Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  lmod1 Structured version   Unicode version

Theorem lmod1 32049
Description: The (smallest) structure representing a zero module over an arbitrary ring. (Contributed by AV, 29-Apr-2019.)
Hypothesis
Ref Expression
lmod1.m  |-  M  =  ( { <. ( Base `  ndx ) ,  { I } >. , 
<. ( +g  `  ndx ) ,  { <. <. I ,  I >. ,  I >. }
>. ,  <. (Scalar `  ndx ) ,  R >. }  u.  { <. ( .s `  ndx ) ,  ( x  e.  (
Base `  R ) ,  y  e.  { I }  |->  y ) >. } )
Assertion
Ref Expression
lmod1  |-  ( ( I  e.  V  /\  R  e.  Ring )  ->  M  e.  LMod )
Distinct variable groups:    x, I,
y    x, R, y    x, V, y    x, M, y

Proof of Theorem lmod1
Dummy variables  r 
q  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2460 . . . . 5  |-  { <. (
Base `  ndx ) ,  { I } >. , 
<. ( +g  `  ndx ) ,  { <. <. I ,  I >. ,  I >. }
>. }  =  { <. (
Base `  ndx ) ,  { I } >. , 
<. ( +g  `  ndx ) ,  { <. <. I ,  I >. ,  I >. }
>. }
21grp1 32037 . . . 4  |-  ( I  e.  V  ->  { <. (
Base `  ndx ) ,  { I } >. , 
<. ( +g  `  ndx ) ,  { <. <. I ,  I >. ,  I >. }
>. }  e.  Grp )
3 fvex 5867 . . . . . . 7  |-  ( Base `  { <. ( Base `  ndx ) ,  { I } >. ,  <. ( +g  `  ndx ) ,  { <. <. I ,  I >. ,  I >. } >. } )  e.  _V
4 lmod1.m . . . . . . . . 9  |-  M  =  ( { <. ( Base `  ndx ) ,  { I } >. , 
<. ( +g  `  ndx ) ,  { <. <. I ,  I >. ,  I >. }
>. ,  <. (Scalar `  ndx ) ,  R >. }  u.  { <. ( .s `  ndx ) ,  ( x  e.  (
Base `  R ) ,  y  e.  { I }  |->  y ) >. } )
5 snex 4681 . . . . . . . . . . . . 13  |-  { I }  e.  _V
61grpbase 14584 . . . . . . . . . . . . 13  |-  ( { I }  e.  _V  ->  { I }  =  ( Base `  { <. ( Base `  ndx ) ,  { I } >. , 
<. ( +g  `  ndx ) ,  { <. <. I ,  I >. ,  I >. }
>. } ) )
75, 6ax-mp 5 . . . . . . . . . . . 12  |-  { I }  =  ( Base `  { <. ( Base `  ndx ) ,  { I } >. ,  <. ( +g  `  ndx ) ,  { <. <. I ,  I >. ,  I >. } >. } )
87opeq2i 4210 . . . . . . . . . . 11  |-  <. ( Base `  ndx ) ,  { I } >.  = 
<. ( Base `  ndx ) ,  ( Base `  { <. ( Base `  ndx ) ,  { I } >. ,  <. ( +g  `  ndx ) ,  { <. <. I ,  I >. ,  I >. } >. } ) >.
9 tpeq1 4108 . . . . . . . . . . 11  |-  ( <.
( Base `  ndx ) ,  { I } >.  = 
<. ( Base `  ndx ) ,  ( Base `  { <. ( Base `  ndx ) ,  { I } >. ,  <. ( +g  `  ndx ) ,  { <. <. I ,  I >. ,  I >. } >. } ) >.  ->  { <. (
Base `  ndx ) ,  { I } >. , 
<. ( +g  `  ndx ) ,  { <. <. I ,  I >. ,  I >. }
>. ,  <. (Scalar `  ndx ) ,  R >. }  =  { <. ( Base `  ndx ) ,  ( Base `  { <. ( Base `  ndx ) ,  { I } >. ,  <. ( +g  `  ndx ) ,  { <. <. I ,  I >. ,  I >. } >. } ) >. ,  <. ( +g  `  ndx ) ,  { <. <. I ,  I >. ,  I >. } >. , 
<. (Scalar `  ndx ) ,  R >. } )
108, 9ax-mp 5 . . . . . . . . . 10  |-  { <. (
Base `  ndx ) ,  { I } >. , 
<. ( +g  `  ndx ) ,  { <. <. I ,  I >. ,  I >. }
>. ,  <. (Scalar `  ndx ) ,  R >. }  =  { <. ( Base `  ndx ) ,  ( Base `  { <. ( Base `  ndx ) ,  { I } >. ,  <. ( +g  `  ndx ) ,  { <. <. I ,  I >. ,  I >. } >. } ) >. ,  <. ( +g  `  ndx ) ,  { <. <. I ,  I >. ,  I >. } >. , 
<. (Scalar `  ndx ) ,  R >. }
1110uneq1i 3647 . . . . . . . . 9  |-  ( {
<. ( Base `  ndx ) ,  { I } >. ,  <. ( +g  `  ndx ) ,  { <. <. I ,  I >. ,  I >. } >. , 
<. (Scalar `  ndx ) ,  R >. }  u.  { <. ( .s `  ndx ) ,  ( x  e.  ( Base `  R
) ,  y  e. 
{ I }  |->  y ) >. } )  =  ( { <. ( Base `  ndx ) ,  ( Base `  { <. ( Base `  ndx ) ,  { I } >. ,  <. ( +g  `  ndx ) ,  { <. <. I ,  I >. ,  I >. } >. } ) >. ,  <. ( +g  `  ndx ) ,  { <. <. I ,  I >. ,  I >. } >. , 
<. (Scalar `  ndx ) ,  R >. }  u.  { <. ( .s `  ndx ) ,  ( x  e.  ( Base `  R
) ,  y  e. 
{ I }  |->  y ) >. } )
124, 11eqtri 2489 . . . . . . . 8  |-  M  =  ( { <. ( Base `  ndx ) ,  ( Base `  { <. ( Base `  ndx ) ,  { I } >. ,  <. ( +g  `  ndx ) ,  { <. <. I ,  I >. ,  I >. } >. } ) >. ,  <. ( +g  `  ndx ) ,  { <. <. I ,  I >. ,  I >. } >. , 
<. (Scalar `  ndx ) ,  R >. }  u.  { <. ( .s `  ndx ) ,  ( x  e.  ( Base `  R
) ,  y  e. 
