  Scalarcsca 14239   .scvsca 14240   0gc0g 14376  CMndccmn 16275   1rcur 16601  SRingcsrg 16605  SLModcslmd 26214
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-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-nul 4419
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3185  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-br 4291  df-iota 5379  df-fv 5424  df-ov 6092  df-slmd 26215
This theorem is referenced by:  slmdlema  26217  lmodslmd  26218  slmdcmn  26219  slmdsrg  26221  xrge0slmod  26310
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