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Theorem mendvscafval 29552
Description: Scalar multiplication in the module endomorphism algebra. (Contributed by Stefan O'Rear, 2-Sep-2015.)
Hypotheses
Ref Expression
mendvscafval.a  |-  A  =  (MEndo `  M )
mendvscafval.v  |-  .x.  =  ( .s `  M )
mendvscafval.b  |-  B  =  ( Base `  A
)
mendvscafval.s  |-  S  =  (Scalar `  M )
mendvscafval.k  |-  K  =  ( Base `  S
)
mendvscafval.e  |-  E  =  ( Base `  M
)
Assertion
Ref Expression
mendvscafval  |-  ( .s
`  A )  =  ( x  e.  K ,  y  e.  B  |->  ( ( E  X.  { x } )  oF  .x.  y
) )
Distinct variable groups:    x, y, B    x, K, y    x, M, y
Allowed substitution hints:    A( x, y)    S( x, y)    .x. ( x, y)    E( x, y)

Proof of Theorem mendvscafval
StepHypRef Expression
1 mendvscafval.a . . 3  |-  A  =  (MEndo `  M )
21fveq2i 5699 . 2  |-  ( .s
`  A )  =  ( .s `  (MEndo `  M ) )
3 mendvscafval.b . . . . . . 7  |-  B  =  ( Base `  A
)
41mendbas 29546 . . . . . . 7  |-  ( M LMHom 
M )  =  (
Base `  A )
53, 4eqtr4i 2466 . . . . . 6  |-  B  =  ( M LMHom  M )
6 eqid 2443 . . . . . 6  |-  ( x  e.  B ,  y  e.  B  |->  ( x  oF ( +g  `  M ) y ) )  =  ( x  e.  B ,  y  e.  B  |->  ( x  oF ( +g  `  M ) y ) )
7 eqid 2443 . . . . . 6  |-  ( x  e.  B ,  y  e.  B  |->  ( x  o.  y ) )  =  ( x  e.  B ,  y  e.  B  |->  ( x  o.  y ) )
8 mendvscafval.s . . . . . 6  |-  S  =  (Scalar `  M )
9 mendvscafval.k . . . . . . 7  |-  K  =  ( Base `  S
)
10 eqid 2443 . . . . . . 7  |-  B  =  B
11 mendvscafval.e . . . . . . . . 9  |-  E  =  ( Base `  M
)
1211xpeq1i 4865 . . . . . . . 8  |-  ( E  X.  { x }
)  =  ( (
Base `  M )  X.  { x } )
13 eqid 2443 . . . . . . . 8  |-  y  =  y
14 mendvscafval.v . . . . . . . . 9  |-  .x.  =  ( .s `  M )
15 ofeq 6327 . . . . . . . . 9  |-  (  .x.  =  ( .s `  M )  ->  oF  .x.  =  oF ( .s `  M
) )
1614, 15ax-mp 5 . . . . . . . 8  |-  oF  .x.  =  oF ( .s `  M
)
1712, 13, 16oveq123i 6110 . . . . . . 7  |-  ( ( E  X.  { x } )  oF  .x.  y )  =  ( ( ( Base `  M )  X.  {
x } )  oF ( .s `  M ) y )
189, 10, 17mpt2eq123i 6154 . . . . . 6  |-  ( x  e.  K ,  y  e.  B  |->  ( ( E  X.  { x } )  oF  .x.  y ) )  =  ( x  e.  ( Base `  S
) ,  y  e.  B  |->  ( ( (
Base `  M )  X.  { x } )  oF ( .s
`  M ) y ) )
195, 6, 7, 8, 18mendval 29545 . . . . 5  |-  ( M  e.  _V  ->  (MEndo `  M )  =  ( { <. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  ( x  e.  B ,  y  e.  B  |->  ( x  oF ( +g  `  M
) y ) )
>. ,  <. ( .r
`  ndx ) ,  ( x  e.  B , 
y  e.  B  |->  ( x  o.  y ) ) >. }  u.  { <. (Scalar `  ndx ) ,  S >. ,  <. ( .s `  ndx ) ,  ( x  e.  K ,  y  e.  B  |->  ( ( E  X.  { x } )  oF  .x.  y
) ) >. } ) )
2019fveq2d 5700 . . . 4  |-  ( M  e.  _V  ->  ( .s `  (MEndo `  M
) )  =  ( .s `  ( {
<. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  ( x  e.  B ,  y  e.  B  |->  ( x  oF ( +g  `  M
) y ) )
>. ,  <. ( .r
`  ndx ) ,  ( x  e.  B , 
y  e.  B  |->  ( x  o.  y ) ) >. }  u.  { <. (Scalar `  ndx ) ,  S >. ,  <. ( .s `  ndx ) ,  ( x  e.  K ,  y  e.  B  |->  ( ( E  X.  { x } )  oF  .x.  y
) ) >. } ) ) )
21 fvex 5706 . . . . . . 7  |-  ( Base `  S )  e.  _V
229, 21eqeltri 2513 . . . . . 6  |-  K  e. 