{ I }  |->  y ) >. } )
1312lmodbase 14609 . . . . . . 7  |-  ( (
Base `  { <. ( Base `  ndx ) ,  { I } >. , 
<. ( +g  `  ndx ) ,  { <. <. I ,  I >. ,  I >. }
>. } )  e.  _V  ->  ( Base `  { <. ( Base `  ndx ) ,  { I } >. ,  <. ( +g  `  ndx ) ,  { <. <. I ,  I >. ,  I >. } >. } )  =  ( Base `  M ) )
143, 13ax-mp 5 . . . . . 6  |-  ( Base `  { <. ( Base `  ndx ) ,  { I } >. ,  <. ( +g  `  ndx ) ,  { <. <. I ,  I >. ,  I >. } >. } )  =  ( Base `  M )
1514eqcomi 2473 . . . . 5  |-  ( Base `  M )  =  (
Base `  { <. ( Base `  ndx ) ,  { I } >. , 
<. ( +g  `  ndx ) ,  { <. <. I ,  I >. ,  I >. }
>. } )
16 fvex 5867 . . . . . . 7  |-  ( +g  `  { <. ( Base `  ndx ) ,  { I } >. ,  <. ( +g  `  ndx ) ,  { <. <. I ,  I >. ,  I >. } >. } )  e.  _V
17 snex 4681 . . . . . . . . . . . . 13  |-  { <. <.
I ,  I >. ,  I >. }  e.  _V
181grpplusg 14585 . . . . . . . . . . . . 13  |-  ( {
<. <. I ,  I >. ,  I >. }  e.  _V  ->  { <. <. I ,  I >. ,  I >. }  =  ( +g  `  { <. ( Base `  ndx ) ,  { I } >. ,  <. ( +g  `  ndx ) ,  { <. <. I ,  I >. ,  I >. } >. } ) )
1917, 18ax-mp 5 . . . . . . . . . . . 12  |-  { <. <.
I ,  I >. ,  I >. }  =  ( +g  `  { <. (
Base `  ndx ) ,  { I } >. , 
<. ( +g  `  ndx ) ,  { <. <. I ,  I >. ,  I >. }
>. } )
2019opeq2i 4210 . . . . . . . . . . 11  |-  <. ( +g  `  ndx ) ,  { <. <. I ,  I >. ,  I >. } >.  = 
<. ( +g  `  ndx ) ,  ( +g  `  { <. ( Base `  ndx ) ,  { I } >. ,  <. ( +g  `  ndx ) ,  { <. <. I ,  I >. ,  I >. } >. } ) >.
21 tpeq2 4109 . . . . . . . . . . 11  |-  ( <.
( +g  `  ndx ) ,  { <. <. I ,  I >. ,  I >. } >.  = 
<. ( +g  `  ndx ) ,  ( +g  `  { <. ( Base `  ndx ) ,  { I } >. ,  <. ( +g  `  ndx ) ,  { <. <. I ,  I >. ,  I >. } >. } ) >.  ->  { <. (
Base `  ndx ) ,  { I } >. , 
<. ( +g  `  ndx ) ,  { <. <. I ,  I >. ,  I >. }
>. ,  <. (Scalar `  ndx ) ,  R >. }  =  { <. ( Base `  ndx ) ,  { I } >. , 
<. ( +g  `  ndx ) ,  ( +g  `  { <. ( Base `  ndx ) ,  { I } >. ,  <. ( +g  `  ndx ) ,  { <. <. I ,  I >. ,  I >. } >. } ) >. ,  <. (Scalar ` 
ndx ) ,  R >. } )
2220, 21ax-mp 5 . . . . . . . . . 10  |-  { <. (
Base `  ndx ) ,  { I } >. , 
<. ( +g  `  ndx ) ,  { <. <. I ,  I >. ,  I >. }
>. ,  <. (Scalar `  ndx ) ,  R >. }  =  { <. ( Base `  ndx ) ,  { I } >. , 
<. ( +g  `  ndx ) ,  ( +g  `  { <. ( Base `  ndx ) ,  { I } >. ,  <. ( +g  `  ndx ) ,  { <. <. I ,  I >. ,  I >. } >. } ) >. ,  <. (Scalar ` 
ndx ) ,  R >. }
2322uneq1i 3647 . . . . . . . . 9  |-  ( {
<. ( Base `  ndx ) ,  { I } >. ,  <. ( +g  `  ndx ) ,  { <. <. I ,  I >. ,  I >. } >. , 
<. (Scalar `  ndx ) ,  R >. }  u.  { <. ( .s `  ndx ) ,  ( x  e.  ( Base `  R
) ,  y  e. 
{ I }  |->  y ) >. } )  =  ( { <. ( Base `  ndx ) ,  { I } >. , 
<. ( +g  `  ndx ) ,  ( +g  `  { <. ( Base `  ndx ) ,  { I } >. ,  <. ( +g  `  ndx ) ,  { <. <. I ,  I >. ,  I >. } >. } ) >. ,  <. (Scalar ` 
ndx ) ,  R >. }  u.  { <. ( .s `  ndx ) ,  ( x  e.  ( Base `  R
) ,  y  e. 
{ I }  |->  y ) >. } )
244, 23eqtri 2489 . . . . . . . 8  |-  M  =  ( { <. ( Base `  ndx ) ,  { I } >. , 
<. ( +g  `  ndx ) ,  ( +g  `  { <. ( Base `  ndx ) ,  { I } >. ,  <. ( +g  `  ndx ) ,  { <. <. I ,  I >. ,  I >. } >. } ) >. ,  <. (Scalar ` 
ndx ) ,  R >. }  u.  { <. ( .s `  ndx ) ,  ( x  e.  ( Base `  R
) ,  y  e. 
