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Theorem lmod1 40793
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 2471 . . . . 5  |-  { <. (
Base `  ndx ) ,  { I } >. , 
<. ( +g  `  ndx ) ,  { <. <. I ,  I >. ,  I >. }
>. }  =  { <. (
Base `  ndx ) ,  { I } >. , 
<. ( +g  `  ndx ) ,  { <. <. I ,  I >. ,  I >. }
>. }
21grp1 16836 . . . 4  |-  ( I  e.  V  ->  { <. (
Base `  ndx ) ,  { I } >. , 
<. ( +g  `  ndx ) ,  { <. <. I ,  I >. ,  I >. }
>. }  e.  Grp )
3 fvex 5889 . . . . . . 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 4641 . . . . . . . . . . . . 13  |-  { I }  e.  _V
61grpbase 15315 . . . . . . . . . . . . 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 4162 . . . . . . . . . . 11  |-  <. ( Base `  ndx ) ,  { I } >.  = 
<. ( Base `  ndx ) ,  ( Base `  { <. ( Base `  ndx ) ,  { I } >. ,  <. ( +g  `  ndx ) ,  { <. <. I ,  I >. ,  I >. } >. } ) >.
9 tpeq1 4051 . . . . . . . . . . 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 3575 . . . . . . . . 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 2493 . . . . . . . 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 15340 . . . . . . 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 2480 . . . . 5  |-  ( Base `  M )  =  (
Base `  { <. ( Base `  ndx ) ,  { I } >. , 
<. ( +g  `  ndx ) ,  { <. <. I ,  I >. ,  I >. }
>. } )
16 fvex 5889 . . . . . . 7  |-  ( +g  `  { <. ( Base `  ndx ) ,  { I } >. ,  <. ( +g  `  ndx ) ,  { <. <. I ,  I >. ,  I >. } >. } )  e.  _V
17 snex 4641 . . . . . . . . . . . . 13  |-  { <. <.
I ,  I >. ,  I >. }  e.  _V
181grpplusg 15316 . . . . . . . . . . . . 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 4162 . . . . . . . . . . 11  |-  <. ( +g  `  ndx ) ,  { <. <. I ,  I >. ,  I >. } >.  = 
<. ( +g  `  ndx ) ,  ( +g  `  { <. ( Base `  ndx ) ,  { I } >. ,  <. ( +g  `  ndx ) ,  { <. <. I ,  I >. ,  I >. } >. } ) >.
21 tpeq2 4052 . . . . . . . . . . 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 3575 . . . . . . . . 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 2493 . . . . . . . 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 15341 . . . . . . 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 2480 . . . . 5  |-  ( +g  `  M )  =  ( +g  `  { <. (
Base `  ndx ) ,  { I } >. , 
<. ( +g  `  ndx ) ,  { <. <. I ,  I >. ,  I >. }
>. } )
2815, 27grpprop 16763 . . . 4  |-  ( M  e.  Grp  <->  { <. ( Base `  ndx ) ,  { I } >. , 
<. ( +g  `  ndx ) ,  { <. <. I ,  I >. ,  I >. }
>. }  e.  Grp )
292, 28sylibr 217 . . 3  |-  ( I  e.  V  ->  M  e.  Grp )
3029adantr 472 . 2  |-  ( ( I  e.  V  /\  R  e.  Ring )  ->  M  e.  Grp )
314lmodsca 15342 . . . . 5  |-  ( R  e.  Ring  ->  R  =  (Scalar `  M )
)
3231eqcomd 2477 . . . 4  |-  ( R  e.  Ring  ->  (Scalar `  M )  =  R )
3332adantl 473 . . 3  |-  ( ( I  e.  V  /\  R  e.  Ring )  -> 
(Scalar `  M )  =  R )
34 simpr 468 . . 3  |-  ( ( I  e.  V  /\  R  e.  Ring )  ->  R  e.  Ring )
3533, 34eqeltrd 2549 . 2  |-  ( ( I  e.  V  /\  R  e.  Ring )  -> 
(Scalar `  M )  e.  Ring )
3633fveq2d 5883 . . . . . . 7  |-  ( ( I  e.  V  /\  R  e.  Ring )  -> 
( Base `  (Scalar `  M
) )  =  (
Base `  R )
)
3736eleq2d 2534 . . . . . 6  |-  ( ( I  e.  V  /\  R  e.  Ring )  -> 
( q  e.  (
Base `  (Scalar `  M
) )  <->  q  e.  ( Base `  R )
) )