_V
23 fvex 5706 . . . . . . 7  |-  ( Base `  A )  e.  _V
243, 23eqeltri 2513 . . . . . 6  |-  B  e. 
_V
2522, 24mpt2ex 6655 . . . . 5  |-  ( x  e.  K ,  y  e.  B  |->  ( ( E  X.  { x } )  oF  .x.  y ) )  e.  _V
26 eqid 2443 . . . . . 6  |-  ( {
<. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  ( x  e.  B ,  y  e.  B  |->  ( x  oF ( +g  `  M
) y ) )
>. ,  <. ( .r
`  ndx ) ,  ( x  e.  B , 
y  e.  B  |->  ( x  o.  y ) ) >. }  u.  { <. (Scalar `  ndx ) ,  S >. ,  <. ( .s `  ndx ) ,  ( x  e.  K ,  y  e.  B  |->  ( ( E  X.  { x } )  oF  .x.  y
) ) >. } )  =  ( { <. (
Base `  ndx ) ,  B >. ,  <. ( +g  `  ndx ) ,  ( x  e.  B ,  y  e.  B  |->  ( x  oF ( +g  `  M
) y ) )
>. ,  <. ( .r
`  ndx ) ,  ( x  e.  B , 
y  e.  B  |->  ( x  o.  y ) ) >. }  u.  { <. (Scalar `  ndx ) ,  S >. ,  <. ( .s `  ndx ) ,  ( x  e.  K ,  y  e.  B  |->  ( ( E  X.  { x } )  oF  .x.  y
) ) >. } )
2726algvsca 29544 . . . . 5  |-  ( ( x  e.  K , 
y  e.  B  |->  ( ( E  X.  {
x } )  oF  .x.  y ) )  e.  _V  ->  ( x  e.  K , 
y  e.  B  |->  ( ( E  X.  {
x } )  oF  .x.  y ) )  =  ( .s
`  ( { <. (
Base `  ndx ) ,  B >. ,  <. ( +g  `  ndx ) ,  ( x  e.  B ,  y  e.  B  |->  ( x  oF ( +g  `  M
) y ) )
>. ,  <. ( .r
`  ndx ) ,  ( x  e.  B , 
y  e.  B  |->  ( x  o.  y ) ) >. }  u.  { <. (Scalar `  ndx ) ,  S >. ,  <. ( .s `  ndx ) ,  ( x  e.  K ,  y  e.  B  |->  ( ( E  X.  { x } )  oF  .x.  y
) ) >. } ) ) )
2825, 27mp1i 12 . . . 4  |-  ( M  e.  _V  ->  (
x  e.  K , 
y  e.  B  |->  ( ( E  X.  {
x } )  oF  .x.  y ) )  =  ( .s
`  ( { <. (
Base `  ndx ) ,  B >. ,  <. ( +g  `  ndx ) ,  ( x  e.  B ,  y  e.  B  |->  ( x  oF ( +g  `  M
) y ) )
>. ,  <. ( .r
`  ndx ) ,  ( x  e.  B , 
y  e.  B  |->  ( x  o.  y ) ) >. }  u.  { <. (Scalar `  ndx ) ,  S >. ,  <. ( .s `  ndx ) ,  ( x  e.  K ,  y  e.  B  |->  ( ( E  X.  { x } )  oF  .x.  y
) ) >. } ) ) )
2920, 28eqtr4d 2478 . . 3  |-  ( M  e.  _V  ->  ( .s `  (MEndo `  M
) )  =  ( x  e.  K , 
y  e.  B  |->  ( ( E  X.  {
x } )  oF  .x.  y ) ) )
30 fvprc 5690 . . . . . 6  |-  ( -.  M  e.  _V  ->  (MEndo `  M )  =  (/) )
3130fveq2d 5700 . . . . 5  |-  ( -.  M  e.  _V  ->  ( .s `  (MEndo `  M ) )  =  ( .s `  (/) ) )
32 df-vsca 14260 . . . . . 6  |-  .s  = Slot  6
3332str0 14217 . . . . 5  |-  (/)  =  ( .s `  (/) )
3431, 33syl6eqr 2493 . . . 4  |-  ( -.  M  e.  _V  ->  ( .s `  (MEndo `  M ) )  =  (/) )