{ I }  |->  y ) >. } )
2524lmodplusg 14610 . . . . . . 7  |-  ( ( +g  `  { <. (
Base `  ndx ) ,  { I } >. , 
<. ( +g  `  ndx ) ,  { <. <. I ,  I >. ,  I >. }
>. } )  e.  _V  ->  ( +g  `  { <. ( Base `  ndx ) ,  { I } >. ,  <. ( +g  `  ndx ) ,  { <. <. I ,  I >. ,  I >. } >. } )  =  ( +g  `  M ) )
2616, 25ax-mp 5 . . . . . 6  |-  ( +g  `  { <. ( Base `  ndx ) ,  { I } >. ,  <. ( +g  `  ndx ) ,  { <. <. I ,  I >. ,  I >. } >. } )  =  ( +g  `  M )
2726eqcomi 2473 . . . . 5  |-  ( +g  `  M )  =  ( +g  `  { <. (
Base `  ndx ) ,  { I } >. , 
<. ( +g  `  ndx ) ,  { <. <. I ,  I >. ,  I >. }
>. } )
2815, 27grpprop 15863 . . . 4  |-  ( M  e.  Grp  <->  { <. ( Base `  ndx ) ,  { I } >. , 
<. ( +g  `  ndx ) ,  { <. <. I ,  I >. ,  I >. }
>. }  e.  Grp )
292, 28sylibr 212 . . 3  |-  ( I  e.  V  ->  M  e.  Grp )
3029adantr 465 . 2  |-  ( ( I  e.  V  /\  R  e.  Ring )  ->  M  e.  Grp )
314lmodsca 14611 . . . . 5  |-  ( R  e.  Ring  ->  R  =  (Scalar `  M )
)
3231eqcomd 2468 . . . 4  |-  ( R  e.  Ring  ->  (Scalar `  M )  =  R )
3332adantl 466 . . 3  |-  ( ( I  e.  V  /\  R  e.  Ring )  -> 
(Scalar `  M )  =  R )
34 simpr 461 . . 3  |-  ( ( I  e.  V  /\  R  e.  Ring )  ->  R  e.  Ring )
3533, 34eqeltrd 2548 . 2  |-  ( ( I  e.  V  /\  R  e.  Ring )  -> 
(Scalar `  M )  e.  Ring )
3633fveq2d 5861 . . . . . . . 8  |-  ( ( I  e.  V  /\  R  e.  Ring )  -> 
( Base `  (Scalar `  M
) )  =  (
Base `  R )
)
3736eleq2d 2530 . . . . . . 7  |-  ( ( I  e.  V  /\  R  e.  Ring )  -> 
( q  e.  (
Base `  (Scalar `  M
) )  <->  q  e.  ( Base `  R )
) )
3836eleq2d 2530 . . . . . . 7  |-  ( ( I  e.  V  /\  R  e.  Ring )  -> 
( r  e.  (
Base `  (Scalar `  M
) )  <->  r  e.  ( Base `  R )
) )
3937, 38anbi12d 710 . . . . . 6  |-  ( ( I  e.  V  /\  R  e.  Ring )  -> 
( ( q  e.  ( Base `  (Scalar `  M ) )  /\  r  e.  ( Base `  (Scalar `  M )
) )  <->  ( q  e.  ( Base `  R
)  /\  r  e.  ( Base `  R )
) ) )
40 simpll 753 . . . . . . . . . . 11  |-  ( ( ( I  e.  V  /\  R  e.  Ring )  /\  ( q  e.  ( Base `  R
)  /\  r  e.  ( Base `  R )
) )  ->  I  e.  V )
41 simplr 754 . . . . . . . . . . 11  |-  ( ( ( I  e.  V  /\  R  e.  Ring )  /\  ( q  e.  ( Base `  R
)  /\  r  e.  ( Base `  R )
) )  ->  R  e.  Ring )
42 simprr 756 . . . . . . . . . . 11  |-  ( ( ( I  e.  V  /\  R  e.  Ring )  /\  ( q  e.  ( Base `  R
)  /\  r  e.  ( Base `  R )
) )  ->  r  e.  ( Base `  R
) )
4340, 41, 423jca 1171 . . . . . . . . . 10  |-  ( ( ( I  e.  V  /\  R  e.  Ring )  /\  ( q  e.  ( Base `  R
)  /\  r  e.  ( Base `  R )
) )  ->  (
I  e.  V  /\  R  e.  Ring  /\  r  e.  ( Base `  R
) ) )
444lmod1lem1 32044 . . . . . . . . . 10  |-  ( ( I  e.  V  /\  R  e.  Ring  /\  r  e.  ( Base `  R
) )  ->  (
r ( .s `  M ) I )  e.  { I }
)
4543, 44syl 16 . . . . . . . . 9  |-  ( ( ( I  e.  V  /\  R  e.  Ring )  /\  ( q  e.  ( Base `  R
)  /\  r  e.  ( Base `  R )
) )  ->  (
r ( .s `  M ) I )  e.  { I }
)
464lmod1lem2 32045 . . . . . . . . . 10  |-  ( ( I  e.  V  /\  R  e.  Ring  /\  r  e.  ( Base `  R
) )  ->  (
r ( .s `  M ) ( I ( +g  `  M
) I ) )  =  ( ( r ( .s `  M
) I ) ( +g  `  M ) ( r ( .s
`  M ) I ) ) )
4743, 46syl 16 . . . . . . . . 9  |-  ( ( ( I  e.  V  /\  R  e.  Ring )  /\  ( q  e.  ( Base `  R
)  /\  r  e.  ( Base `  R )
) )  ->  (
r ( .s `  M ) ( I ( +g  `  M
) I ) )  =  ( ( r ( .s `  M
) I ) ( +g  `  M ) ( r ( .s
`  M ) I ) ) )
484lmod1lem3 32046 . . . . . . . . 9  |-  ( ( ( I  e.  V  /\  R  e.  Ring )  /\  ( q  e.  ( Base `  R
)  /\  r  e.  ( Base `  R )
) )  ->  (
( q ( +g  `  (Scalar `  M )
) r ) ( .s `  M ) I )  =  ( ( q ( .s
`  M ) I ) ( +g  `  M
) ( r ( .s `  M ) I ) ) )
4945, 47, 483jca 1171 . . . . . . . 8  |-  ( ( ( I  e.  V  /\  R  e.  Ring )  /\  ( q  e.  ( Base `  R
)  /\  r  e.  ( Base `  R )
) )  ->  (
( r ( .s
`  M ) I )  e.  { I }  /\  ( r ( .s `  M ) ( I ( +g  `  M ) I ) )  =  ( ( r ( .s `  M ) I ) ( +g  `  M
) ( r ( .s `  M ) I ) )  /\  ( ( q ( +g  `  (Scalar `  M ) ) r ) ( .s `  M ) I )  =  ( ( q ( .s `  M
) I ) ( +g  `  M ) ( r ( .s
`  M ) I ) ) ) )
504lmod1lem4 32047 . . . . . . . 8  |-  ( ( ( I  e.  V  /\  R  e.  Ring )  /\  ( q  e.  ( Base `  R
)  /\  r  e.  ( Base `  R )
) )  ->  (
( q ( .r
`  (Scalar `  M )
) r ) ( .s `  M ) I )  =  ( q ( .s `  M ) ( r ( .s `  M
) I ) ) )
514lmod1lem5 32048 . . . . . . . . 9  |-  ( ( I  e.  V  /\  R  e.  Ring )  -> 
( ( 1r `  (Scalar `  M ) ) ( .s `  M
) I )  =  I )
5251adantr 465 . . . . . . . 8  |-  ( ( ( I  e.  V  /\  R  e.  Ring )  /\  ( q  e.  ( Base `  R