3836eleq2d 2534 . . . . . 6  |-  ( ( I  e.  V  /\  R  e.  Ring )  -> 
( r  e.  (
Base `  (Scalar `  M
) )  <->  r  e.  ( Base `  R )
) )
3937, 38anbi12d 725 . . . . 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 768 . . . . . . . . . 10  |-  ( ( ( I  e.  V  /\  R  e.  Ring )  /\  ( q  e.  ( Base `  R
)  /\  r  e.  ( Base `  R )
) )  ->  I  e.  V )
41 simplr 770 . . . . . . . . . 10  |-  ( ( ( I  e.  V  /\  R  e.  Ring )  /\  ( q  e.  ( Base `  R
)  /\  r  e.  ( Base `  R )
) )  ->  R  e.  Ring )
42 simprr 774 . . . . . . . . . 10  |-  ( ( ( I  e.  V  /\  R  e.  Ring )  /\  ( q  e.  ( Base `  R
)  /\  r  e.  ( Base `  R )
) )  ->  r  e.  ( Base `  R
) )
4340, 41, 423jca 1210 . . . . . . . . 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 40788 . . . . . . . . 9  |-  ( ( I  e.  V  /\  R  e.  Ring  /\  r  e.  ( Base `  R
) )  ->  (
r ( .s `  M ) I )  e.  { I }
)
4543, 44syl 17 . . . . . . . 8  |-  ( ( ( I  e.  V  /\  R  e.  Ring )  /\  ( q  e.  ( Base `  R
)  /\  r  e.  ( Base `  R )
) )  ->  (
r ( .s `  M ) I )  e.  { I }
)
464lmod1lem2 40789 . . . . . . . . 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 17 . . . . . . . 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 40790 . . . . . . . 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 1210 . . . . . . 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 40791 . . . . . . 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 40792 . . . . . . . 8  |-  ( ( I  e.  V  /\  R  e.  Ring )  -> 
( ( 1r `  (Scalar `  M ) ) ( .s `  M
) I )  =  I )
5251adantr 472 . . . . . . 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 544 . . . . . 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 441 . . . . 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 223 . . . 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 2813 . . 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 6316 . . . . . . . . . . . 12  |-  ( x  =  I  ->  (
w ( +g  `  M
) x )  =  ( w ( +g  `  M ) I ) )
5857oveq2d 6324 . . . . . . . . . . 11  |-  ( x  =  I  ->  (
r ( .s `  M ) ( w ( +g  `  M
) x ) )  =  ( r ( .s `  M ) ( w ( +g  `  M ) I ) ) )
59 oveq2 6316 . . . . . . . . . . . 12  |-  ( x  =  I  ->  (
r ( .s `  M ) x )  =  ( r ( .s `  M ) I ) )
6059oveq2d 6324 . . . . . . . . . . 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 2486 . . . . . . . . . 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 1370 . . . . . . . . 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 719 . . . . . . . 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 2829 . . . . . . 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 3997 . . . . . 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 472 . . . . 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 6316 . . . . . . . . . 10  |-  ( w  =  I  ->  (
r ( .s `  M ) w )  =  ( r ( .s `  M ) I ) )
6867eleq1d 2533 . . . . . . . . 9  |-  ( w  =  I  ->  (
( r ( .s
`  M ) w )  e.  { I } 
<->  ( r ( .s
`  M ) I )  e.  { I } ) )
69 oveq1 6315 . . . . . . . . . . 11  |-  ( w  =  I  ->  (
w ( +g  `  M
) I )  =  ( I ( +g  `  M ) I ) )
7069oveq2d 6324 . . . . . . . . . 10  |-  ( w  =  I  ->  (
r ( .s `  M ) ( w ( +g  `  M
) I ) )  =  ( r ( .s `  M ) ( I ( +g  `  M ) I ) ) )
7167oveq1d 6323 . . . . . . . . . 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 2486 . . . . . . . . 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 6316 . . . . . . . . . 10  |-  ( w  =  I  ->  (
