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Theorem lmod1 32803
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 2441 . . . . 5  |-  { <. (
Base `  ndx ) ,  { I } >. , 
<. ( +g  `  ndx ) ,  { <. <. I ,  I >. ,  I >. }
>. }  =  { <. (
Base `  ndx ) ,  { I } >. , 
<. ( +g  `  ndx ) ,  { <. <. I ,  I >. ,  I >. }
>. }
21grp1 16011 . . . 4  |-  ( I  e.  V  ->  { <. (
Base `  ndx ) ,  { I } >. , 
<. ( +g  `  ndx ) ,  { <. <. I ,  I >. ,  I >. }
>. }  e.  Grp )
3 fvex 5862 . . . . . . 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 4674 . . . . . . . . . . . . 13  |-  { I }  e.  _V
61grpbase 14609 . . . . . . . . . . . . 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 4202 . . . . . . . . . . 11  |-  <. ( Base `  ndx ) ,  { I } >.  = 
<. ( Base `  ndx ) ,  ( Base `  { <. ( Base `  ndx ) ,  { I } >. ,  <. ( +g  `  ndx ) ,  { <. <. I ,  I >. ,  I >. } >. } ) >.
9 tpeq1 4099 . . . . . . . . . . 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 3636 . . . . . . . . 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 2470 . . . . . . . 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 14634 . . . . . . 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 2454 . . . . 5  |-  ( Base `  M )  =  (
Base `  { <. ( Base `  ndx ) ,  { I } >. , 
<. ( +g  `  ndx ) ,  { <. <. I ,  I >. ,  I >. }
>. } )
16 fvex 5862 . . . . . . 7  |-  ( +g  `  { <. ( Base `  ndx ) ,  { I } >. ,  <. ( +g  `  ndx ) ,  { <. <. I ,  I >. ,  I >. } >. } )  e.  _V
17 snex 4674 . . . . . . . . . . . . 13  |-  { <. <.
I ,  I >. ,  I >. }  e.  _V
181grpplusg 14610 . . . . . . . . . . . . 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 4202 . . . . . . . . . . 11  |-  <. ( +g  `  ndx ) ,  { <. <. I ,  I >. ,  I >. } >.  = 
<. ( +g  `  ndx ) ,  ( +g  `  { <. ( Base `  ndx ) ,  { I } >. ,  <. ( +g  `  ndx ) ,  { <. <. I ,  I >. ,  I >. } >. } ) >.
21 tpeq2 4100 . . . . . . . . . . 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 3636 . . . . . . . . 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 2470 . . . . . . . 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 14635 . . . . . . 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 2454 . . . . 5  |-  ( +g  `  M )  =  ( +g  `  { <. (
Base `  ndx ) ,  { I } >. , 
<. ( +g  `  ndx ) ,  { <. <. I ,  I >. ,  I >. }
>. } )
2815, 27grpprop 15938 . . . 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 14636 . . . . 5  |-  ( R  e.  Ring  ->  R  =  (Scalar `  M )
)
3231eqcomd 2449 . . . 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 2529 . 2  |-  ( ( I  e.  V  /\  R  e.  Ring )  -> 
(Scalar `  M )  e.  Ring )
3633fveq2d 5856 . . . . . . 7  |-  ( ( I  e.  V  /\  R  e.  Ring )  -> 
( Base `  (Scalar `  M
) )  =  (
Base `  R )
)
3736eleq2d 2511 . . . . . 6  |-  ( ( I  e.  V  /\  R  e.  Ring )  -> 
( q  e.  (
Base `  (Scalar `  M
) )  <->  q  e.  ( Base `  R )
) )
3836eleq2d 2511 . . . . . 6  |-  ( ( I  e.  V  /\  R  e.  Ring )  -> 
( r  e.  (
Base `  (Scalar `  M
) )  <->  r  e.  ( Base `  R )
) )
3937, 38anbi12d 710 . . . . 5  |-  ( ( 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 . . . . . . . . . 10  |-  ( ( ( I  e.  V  /\  R  e.  Ring )  /\  ( q  e.  ( Base `  R
)  /\  r  e.  ( Base `  R )
) )  ->  I  e.  V )
41 simplr 754 . . . . . . . . . 10  |-  ( ( ( I  e.  V  /\  R  e.  Ring )  /\  ( q  e.  ( Base `  R
)  /\  r  e.  ( Base `  R )
) )  ->  R  e.  Ring )
42 simprr 756 . . . . . . . . . 10  |-  ( ( ( I  e.  V  /\  R  e.  Ring )  /\  ( q  e.  ( Base `  R
)  /\  r  e.  ( Base `  R )
) )  ->  r  e.  ( Base `  R
) )
4340, 41, 423jca 1175 . . . . . . . . 9  |-  ( ( ( 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 32798 . . . . . . . . 9  |-  ( ( I  e.  V  /\  R  e.  Ring  /\  r  e.  ( Base `  R