)  /\  r  e.  ( Base `  R )
) )  ->  (
( 1r `  (Scalar `  M ) ) ( .s `  M ) I )  =  I )
5349, 50, 52jca32 535 . . . . . . 7  |-  ( ( ( I  e.  V  /\  R  e.  Ring )  /\  ( q  e.  ( Base `  R
)  /\  r  e.  ( Base `  R )
) )  ->  (
( ( r ( .s `  M ) I )  e.  {
I }  /\  (
r ( .s `  M ) ( I ( +g  `  M
) I ) )  =  ( ( r ( .s `  M
) I ) ( +g  `  M ) ( r ( .s
`  M ) I ) )  /\  (
( q ( +g  `  (Scalar `  M )
) r ) ( .s `  M ) I )  =  ( ( q ( .s
`  M ) I ) ( +g  `  M
) ( r ( .s `  M ) I ) ) )  /\  ( ( ( q ( .r `  (Scalar `  M ) ) r ) ( .s
`  M ) I )  =  ( q ( .s `  M
) ( r ( .s `  M ) I ) )  /\  ( ( 1r `  (Scalar `  M ) ) ( .s `  M
) I )  =  I ) ) )
5453ex 434 . . . . . 6  |-  ( ( I  e.  V  /\  R  e.  Ring )  -> 
( ( q  e.  ( Base `  R
)  /\  r  e.  ( Base `  R )
)  ->  ( (
( r ( .s
`  M ) I )  e.  { I }  /\  ( r ( .s `  M ) ( I ( +g  `  M ) I ) )  =  ( ( r ( .s `  M ) I ) ( +g  `  M
) ( r ( .s `  M ) I ) )  /\  ( ( q ( +g  `  (Scalar `  M ) ) r ) ( .s `  M ) I )  =  ( ( q ( .s `  M
) I ) ( +g  `  M ) ( r ( .s
`  M ) I ) ) )  /\  ( ( ( q ( .r `  (Scalar `  M ) ) r ) ( .s `  M ) I )  =  ( q ( .s `  M ) ( r ( .s
`  M ) I ) )  /\  (
( 1r `  (Scalar `  M ) ) ( .s `  M ) I )  =  I ) ) ) )
5539, 54sylbid 215 . . . . 5  |-  ( ( I  e.  V  /\  R  e.  Ring )  -> 
( ( q  e.  ( Base `  (Scalar `  M ) )  /\  r  e.  ( Base `  (Scalar `  M )
) )  ->  (
( ( r ( .s `  M ) I )  e.  {
I }  /\  (
r ( .s `  M ) ( I ( +g  `  M
) I ) )  =  ( ( r ( .s `  M
) I ) ( +g  `  M ) ( r ( .s
`  M ) I ) )  /\  (
( q ( +g  `  (Scalar `  M )
) r ) ( .s `  M ) I )  =  ( ( q ( .s
`  M ) I ) ( +g  `  M
) ( r ( .s `  M ) I ) ) )  /\  ( ( ( q ( .r `  (Scalar `  M ) ) r ) ( .s
`  M ) I )  =  ( q ( .s `  M
) ( r ( .s `  M ) I ) )  /\  ( ( 1r `  (Scalar `  M ) ) ( .s `  M
) I )  =  I ) ) ) )
5655imp 429 . . . 4  |-  ( ( ( I  e.  V  /\  R  e.  Ring )  /\  ( q  e.  ( Base `  (Scalar `  M ) )  /\  r  e.  ( Base `  (Scalar `  M )
) ) )  -> 
( ( ( r ( .s `  M
) I )  e. 
{ I }  /\  ( r ( .s
`  M ) ( I ( +g  `  M
) I ) )  =  ( ( r ( .s `  M
) I ) ( +g  `  M ) ( r ( .s
`  M ) I ) )  /\  (
( q ( +g  `  (Scalar `  M )
) r ) ( .s `  M ) I )  =  ( ( q ( .s
`  M ) I ) ( +g  `  M
) ( r ( .s `  M ) I ) ) )  /\  ( ( ( q ( .r `  (Scalar `  M ) ) r ) ( .s
`  M ) I )  =  ( q ( .s `  M
) ( r ( .s `  M ) I ) )  /\  ( ( 1r `  (Scalar `  M ) ) ( .s `  M
) I )  =  I ) ) )
5756ralrimivva 2878 . . 3  |-  ( ( I  e.  V  /\  R  e.  Ring )  ->  A. q  e.  ( Base `  (Scalar `  M
) ) A. r  e.  ( Base `  (Scalar `  M ) ) ( ( ( r ( .s `  M ) I )  e.  {
I }  /\  (
r ( .s `  M ) ( I ( +g  `  M
) I ) )  =  ( ( r ( .s `  M
) I ) ( +g  `  M ) ( r ( .s
`  M ) I ) )  /\  (
( q ( +g  `  (Scalar `  M )
) r ) ( .s `  M ) I )  =  ( ( q ( .s
`  M ) I ) ( +g  `  M
) ( r ( .s `  M ) I ) ) )  /\  ( ( ( q ( .r `  (Scalar `  M ) ) r ) ( .s
`  M ) I )  =  ( q ( .s `  M
) ( r ( .s `  M ) I ) )  /\  ( ( 1r `  (Scalar `  M ) ) ( .s `  M
) I )  =  I ) ) )
58 oveq2 6283 . . . . . . . . . . . . 13  |-  ( x  =  I  ->  (
w ( +g  `  M
) x )  =  ( w ( +g  `  M ) I ) )
5958oveq2d 6291 . . . . . . . . . . . 12  |-  ( x  =  I  ->  (
r ( .s `  M ) ( w ( +g  `  M
) x ) )  =  ( r ( .s `  M ) ( w ( +g  `  M ) I ) ) )
60 oveq2 6283 . . . . . . . . . . . . 13  |-  ( x  =  I  ->  (
r ( .s `  M ) x )  =  ( r ( .s `  M ) I ) )
6160oveq2d 6291 . . . . . . . . . . . 12  |-  ( x  =  I  ->  (
( r ( .s
`  M ) w ) ( +g  `  M
) ( r ( .s `  M ) x ) )  =  ( ( r ( .s `  M ) w ) ( +g  `  M ) ( r ( .s `  M
) I ) ) )
6259, 61eqeq12d 2482 . . . . . . . . . . 11  |-  ( x  =  I  ->  (
( r ( .s
`  M ) ( w ( +g  `  M
) x ) )  =  ( ( r ( .s `  M
) w ) ( +g  `  M ) ( r ( .s
`  M ) x ) )  <->  ( r
( .s `  M
) ( w ( +g  `  M ) I ) )  =  ( ( r ( .s `  M ) w ) ( +g  `  M ) ( r ( .s `  M
) I ) ) ) )
63623anbi2d 1299 . . . . . . . . . 10  |-  ( x  =  I  ->  (
( ( r ( .s `  M ) w )  e.  {
I }  /\  (
r ( .s `  M ) ( w ( +g  `  M
) x ) )  =  ( ( r ( .s `  M
) w ) ( +g  `  M ) ( r ( .s
`  M ) x ) )  /\  (
( q ( +g  `  (Scalar `  M )
) r ) ( .s `  M ) w )  =  ( ( q ( .s
`  M ) w ) ( +g  `  M
) ( r ( .s `  M ) w ) ) )  <-> 
( ( r ( .s `  M ) w )  e.  {
I }  /\  (
r ( .s `  M ) ( w ( +g  `  M
) I ) )  =  ( ( r ( .s `  M
) w ) ( +g  `  M ) ( r ( .s
`  M ) I ) )  /\  (
( q ( +g  `  (Scalar `  M )
) r ) ( .s `  M ) w )  =  ( ( q ( .s
`  M ) w ) ( +g  `  M
) ( r ( .s `  M ) w ) ) ) ) )
6463anbi1d 704 . . . . . . . . 9  |-  ( x  =  I  ->  (
( ( ( r ( .s `  M
) w )  e. 