( q ( +g  `  (Scalar `  M )
) r ) ( .s `  M ) w )  =  ( ( q ( +g  `  (Scalar `  M )
) r ) ( .s `  M ) I ) )
74 oveq2 6316 . . . . . . . . . . 11  |-  ( w  =  I  ->  (
q ( .s `  M ) w )  =  ( q ( .s `  M ) I ) )
7574, 67oveq12d 6326 . . . . . . . . . 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 2486 . . . . . . . . 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 1365 . . . . . . . 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 6316 . . . . . . . . . 10  |-  ( w  =  I  ->  (
( q ( .r
`  (Scalar `  M )
) r ) ( .s `  M ) w )  =  ( ( q ( .r
`  (Scalar `  M )
) r ) ( .s `  M ) I ) )
7967oveq2d 6324 . . . . . . . . . 10  |-  ( w  =  I  ->  (
q ( .s `  M ) ( r ( .s `  M
) w ) )  =  ( q ( .s `  M ) ( r ( .s
`  M ) I ) ) )
8078, 79eqeq12d 2486 . . . . . . . . 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 6316 . . . . . . . . . 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 2486 . . . . . . . . 9  |-  ( w  =  I  ->  (
( ( 1r `  (Scalar `  M ) ) ( .s `  M
) w )  =  w  <->  ( ( 1r
`  (Scalar `  M )
) ( .s `  M ) I )  =  I ) )
8480, 83anbi12d 725 . . . . . . . 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 725 . . . . . . 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 3997 . . . . . 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 472 . . . . 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 261 . . . 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 2832 . . 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 240 . 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 15340 . . . 4  |-  ( { I }  e.  _V  ->  { I }  =  ( Base `  M )
)
925, 91ax-mp 5 . . 3  |-  { I }  =  ( Base `  M )
93 eqid 2471 . . 3  |-  ( +g  `  M )  =  ( +g  `  M )
94 eqid 2471 . . 3  |-  ( .s
`  M )  =  ( .s `  M
)
95 eqid 2471 . . 3  |-  (Scalar `  M )  =  (Scalar `  M )
96 eqid 2471 . . 3  |-  ( Base `  (Scalar `  M )
)  =  ( Base `  (Scalar `  M )
)
97 eqid 2471 . . 3  |-  ( +g  `  (Scalar `  M )
)  =  ( +g  `  (Scalar `  M )
)
98 eqid 2471 . . 3  |-  ( .r
`  (Scalar `  M )
)  =  ( .r
`  (Scalar `  M )
)
99 eqid 2471 . . 3  |-  ( 1r
`  (Scalar `  M )
)  =  ( 1r
`  (Scalar `  M )
)
10092, 93, 94, 95, 96, 97, 98, 99islmod 18173 . 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 1214 1  |-  ( ( I  e.  V  /\  R  e.  Ring )  ->  M  e.  LMod )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904   A.wral 2756   _Vcvv 3031    u. cun 3388   {csn 3959   {cpr 3961   {ctp 3963   <.cop 3965   ` cfv 5589  (class class class)co 6308    |-> cmpt2 6310   ndxcnx 15196   Basecbs 15199   +g cplusg 15268   .rcmulr 15269  Scalarcsca 15271   .scvsca 15272   Grpcgrp 16747   1rcur 17813   Ringcrg 17858   LModclmod 18169
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-2 10690  df-3 10691  df-4 10692  df-5 10693  df-6 10694  df-n0 10894  df-z 10962  df-uz 11183  df-fz 11811  df-struct 15201  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-plusg 15281  df-sca 15284  df-vsca 15285  df-0g 15418  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-grp 16751  df-mgp 17802  df-ur 17814  df-ring 17860  df-lmod 18171
This theorem is referenced by:  lmod1zr  40794
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