) )  ->  (
r ( .s `  M ) I )  e.  { I }
)
4543, 44syl 16 . . . . . . . 8  |-  ( ( ( I  e.  V  /\  R  e.  Ring )  /\  ( q  e.  ( Base `  R
)  /\  r  e.  ( Base `  R )
) )  ->  (
r ( .s `  M ) I )  e.  { I }
)
464lmod1lem2 32799 . . . . . . . . 9  |-  ( ( 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 . . . . . . . 8  |-  ( ( ( 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 32800 . . . . . . . 8  |-  ( ( ( 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 1175 . . . . . . 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 ) ) ) )
504lmod1lem4 32801 . . . . . . 7  |-  ( ( ( 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 32802 . . . . . . . 8  |-  ( ( I  e.  V  /\  R  e.  Ring )  -> 
( ( 1r `  (Scalar `  M ) ) ( .s `  M
) I )  =  I )
5251adantr 465 . . . . . . 7  |-  ( ( ( 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 . . . . . 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 ) ) )
5453ex 434 . . . . 5  |-  ( ( 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 . . . 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 ) ) ) )
5655ralrimivv 2861 . . 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 ) ) )
57 oveq2 6285 . . . . . . . . . . . 12  |-  ( x  =  I  ->  (
w ( +g  `  M
) x )  =  ( w ( +g  `  M ) I ) )
5857oveq2d 6293 . . . . . . . . . . 11  |-  ( x  =  I  ->  (
r ( .s `  M ) ( w ( +g  `  M
) x ) )  =  ( r ( .s `  M ) ( w ( +g  `  M ) I ) ) )
59 oveq2 6285 . . . . . . . . . . . 12  |-  ( x  =  I  ->  (
r ( .s `  M ) x )  =  ( r ( .s `  M ) I ) )
6059oveq2d 6293 . . . . . . . . . . 11  |-  ( x  =  I  ->  (
( r ( .s
`  M ) w ) ( +g  `  M
) ( r ( .s `  M ) x ) )  =  ( ( r ( .s `  M ) w ) ( +g  `  M ) ( r ( .s `  M
) I ) ) )
6158, 60eqeq12d 2463 . . . . . . . . . 10  |-  ( 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 ) ) ) )
62613anbi2d 1303 . . . . . . . . 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 ) ) )  <-> 
( ( 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 ) ) ) ) )
6362anbi1d 704 . . . . . . . 8  |-  ( 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 ) ) ) )
6463ralbidv 2880 . . . . . . 7  |-  ( 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 ) ) ) )
6564ralsng 4045 . . . . . 6  |-  ( 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 ) ) ) )
6665adantr 465 . . . . 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 ) )  <->  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 ) ) ) )
67 oveq2 6285 . . . . . . . . . 10  |-  ( w  =  I  ->  (
r ( .s `  M ) w )  =  ( r ( .s `  M ) I ) )
6867eleq1d 2510 . . . . . . . . 9  |-  ( w  =  I  ->  (
( r ( .s
`  M ) w )  e.  { I } 
<->  ( r ( .s
`  M ) I )  e.  { I } ) )
69 oveq1 6284 . . . . . . . . . . 11  |-  ( w  =  I  ->  (
w ( +g  `  M
) I )  =  ( I ( +g  `  M ) I ) )
7069oveq2d 6293 . . . . . . . . . 10  |-  ( w  =  I  ->  (
r ( .s `  M ) ( w ( +g  `  M
) I ) )  =  ( r ( .s `  M ) ( I ( +g  `  M ) I ) ) )
7167oveq1d 6292 . . . . . . . . . 10  |-  ( w  =  I  ->  (
( r ( .s
`  M ) w ) ( +g  `  M
) ( r ( .s `  M ) I ) )  =  ( ( r ( .s `  M ) I ) ( +g  `  M ) ( r ( .s `  M
) I ) ) )
7270, 71eqeq12d 2463 . . . . . . . . 9  |-  ( 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 ) ) ) )
73 oveq2 6285 . . . . . . . . . 10  |-  ( w  =  I  ->  (
( q ( +g  `  (Scalar `  M )
) r ) ( .s `  M ) w )  =  ( ( q ( +g  `  (Scalar `  M )
) r ) ( .s `  M ) I ) )
74 oveq2 6285 . . . . . . . . . . 11  |-  ( w  =  I  ->  (
q ( .s `  M ) w )  =  ( q ( .s `  M ) I ) )
7574, 67oveq12d 6295 . . . . . . . . . 10  |-  ( w  =  I  ->  (
( q ( .s
`  M ) w ) ( +g  `  M
) ( r ( .s `  M ) w ) )  =  ( ( q ( .s `  M ) I ) ( +g  `  M ) ( r ( .s `  M
) I ) ) )