{ I }  /\  ( r ( .s
`  M ) ( w ( +g  `  M
) x ) )  =  ( ( r ( .s `  M
) w ) ( +g  `  M ) ( r ( .s
`  M ) x ) )  /\  (
( q ( +g  `  (Scalar `  M )
) r ) ( .s `  M ) w )  =  ( ( q ( .s
`  M ) w ) ( +g  `  M
) ( r ( .s `  M ) w ) ) )  /\  ( ( ( q ( .r `  (Scalar `  M ) ) r ) ( .s
`  M ) w )  =  ( q ( .s `  M
) ( r ( .s `  M ) w ) )  /\  ( ( 1r `  (Scalar `  M ) ) ( .s `  M
) w )  =  w ) )  <->  ( (
( r ( .s
`  M ) w )  e.  { I }  /\  ( r ( .s `  M ) ( w ( +g  `  M ) I ) )  =  ( ( r ( .s `  M ) w ) ( +g  `  M
) ( r ( .s `  M ) I ) )  /\  ( ( q ( +g  `  (Scalar `  M ) ) r ) ( .s `  M ) w )  =  ( ( q ( .s `  M
) w ) ( +g  `  M ) ( r ( .s
`  M ) w ) ) )  /\  ( ( ( q ( .r `  (Scalar `  M ) ) r ) ( .s `  M ) w )  =  ( q ( .s `  M ) ( r ( .s
`  M ) w ) )  /\  (
( 1r `  (Scalar `  M ) ) ( .s `  M ) w )  =  w ) ) ) )
6564ralbidv 2896 . . . . . . . 8  |-  ( x  =  I  ->  ( A. w  e.  { I }  ( ( ( r ( .s `  M ) w )  e.  { I }  /\  ( r ( .s
`  M ) ( w ( +g  `  M
) x ) )  =  ( ( r ( .s `  M
) w ) ( +g  `  M ) ( r ( .s
`  M ) x ) )  /\  (
( q ( +g  `  (Scalar `  M )
) r ) ( .s `  M ) w )  =  ( ( q ( .s
`  M ) w ) ( +g  `  M
) ( r ( .s `  M ) w ) ) )  /\  ( ( ( q ( .r `  (Scalar `  M ) ) r ) ( .s
`  M ) w )  =  ( q ( .s `  M
) ( r ( .s `  M ) w ) )  /\  ( ( 1r `  (Scalar `  M ) ) ( .s `  M
) w )  =  w ) )  <->  A. w  e.  { I }  (
( ( r ( .s `  M ) w )  e.  {
I }  /\  (
r ( .s `  M ) ( w ( +g  `  M
) I ) )  =  ( ( r ( .s `  M
) w ) ( +g  `  M ) ( r ( .s
`  M ) I ) )  /\  (
( q ( +g  `  (Scalar `  M )
) r ) ( .s `  M ) w )  =  ( ( q ( .s
`  M ) w ) ( +g  `  M
) ( r ( .s `  M ) w ) ) )  /\  ( ( ( q ( .r `  (Scalar `  M ) ) r ) ( .s
`  M ) w )  =  ( q ( .s `  M
) ( r ( .s `  M ) w ) )  /\  ( ( 1r `  (Scalar `  M ) ) ( .s `  M
) w )  =  w ) ) ) )
6665ralsng 4055 . . . . . . 7  |-  ( I  e.  V  ->  ( A. x  e.  { I } A. w  e.  {
I }  ( ( ( r ( .s
`  M ) w )  e.  { I }  /\  ( r ( .s `  M ) ( w ( +g  `  M ) x ) )  =  ( ( r ( .s `  M ) w ) ( +g  `  M
) ( r ( .s `  M ) x ) )  /\  ( ( q ( +g  `  (Scalar `  M ) ) r ) ( .s `  M ) w )  =  ( ( q ( .s `  M
) w ) ( +g  `  M ) ( r ( .s
`  M ) w ) ) )  /\  ( ( ( q ( .r `  (Scalar `  M ) ) r ) ( .s `  M ) w )  =  ( q ( .s `  M ) ( r ( .s
`  M ) w ) )  /\  (
( 1r `  (Scalar `  M ) ) ( .s `  M ) w )  =  w ) )  <->  A. w  e.  { I }  (
( ( r ( .s `  M ) w )  e.  {
I }  /\  (
r ( .s `  M ) ( w ( +g  `  M
) I ) )  =  ( ( r ( .s `  M
) w ) ( +g  `  M ) ( r ( .s
`  M ) I ) )  /\  (
( q ( +g  `  (Scalar `  M )
) r ) ( .s `  M ) w )  =  ( ( q ( .s
`  M ) w ) ( +g  `  M
) ( r ( .s `  M ) w ) ) )  /\  ( ( ( q ( .r `  (Scalar `  M ) ) r ) ( .s
`  M ) w )  =  ( q ( .s `  M
) ( r ( .s `  M ) w ) )  /\  ( ( 1r `  (Scalar `  M ) ) ( .s `  M
) w )  =  w ) ) ) )
6766adantr 465 . . . . . 6  |-  ( ( I  e.  V  /\  R  e.  Ring )  -> 
( A. x  e. 
{ I } A. w  e.  { I }  ( ( ( r ( .s `  M ) w )  e.  { I }  /\  ( r ( .s
`  M ) ( w ( +g  `  M
) x ) )  =  ( ( r ( .s `  M
) w ) ( +g  `  M ) ( r ( .s
`  M ) x ) )  /\  (
( q ( +g  `  (Scalar `  M )
) r ) ( .s `  M ) w )  =  ( ( q ( .s
`  M ) w ) ( +g  `  M
) ( r ( .s `  M ) w ) ) )  /\  ( ( ( q ( .r `  (Scalar `  M ) ) r ) ( .s
`  M ) w )  =  ( q ( .s `  M
) ( r ( .s `  M ) w ) )  /\  ( ( 1r `  (Scalar `  M ) ) ( .s `  M
) w )  =  w ) )  <->  A. w  e.  { I }  (
( ( r ( .s `  M ) w )  e.  {
I }  /\  (
r ( .s `  M ) ( w ( +g  `  M
) I ) )  =  ( ( r ( .s `  M
) w ) ( +g  `  M ) ( r ( .s
`  M ) I ) )  /\  (
( q ( +g  `  (Scalar `  M )
) r ) ( .s `  M ) w )  =  ( ( q ( .s
`  M ) w ) ( +g  `  M
) ( r ( .s `  M ) w ) ) )  /\  ( ( ( q ( .r `  (Scalar `  M ) ) r ) ( .s
`  M ) w )  =  ( q ( .s `  M
) ( r ( .s `  M ) w ) )  /\  ( ( 1r `  (Scalar `  M ) ) ( .s `  M
) w )  =  w ) ) ) )
