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Theorem lmod1 31143
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 2451 . . . . 5  |-  { <. (
Base `  ndx ) ,  { I } >. , 
<. ( +g  `  ndx ) ,  { <. <. I ,  I >. ,  I >. }
>. }  =  { <. (
Base `  ndx ) ,  { I } >. , 
<. ( +g  `  ndx ) ,  { <. <. I ,  I >. ,  I >. }
>. }
21grp1 31131 . . . 4  |-  ( I  e.  V  ->  { <. (
Base `  ndx ) ,  { I } >. , 
<. ( +g  `  ndx ) ,  { <. <. I ,  I >. ,  I >. }
>. }  e.  Grp )
3 fvex 5801 . . . . . . 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 4633 . . . . . . . . . . . . 13  |-  { I }  e.  _V
61grpbase 14382 . . . . . . . . . . . . 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 4163 . . . . . . . . . . 11  |-  <. ( Base `  ndx ) ,  { I } >.  = 
<. ( Base `  ndx ) ,  ( Base `  { <. ( Base `  ndx ) ,  { I } >. ,  <. ( +g  `  ndx ) ,  { <. <. I ,  I >. ,  I >. } >. } ) >.
9 tpeq1 4063 . . . . . . . . . . 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 3606 . . . . . . . . 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 2480 . . . . . . . 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 14407 . . . . . . 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 2464 . . . . 5  |-  ( Base `  M )  =  (
Base `  { <. ( Base `  ndx ) ,  { I } >. , 
<. ( +g  `  ndx ) ,  { <. <. I ,  I >. ,  I >. }
>. } )
16 fvex 5801 . . . . . . 7  |-  ( +g  `  { <. ( Base `  ndx ) ,  { I } >. ,  <. ( +g  `  ndx ) ,  { <. <. I ,  I >. ,  I >. } >. } )  e.  _V
17 snex 4633 . . . . . . . . . . . . 13  |-  { <. <.
I ,  I >. ,  I >. }  e.  _V
181grpplusg 14383 . . . . . . . . . . . . 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 4163 . . . . . . . . . . 11  |-  <. ( +g  `  ndx ) ,  { <. <. I ,  I >. ,  I >. } >.  = 
<. ( +g  `  ndx ) ,  ( +g  `  { <. ( Base `  ndx ) ,  { I } >. ,  <. ( +g  `  ndx ) ,  { <. <. I ,  I >. ,  I >. } >. } ) >.
21 tpeq2 4064 . . . . . . . . . . 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 3606 . . . . . . . . 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 2480 . . . . . . . 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 14408 . . . . . . 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 2464 . . . . 5  |-  ( +g  `  M )  =  ( +g  `  { <. (
Base `  ndx ) ,  { I } >. , 
<. ( +g  `  ndx ) ,  { <. <. I ,  I >. ,  I >. }
>. } )
2815, 27grpprop 15661 . . . 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 14409 . . . . 5  |-  ( R  e.  Ring  ->  R  =  (Scalar `  M )
)
3231eqcomd 2459 . . . 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 2539 . 2  |-  ( ( I  e.  V  /\  R  e.  Ring )  -> 
(Scalar `  M )  e.  Ring )
3633fveq2d 5795 . . . . . . . 8  |-  ( ( I  e.  V  /\  R  e.  Ring )  -> 
( Base `  (Scalar `  M
) )  =  (
Base `  R )
)
3736eleq2d 2521 . . . . . . 7  |-  ( ( I  e.  V  /\  R  e.  Ring )  -> 
( q  e.  (
Base `  (Scalar `  M
) )  <->  q  e.  ( Base `  R )
) )
3836eleq2d 2521 . . . . . . 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 1168 . . . . . . . . . 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 31138 . . . . . . . . . 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 31139 . . . . . . . . . 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 31140 . . . . . . . . 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 1168 . . . . . . . 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 31141 . . . . . . . 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 31142 . . . . . . . . 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 2906 . . 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 6200 . . . . . . . . . . . . 13  |-  ( x  =  I  ->  (
w ( +g  `  M
) x )  =  ( w ( +g  `  M ) I ) )
5958oveq2d 6208 . . . . . . . . . . . 12  |-  ( x  =  I  ->  (
r ( .s `  M ) ( w ( +g  `  M
) x ) )  =  ( r ( .s `  M ) ( w ( +g  `  M ) I ) ) )
60 oveq2 6200 . . . . . . . . . . . . 13  |-  ( x  =  I  ->  (
r ( .s `  M ) x )  =  ( r ( .s `  M ) I ) )
6160oveq2d 6208 . . . . . . . . . . . 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 2473 . . . . . . . . . . 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 1295 . . . . . . . . . 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 2838 . . . . . . . 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 4012 . . . . . . 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 6200 . . . . . . . . . . 11  |-  ( w  =  I  ->  (
r ( .s `  M ) w )  =  ( r ( .s `  M ) I ) )