35 fvprc 5690 . . . . . . . . 9  |-  ( -.  M  e.  _V  ->  (Scalar `  M )  =  (/) )
368, 35syl5eq 2487 . . . . . . . 8  |-  ( -.  M  e.  _V  ->  S  =  (/) )
3736fveq2d 5700 . . . . . . 7  |-  ( -.  M  e.  _V  ->  (
Base `  S )  =  ( Base `  (/) ) )
38 base0 14218 . . . . . . 7  |-  (/)  =  (
Base `  (/) )
3937, 9, 383eqtr4g 2500 . . . . . 6  |-  ( -.  M  e.  _V  ->  K  =  (/) )
40 mpt2eq12 6151 . . . . . 6  |-  ( ( K  =  (/)  /\  B  =  B )  ->  (
x  e.  K , 
y  e.  B  |->  ( ( E  X.  {
x } )  oF  .x.  y ) )  =  ( x  e.  (/) ,  y  e.  B  |->  ( ( E  X.  { x }
)  oF  .x.  y ) ) )
4139, 10, 40sylancl 662 . . . . 5  |-  ( -.  M  e.  _V  ->  ( x  e.  K , 
y  e.  B  |->  ( ( E  X.  {
x } )  oF  .x.  y ) )  =  ( x  e.  (/) ,  y  e.  B  |->  ( ( E  X.  { x }
)  oF  .x.  y ) ) )
42 mpt20 6161 . . . . 5  |-  ( x  e.  (/) ,  y  e.  B  |->  ( ( E  X.  { x }
)  oF  .x.  y ) )  =  (/)
4341, 42syl6eq 2491 . . . 4  |-  ( -.  M  e.  _V  ->  ( x  e.  K , 
y  e.  B  |->  ( ( E  X.  {
x } )  oF  .x.  y ) )  =  (/) )
4434, 43eqtr4d 2478 . . 3  |-  ( -.  M  e.  _V  ->  ( .s `  (MEndo `  M ) )  =  ( x  e.  K ,  y  e.  B  |->  ( ( E  X.  { x } )  oF  .x.  y
) ) )
4529, 44pm2.61i 164 . 2  |-  ( .s
`  (MEndo `  M
) )  =  ( x  e.  K , 
y  e.  B  |->  ( ( E  X.  {
x } )  oF  .x.  y ) )
462, 45eqtri 2463 1  |-  ( .s
`  A )  =  ( x  e.  K ,  y  e.  B  |->  ( ( E  X.  { x } )  oF  .x.  y
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1369    e. wcel 1756   _Vcvv 2977    u. cun 3331   (/)c0 3642   {csn 3882   {cpr 3884   {ctp 3886   <.cop 3888    X. cxp 4843    o. ccom 4849   ` cfv 5423  (class class class)co 6096    e. cmpt2 6098    oFcof 6323   6c6 10380   ndxcnx 14176   Basecbs 14179   +g cplusg 14243   .rcmulr 14244  Scalarcsca 14246   .scvsca 14247   LMHom clmhm 17105  MEndocmend 29537
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-of 6325  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-1o 6925  df-oadd 6929  df-er 7106  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-nn 10328  df-2 10385  df-3 10386  df-4 10387  df-5 10388  df-6 10389  df-n0 10585  df-z 10652  df-uz 10867  df-fz 11443  df-struct 14181  df-ndx 14182  df-slot 14183  df-base 14184  df-plusg 14256  df-mulr 14257  df-sca 14259  df-vsca 14260  df-lmhm 17108  df-mend 29538
This theorem is referenced by:  mendvsca  29553
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