68 oveq2 6283 . . . . . . . . . . 11  |-  ( w  =  I  ->  (
r ( .s `  M ) w )  =  ( r ( .s `  M ) I ) )
6968eleq1d 2529 . . . . . . . . . 10  |-  ( w  =  I  ->  (
( r ( .s
`  M ) w )  e.  { I } 
<->  ( r ( .s
`  M ) I )  e.  { I } ) )
70 oveq1 6282 . . . . . . . . . . . 12  |-  ( w  =  I  ->  (
w ( +g  `  M
) I )  =  ( I ( +g  `  M ) I ) )
7170oveq2d 6291 . . . . . . . . . . 11  |-  ( w  =  I  ->  (
r ( .s `  M ) ( w ( +g  `  M
) I ) )  =  ( r ( .s `  M ) ( I ( +g  `  M ) I ) ) )
7268oveq1d 6290 . . . . . . . . . . 11  |-  ( w  =  I  ->  (
( r ( .s
`  M ) w ) ( +g  `  M
) ( r ( .s `  M ) I ) )  =  ( ( r ( .s `  M ) I ) ( +g  `  M ) ( r ( .s `  M
) I ) ) )
7371, 72eqeq12d 2482 . . . . . . . . . 10  |-  ( w  =  I  ->  (
( r ( .s
`  M ) ( w ( +g  `  M
) I ) )  =  ( ( r ( .s `  M
) w ) ( +g  `  M ) ( r ( .s
`  M ) I ) )  <->  ( r
( .s `  M
) ( I ( +g  `  M ) I ) )  =  ( ( r ( .s `  M ) I ) ( +g  `  M ) ( r ( .s `  M
) I ) ) ) )
74 oveq2 6283 . . . . . . . . . . 11  |-  ( w  =  I  ->  (
( q ( +g  `  (Scalar `  M )
) r ) ( .s `  M ) w )  =  ( ( q ( +g  `  (Scalar `  M )
) r ) ( .s `  M ) I ) )
75 oveq2 6283 . . . . . . . . . . . 12  |-  ( w  =  I  ->  (
q ( .s `  M ) w )  =  ( q ( .s `  M ) I ) )
7675, 68oveq12d 6293 . . . . . . . . . . 11  |-  ( w  =  I  ->  (
( q ( .s
`  M ) w ) ( +g  `  M
) ( r ( .s `  M ) w ) )  =  ( ( q ( .s `  M ) I ) ( +g  `  M ) ( r ( .s `  M
) I ) ) )
7774, 76eqeq12d 2482 . . . . . . . . . 10  |-  ( w  =  I  ->  (
( ( q ( +g  `  (Scalar `  M ) ) r ) ( .s `  M ) w )  =  ( ( q ( .s `  M
) w ) ( +g  `  M ) ( r ( .s
`  M ) w ) )  <->  ( (
q ( +g  `  (Scalar `  M ) ) r ) ( .s `  M ) I )  =  ( ( q ( .s `  M
) I ) ( +g  `  M ) ( r ( .s
`  M ) I ) ) ) )
7869, 73, 773anbi123d 1294 . . . . . . . . 9  |-  ( w  =  I  ->  (
( ( r ( .s `  M ) w )  e.  {
I }  /\  (
r ( .s `  M ) ( w ( +g  `  M
) I ) )  =  ( ( r ( .s `  M
) w ) ( +g  `  M ) ( r ( .s
`  M ) I ) )  /\  (
( q ( +g  `  (Scalar `  M )
) r ) ( .s `  M ) w )  =  ( ( q ( .s
`  M ) w ) ( +g  `  M
) ( r ( .s `  M ) w ) ) )  <-> 
( ( r ( .s `  M ) I )  e.  {
I }  /\  (
r ( .s `  M ) ( I ( +g  `  M
) I ) )  =  ( ( r ( .s `  M
) I ) ( +g  `  M ) ( r ( .s
`  M ) I ) )  /\  (
( q ( +g  `  (Scalar `  M )
) r ) ( .s `  M ) I )  =  ( ( q ( .s
`  M ) I ) ( +g  `  M
) ( r ( .s `  M ) I ) ) ) ) )
79 oveq2 6283 . . . . . . . . . . 11  |-  ( w  =  I  ->  (
( q ( .r
`  (Scalar `  M )
) r ) ( .s `  M ) w )  =  ( ( q ( .r
`  (Scalar `  M )
) r ) ( .s `  M ) I ) )
8068oveq2d 6291 . . . . . . . . . . 11  |-  ( w  =  I  ->  (
q ( .s `  M ) ( r ( .s `  M
) w ) )  =  ( q ( .s `  M ) ( r ( .s
`  M ) I ) ) )
8179, 80eqeq12d 2482 . . . . . . . . . 10  |-  ( w  =  I  ->  (
( ( q ( .r `  (Scalar `  M ) ) r ) ( .s `  M ) w )  =  ( q ( .s `  M ) ( r ( .s
`  M ) w ) )  <->  ( (
q ( .r `  (Scalar `  M ) ) r ) ( .s
`  M ) I )  =  ( q ( .s `  M
) ( r ( .s `  M ) I ) ) ) )
82 oveq2 6283 . . . . . . . . . . 11  |-  ( w  =  I  ->  (
( 1r `  (Scalar `  M ) ) ( .s `  M ) w )  =  ( ( 1r `  (Scalar `  M ) ) ( .s `  M ) I ) )
83 id 22 . . . . . . . . . . 11  |-  ( w  =  I  ->  w  =  I )
8482, 83eqeq12d 2482 . . . . . . . . . 10  |-  ( w  =  I  ->  (
( ( 1r `  (Scalar `  M ) ) ( .s `  M
) w )  =  w  <->  ( ( 1r
`  (Scalar `  M )
) ( .s `  M ) I )  =  I ) )
8581, 84anbi12d 710 . . . . . . . . 9  |-  ( w  =  I  ->  (
( ( ( q ( .r `  (Scalar `  M ) ) r ) ( .s `  M ) w )  =  ( q ( .s `  M ) ( r ( .s
`  M ) w ) )  /\  (
( 1r `  (Scalar `  M ) ) ( .s `  M ) w )  =  w )  <->  ( ( ( q ( .r `  (Scalar `  M ) ) r ) ( .s
`  M ) I )  =  ( q ( .s `  M
) ( r ( .s `  M ) I ) )  /\  ( ( 1r `  (Scalar `  M ) ) ( .s `  M
) I )  =  I ) ) )
8678, 85anbi12d 710 . . . . . . . 8  |-  ( w  =  I  ->  (
( ( ( r ( .s `  M
) w )  e. 