6968eleq1d 2520 . . . . . . . . . 10  |-  ( w  =  I  ->  (
( r ( .s
`  M ) w )  e.  { I } 
<->  ( r ( .s
`  M ) I )  e.  { I } ) )
70 oveq1 6199 . . . . . . . . . . . 12  |-  ( w  =  I  ->  (
w ( +g  `  M
) I )  =  ( I ( +g  `  M ) I ) )
7170oveq2d 6208 . . . . . . . . . . 11  |-  ( w  =  I  ->  (
r ( .s `  M ) ( w ( +g  `  M
) I ) )  =  ( r ( .s `  M ) ( I ( +g  `  M ) I ) ) )
7268oveq1d 6207 . . . . . . . . . . 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 2473 . . . . . . . . . 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 6200 . . . . . . . . . . 11  |-  ( w  =  I  ->  (
( q ( +g  `  (Scalar `  M )
) r ) ( .s `  M ) w )  =  ( ( q ( +g  `  (Scalar `  M )
) r ) ( .s `  M ) I ) )
75 oveq2 6200 . . . . . . . . . . . 12  |-  ( w  =  I  ->  (
q ( .s `  M ) w )  =  ( q ( .s `  M ) I ) )
7675, 68oveq12d 6210 . . . . . . . . . . 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 2473 . . . . . . . . . 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 1290 . . . . . . . . 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 6200 . . . . . . . . . . 11  |-  ( w  =  I  ->  (
( q ( .r
`  (Scalar `  M )
) r ) ( .s `  M ) w )  =  ( ( q ( .r
`  (Scalar `  M )
) r ) ( .s `  M ) I ) )
8068oveq2d 6208 . . . . . . . . . . 11  |-  ( w  =  I  ->  (
q ( .s `  M ) ( r ( .s `  M
) w ) )  =  ( q ( .s `  M ) ( r ( .s
`  M ) I ) ) )
8179, 80eqeq12d 2473 . . . . . . . . . 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 6200 . . . . . . . . . . 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 2473 . . . . . . . . . 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 4012 . . . . . . 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 2838 . . . 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 2838 . . 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 14407 . . . 4  |-  ( { I }  e.  _V  ->  { I }  =  ( Base `  M )
)
945, 93ax-mp 5 . . 3  |-  { I }  =  ( Base `  M )
95 eqid 2451 . . 3  |-  ( +g  `  M )  =  ( +g  `  M )
96 eqid 2451 . . 3  |-  ( .s
`  M )  =  ( .s `  M
)
97 eqid 2451 . . 3  |-  (Scalar `  M )  =  (Scalar `  M )
98 eqid 2451 . . 3  |-  ( Base `  (Scalar `  M )
)  =  ( Base `  (Scalar `  M )
)
99 eqid 2451 . . 3  |-  ( +g  `  (Scalar `  M )
)  =  ( +g  `  (Scalar `  M )
)
100 eqid 2451 . . 3  |-  ( .r
`  (Scalar `  M )
)  =  ( .r
`  (Scalar `  M )
)
101 eqid 2451 . . 3  |-  ( 1r
`  (Scalar `  M )
)  =  ( 1r
`  (Scalar `  M )
)
10294, 95, 96, 97, 98, 99, 100, 101islmod 17060 . 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 1172 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 965    = wceq 1370    e. wcel 1758   A.wral 2795   _Vcvv 3070    u. cun 3426   {csn 3977   {cpr 3979   {ctp 3981   <.cop 3983   ` cfv 5518  (class class class)co 6192    |-> cmpt2 6194   ndxcnx 14275   Basecbs 14278   +g cplusg 14342   .rcmulr 14343  Scalarcsca 14345   .scvsca 14346   Grpcgrp 15514   1rcur 16710   Ringcrg 16753   LModclmod 17056
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4503  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474  ax-cnex 9441  ax-resscn 9442  ax-1cn 9443  ax-icn 9444  ax-addcl 9445  ax-addrcl 9446  ax-mulcl 9447  ax-mulrcl 9448  ax-mulcom 9449  ax-addass 9450  ax-mulass 9451  ax-distr 9452  ax-i2m1 9453  ax-1ne0 9454  ax-1rid 9455  ax-rnegex 9456  ax-rrecex 9457  ax-cnre 9458  ax-pre-lttri 9459  ax-pre-lttrn 9460  ax-pre-ltadd 9461  ax-pre-mulgt0 9462
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-pss 3444  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-tp 3982  df-op 3984  df-uni 4192  df-int 4229  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4486  df-eprel 4732  df-id 4736  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-riota 6153  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-om 6579  df-1st 6679  df-2nd 6680  df-recs 6934  df-rdg 6968  df-1o 7022  df-oadd 7026  df-er 7203  df-en 7413  df-dom 7414  df-sdom 7415  df-fin 7416  df-pnf 9523  df-mnf 9524  df-xr 9525  df-ltxr 9526  df-le 9527  df-sub 9700  df-neg 9701  df-nn 10426  df-2 10483  df-3 10484  df-4 10485  df-5 10486  df-6 10487  df-n0 10683  df-z 10750  df-uz 10965  df-fz 11541  df-struct 14280  df-ndx 14281  df-slot 14282  df-base 14283  df-sets 14284  df-plusg 14355  df-sca 14358  df-vsca 14359  df-0g 14484  df-mnd 15519  df-grp 15649  df-mgp 16699  df-ur 16711  df-rng 16755  df-lmod 17058
This theorem is referenced by:  lmod1zr  31144
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