7673, 75eqeq12d 2463 . . . . . . . . 9  |-  ( 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 ) ) ) )
7768, 72, 763anbi123d 1298 . . . . . . . 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 ) ) )  <-> 
( ( 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 ) ) ) ) )
78 oveq2 6285 . . . . . . . . . 10  |-  ( w  =  I  ->  (
( q ( .r
`  (Scalar `  M )
) r ) ( .s `  M ) w )  =  ( ( q ( .r
`  (Scalar `  M )
) r ) ( .s `  M ) I ) )
7967oveq2d 6293 . . . . . . . . . 10  |-  ( w  =  I  ->  (
q ( .s `  M ) ( r ( .s `  M
) w ) )  =  ( q ( .s `  M ) ( r ( .s
`  M ) I ) ) )
8078, 79eqeq12d 2463 . . . . . . . . 9  |-  ( 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 ) ) ) )
81 oveq2 6285 . . . . . . . . . 10  |-  ( w  =  I  ->  (
( 1r `  (Scalar `  M ) ) ( .s `  M ) w )  =  ( ( 1r `  (Scalar `  M ) ) ( .s `  M ) I ) )
82 id 22 . . . . . . . . . 10  |-  ( w  =  I  ->  w  =  I )
8381, 82eqeq12d 2463 . . . . . . . . 9  |-  ( w  =  I  ->  (
( ( 1r `  (Scalar `  M ) ) ( .s `  M
) w )  =  w  <->  ( ( 1r
`  (Scalar `  M )
) ( .s `  M ) I )  =  I ) )
8480, 83anbi12d 710 . . . . . . . 8  |-  ( 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 ) ) )
8577, 84anbi12d 710 . . . . . . 7  |-  ( 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 ) ) ) )
8685ralsng 4045 . . . . . 6  |-  ( 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 ) ) ) )
8786adantr 465 . . . . 5  |-  ( ( 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 ) ) ) )
8866, 87bitrd 253 . . . 4  |-  ( ( 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 ) ) ) )
89882ralbidv 2885 . . 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 ) ) ) )
9056, 89mpbird 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 ) ) )
914lmodbase 14634 . . . 4  |-  ( { I }  e.  _V  ->  { I }  =  ( Base `  M )
)
925, 91ax-mp 5 . . 3  |-  { I }  =  ( Base `  M )
93 eqid 2441 . . 3  |-  ( +g  `  M )  =  ( +g  `  M )
94 eqid 2441 . . 3  |-  ( .s
`  M )  =  ( .s `  M
)
95 eqid 2441 . . 3  |-  (Scalar `  M )  =  (Scalar `  M )
96 eqid 2441 . . 3  |-  ( Base `  (Scalar `  M )
)  =  ( Base `  (Scalar `  M )
)
97 eqid 2441 . . 3  |-  ( +g  `  (Scalar `  M )
)  =  ( +g  `  (Scalar `  M )
)
98 eqid 2441 . . 3  |-  ( .r
`  (Scalar `  M )
)  =  ( .r
`  (Scalar `  M )
)
99 eqid 2441 . . 3  |-  ( 1r
`  (Scalar `  M )
)  =  ( 1r
`  (Scalar `  M )
)
10092, 93, 94, 95, 96, 97, 98, 99islmod 17384 . 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 ) ) ) )
10130, 35, 90, 100syl3anbrc 1179 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 972    = wceq 1381    e. wcel 1802   A.wral 2791   _Vcvv 3093    u. cun 3456   {csn 4010   {cpr 4012   {ctp 4014   <.cop 4016   ` cfv 5574  (class class class)co 6277    |-> cmpt2 6279   ndxcnx 14501   Basecbs 14504   +g cplusg 14569   .rcmulr 14570  Scalarcsca 14572   .scvsca 14573   Grpcgrp 15922   1rcur 17021   Ringcrg 17066   LModclmod 17380
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-int 4268  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6682  df-1st 6781  df-2nd 6782  df-recs 7040  df-rdg 7074  df-1o 7128  df-oadd 7132  df-er 7309  df-en 7515  df-dom 7516  df-sdom 7517  df-fin 7518  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9807  df-neg 9808  df-nn 10538  df-2 10595  df-3 10596  df-4 10597  df-5 10598  df-6 10599  df-n0 10797  df-z 10866  df-uz 11086  df-fz 11677  df-struct 14506  df-ndx 14507  df-slot 14508  df-base 14509  df-sets 14510  df-plusg 14582  df-sca 14585  df-vsca 14586  df-0g 14711  df-mgm 15741  df-sgrp 15780  df-mnd 15790  df-grp 15926  df-mgp 17010  df-ur 17022  df-ring 17068  df-lmod 17382
This theorem is referenced by:  lmod1zr  32804
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