{ I }  /\  ( r ( .s
`  M ) ( w ( +g  `  M
) I ) )  =  ( ( r ( .s `  M
) w ) ( +g  `  M ) ( r ( .s
`  M ) I ) )  /\  (
( q ( +g  `  (Scalar `  M )
) r ) ( .s `  M ) w )  =  ( ( q ( .s
`  M ) w ) ( +g  `  M
) ( r ( .s `  M ) w ) ) )  /\  ( ( ( q ( .r `  (Scalar `  M ) ) r ) ( .s
`  M ) w )  =  ( q ( .s `  M
) ( r ( .s `  M ) w ) )  /\  ( ( 1r `  (Scalar `  M ) ) ( .s `  M
) w )  =  w ) )  <->  ( (
( r ( .s
`  M ) I )  e.  { I }  /\  ( r ( .s `  M ) ( I ( +g  `  M ) I ) )  =  ( ( r ( .s `  M ) I ) ( +g  `  M
) ( r ( .s `  M ) I ) )  /\  ( ( q ( +g  `  (Scalar `  M ) ) r ) ( .s `  M ) I )  =  ( ( q ( .s `  M
) I ) ( +g  `  M ) ( r ( .s
`  M ) I ) ) )  /\  ( ( ( q ( .r `  (Scalar `  M ) ) r ) ( .s `  M ) I )  =  ( q ( .s `  M ) ( r ( .s
`  M ) I ) )  /\  (
( 1r `  (Scalar `  M ) ) ( .s `  M ) I )  =  I ) ) ) )
8786ralsng 4055 . . . . . . 7  |-  ( I  e.  V  ->  ( A. w  e.  { I }  ( ( ( r ( .s `  M ) w )  e.  { I }  /\  ( r ( .s
`  M ) ( w ( +g  `  M
) I ) )  =  ( ( r ( .s `  M
) w ) ( +g  `  M ) ( r ( .s
`  M ) I ) )  /\  (
( q ( +g  `  (Scalar `  M )
) r ) ( .s `  M ) w )  =  ( ( q ( .s
`  M ) w ) ( +g  `  M
) ( r ( .s `  M ) w ) ) )  /\  ( ( ( q ( .r `  (Scalar `  M ) ) r ) ( .s
`  M ) w )  =  ( q ( .s `  M
) ( r ( .s `  M ) w ) )  /\  ( ( 1r `  (Scalar `  M ) ) ( .s `  M
) w )  =  w ) )  <->  ( (
( r ( .s
`  M ) I )  e.  { I }  /\  ( r ( .s `  M ) ( I ( +g  `  M ) I ) )  =  ( ( r ( .s `  M ) I ) ( +g  `  M
) ( r ( .s `  M ) I ) )  /\  ( ( q ( +g  `  (Scalar `  M ) ) r ) ( .s `  M ) I )  =  ( ( q ( .s `  M
) I ) ( +g  `  M ) ( r ( .s
`  M ) I ) ) )  /\  ( ( ( q ( .r `  (Scalar `  M ) ) r ) ( .s `  M ) I )  =  ( q ( .s `  M ) ( r ( .s
`  M ) I ) )  /\  (
( 1r `  (Scalar `  M ) ) ( .s `  M ) I )  =  I ) ) ) )
8887adantr 465 . . . . . 6  |-  ( ( I  e.  V  /\  R  e.  Ring )  -> 
( A. w  e. 
{ I }  (
( ( r ( .s `  M ) w )  e.  {
I }  /\  (
r ( .s `  M ) ( w ( +g  `  M
) I ) )  =  ( ( r ( .s `  M
) w ) ( +g  `  M ) ( r ( .s
`  M ) I ) )  /\  (
( q ( +g  `  (Scalar `  M )
) r ) ( .s `  M ) w )  =  ( ( q ( .s
`  M ) w ) ( +g  `  M
) ( r ( .s `  M ) w ) ) )  /\  ( ( ( q ( .r `  (Scalar `  M ) ) r ) ( .s
`  M ) w )  =  ( q ( .s `  M
) ( r ( .s `  M ) w ) )  /\  ( ( 1r `  (Scalar `  M ) ) ( .s `  M
) w )  =  w ) )  <->  ( (
( r ( .s
`  M ) I )  e.  { I }  /\  ( r ( .s `  M ) ( I ( +g  `  M ) I ) )  =  ( ( r ( .s `  M ) I ) ( +g  `  M
) ( r ( .s `  M ) I ) )  /\  ( ( q ( +g  `  (Scalar `  M ) ) r ) ( .s `  M ) I )  =  ( ( q ( .s `  M
) I ) ( +g  `  M ) ( r ( .s
`  M ) I ) ) )  /\  ( ( ( q ( .r `  (Scalar `  M ) ) r ) ( .s `  M ) I )  =  ( q ( .s `  M ) ( r ( .s
`  M ) I ) )  /\  (
( 1r `  (Scalar `  M ) ) ( .s `  M ) I )  =  I ) ) ) )
8967, 88bitrd 253 . . . . 5  |-  ( ( I  e.  V  /\  R  e.  Ring )  -> 
( A. x  e. 
{ I } A. w  e.  { I }  ( ( ( r ( .s `  M ) w )  e.  { I }  /\  ( r ( .s
`  M ) ( w ( +g  `  M
) x ) )  =  ( ( r ( .s `  M
) w ) ( +g  `  M ) ( r ( .s
`  M ) x ) )  /\  (
( q ( +g  `  (Scalar `  M )
) r ) ( .s `  M ) w )  =  ( ( q ( .s
`  M ) w ) ( +g  `  M
) ( r ( .s `  M ) w ) ) )  /\  ( ( ( q ( .r `  (Scalar `  M ) ) r ) ( .s
`  M ) w )  =  ( q ( .s `  M
) ( r ( .s `  M ) w ) )  /\  ( ( 1r `  (Scalar `  M ) ) ( .s `  M
) w )  =  w ) )  <->  ( (
( r ( .s
`  M ) I )  e.  { I }  /\  ( r ( .s `  M ) ( I ( +g  `  M ) I ) )  =  ( ( r ( .s `  M ) I ) ( +g  `  M
) ( r ( .s `  M ) I ) )  /\  ( ( q ( +g  `  (Scalar `  M ) ) r ) ( .s `  M ) I )  =  ( ( q ( .s `  M
) I ) ( +g  `  M ) ( r ( .s
`  M ) I ) ) )  /\  ( ( ( q ( .r `  (Scalar `  M ) ) r ) ( .s `  M ) I )  =  ( q ( .s `  M ) ( r ( .s
`  M ) I ) )  /\  (
( 1r `  (Scalar `  M ) ) ( .s `  M ) I )  =  I ) ) ) )
9089ralbidv 2896 . . . 4  |-  ( ( I  e.  V  /\  R  e.  Ring )  -> 
( A. r  e.  ( Base `  (Scalar `  M ) ) A. x  e.  { I } A. w  e.  {
I }  ( ( ( r ( .s
`  M ) w )  e.  { I }  /\  ( r ( .s `  M ) ( w ( +g  `  M ) x ) )  =  ( ( r ( .s `  M ) w ) ( +g  `  M
) ( r ( .s `  M ) x ) )  /\  ( ( q ( +g  `  (Scalar `  M ) ) r ) ( .s `  M ) w )  =  ( ( q ( .s `  M
) w ) ( +g  `  M ) ( r ( .s
`  M ) w ) ) )  /\  ( ( ( q ( .r `  (Scalar `  M ) ) r ) ( .s `  M ) w )  =  ( q ( .s `  M ) ( r ( .s
`  M ) w ) )  /\  (
( 1r `  (Scalar `  M ) ) ( .s `  M ) w )  =  w ) )  <->  A. r  e.  ( Base `  (Scalar `  M ) ) ( ( ( r ( .s `  M ) I )  e.  {
I }  /\  (
r ( .s `  M ) ( I ( +g  `  M
) I ) )  =  ( ( r ( .s `  M
) I ) ( +g  `  M ) ( r ( .s
`  M ) I ) )  /\  (
( q ( +g  `  (Scalar `  M )
) r ) ( .s `  M ) I )  =  ( ( q ( .s
`  M ) I ) ( +g  `  M
) ( r ( .s `  M ) I ) ) )  /\  ( ( ( q ( .r `  (Scalar `  M ) ) r ) ( .s
`  M ) I )  =  ( q ( .s `  M
) ( r ( .s `  M ) I ) )  /\  ( ( 1r `  (Scalar `  M ) ) ( .s `  M
) I )  =  I ) ) ) )
9190ralbidv 2896 . . 3  |-  ( ( I  e.  V  /\  R  e.  Ring )  -> 
( A. q  e.  ( Base `  (Scalar `  M ) ) A. r  e.  ( Base `  (Scalar `  M )
) A. x  e. 
{ I } A. w  e.  { I }  ( ( ( r ( .s `  M ) w )  e.  { I }  /\  ( r ( .s
`  M ) ( w ( +g  `  M
) x ) )  =  ( ( r ( .s `  M
) w ) ( +g  `  M ) ( r ( .s
`  M ) x ) )  /\  (
( q ( +g  `  (Scalar `  M )
) r ) ( .s `  M ) w )  =  ( ( q ( .s
`  M ) w ) ( +g  `  M
) ( r ( .s `  M ) w ) ) )  /\  ( ( ( q ( .r `  (Scalar `  M ) ) r ) ( .s
`  M ) w )  =  ( q ( .s `  M
) ( r ( .s `  M ) w ) )  /\  ( ( 1r `  (Scalar `  M ) ) ( .s `  M
) w )  =  w ) )  <->  A. q  e.  ( Base `  (Scalar `  M ) ) A. r  e.  ( Base `  (Scalar `  M )
) ( ( ( r ( .s `  M ) I )  e.  { I }  /\  ( r ( .s
`  M ) ( I ( +g  `  M
) I ) )  =  ( ( r ( .s `  M
) I ) ( +g  `  M ) ( r ( .s
`  M ) I ) )  /\  (
( q ( +g  `  (Scalar `  M )
) r ) ( .s `  M ) I )  =  ( ( q ( .s
`  M ) I ) ( +g  `  M
) ( r ( .s `  M ) I ) ) )  /\  ( ( ( q ( .r `  (Scalar `  M ) ) r ) ( .s
`  M ) I )  =  ( q ( .s `  M
) ( r ( .s `  M ) I ) )  /\  ( ( 1r `  (Scalar `  M ) ) ( .s `  M
) I )  =  I ) ) ) )
9257, 91mpbird 232 . 2  |-  ( ( I  e.  V  /\  R  e.  Ring )  ->  A. q  e.  ( Base `  (Scalar `  M
) ) A. r  e.  ( Base `  (Scalar `  M ) ) A. x  e.  { I } A. w  e.  {
I }  ( ( ( r ( .s
`  M ) w )  e.  { I }  /\  ( r ( .s `  M ) ( w ( +g  `  M ) x ) )  =  ( ( r ( .s `  M ) w ) ( +g  `  M
) ( r ( .s `  M ) x ) )  /\  ( ( q ( +g  `  (Scalar `  M ) ) r ) ( .s `  M ) w )  =  ( ( q ( .s `  M
) w ) ( +g  `  M ) ( r ( .s
`  M ) w ) ) )  /\  ( ( ( q ( .r `  (Scalar `  M ) ) r ) ( .s `  M ) w )  =  ( q ( .s `  M ) ( r ( .s
`  M ) w ) )  /\  (
( 1r `  (Scalar `  M ) ) ( .s `  M ) w )  =  w ) ) )
934lmodbase 14609 . . . 4  |-  ( { I }  e.  _V  ->  { I }  =  ( Base `  M )
)
945, 93ax-mp 5 . . 3  |-  { I }  =  ( Base `  M )
95 eqid 2460 . . 3  |-  ( +g  `  M )  =  ( +g  `  M )
96 eqid 2460 . . 3  |-  ( .s
`  M )  =  ( .s `  M
)
97 eqid 2460 . . 3  |-  (Scalar `  M )  =  (Scalar `  M )
98 eqid 2460 . . 3  |-  ( Base `  (Scalar `  M )
)  =  ( Base `  (Scalar `  M )
)
99 eqid 2460 . . 3  |-  ( +g  `  (Scalar `  M )
)  =  ( +g  `  (Scalar `  M )
)
100 eqid 2460 . . 3  |-  ( .r
`  (Scalar `  M )
)  =  ( .r
`  (Scalar `  M )
)
101 eqid 2460 . . 3  |-  ( 1r
`  (Scalar `  M )
)  =  ( 1r
`  (Scalar `  M )
)
10294, 95, 96, 97, 98, 99, 100, 101islmod 17292 . 2  |-  ( M  e.  LMod  <->  ( M  e. 
Grp  /\  (Scalar `  M
)  e.  Ring  /\  A. q  e.  ( Base `  (Scalar `  M )
) A. r  e.  ( Base `  (Scalar `  M ) ) A. x  e.  { I } A. w  e.  {
I }  ( ( ( r ( .s
`  M ) w )  e.  { I }  /\  ( r ( .s `  M ) ( w ( +g  `  M ) x ) )  =  ( ( r ( .s `  M ) w ) ( +g  `  M
) ( r ( .s `  M ) x ) )  /\  ( ( q ( +g  `  (Scalar `  M ) ) r ) ( .s `  M ) w )  =  ( ( q ( .s `  M
) w ) ( +g  `  M ) ( r ( .s
`  M ) w ) ) )  /\  ( ( ( q ( .r `  (Scalar `  M ) ) r ) ( .s `  M ) w )  =  ( q ( .s `  M ) ( r ( .s
`  M ) w ) )  /\  (
( 1r `  (Scalar `  M ) ) ( .s `  M ) w )  =  w ) ) ) )
10330, 35, 92, 102syl3anbrc 1175 1  |-  ( ( I  e.  V  /\  R  e.  Ring )  ->  M  e.  LMod )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762   A.wral 2807   _Vcvv 3106    u. cun 3467   {csn 4020   {cpr 4022   {ctp 4024   <.cop 4026   ` cfv 5579  (class class class)co 6275    |-> cmpt2 6277   ndxcnx 14476   Basecbs 14479   +g cplusg 14544   .rcmulr 14545  Scalarcsca 14547   .scvsca 14548   Grpcgrp 15716   1rcur 16936   Ringcrg 16979   LModclmod 17288
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-recs 7032  df-rdg 7066  df-1o 7120  df-oadd 7124  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-nn 10526  df-2 10583  df-3 10584  df-4 10585  df-5 10586  df-6 10587  df-n0 10785  df-z 10854  df-uz 11072  df-fz 11662  df-struct 14481  df-ndx 14482  df-slot 14483  df-base 14484  df-sets 14485  df-plusg 14557  df-sca 14560  df-vsca 14561  df-0g 14686  df-mnd 15721  df-grp 15851  df-mgp 16925  df-ur 16937  df-rng 16981  df-lmod 17290
This theorem is referenced by:  lmod1zr  32050
  Copyright terms: Public domain W3C validator