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Theorem rrgsupp 16310
Description: Left multiplication by a left regular element does not change the support set of a vector. (Contributed by Stefan O'Rear, 28-Mar-2015.)
Hypotheses
Ref Expression
rrgval.e  |-  E  =  (RLReg `  R )
rrgval.b  |-  B  =  ( Base `  R
)
rrgval.t  |-  .x.  =  ( .r `  R )
rrgval.z  |-  .0.  =  ( 0g `  R )
rrgsupp.i  |-  ( ph  ->  I  e.  V )
rrgsupp.r  |-  ( ph  ->  R  e.  Ring )
rrgsupp.x  |-  ( ph  ->  X  e.  E )
rrgsupp.y  |-  ( ph  ->  Y : I --> B )
Assertion
Ref Expression
rrgsupp  |-  ( ph  ->  ( `' ( ( I  X.  { X } )  o F 
.x.  Y ) "
( _V  \  {  .0.  } ) )  =  ( `' Y "
( _V  \  {  .0.  } ) ) )

Proof of Theorem rrgsupp
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 rrgsupp.r . . . . . . . 8  |-  ( ph  ->  R  e.  Ring )
21adantr 452 . . . . . . 7  |-  ( (
ph  /\  x  e.  I )  ->  R  e.  Ring )
3 rrgsupp.x . . . . . . . 8  |-  ( ph  ->  X  e.  E )
43adantr 452 . . . . . . 7  |-  ( (
ph  /\  x  e.  I )  ->  X  e.  E )
5 rrgsupp.y . . . . . . . 8  |-  ( ph  ->  Y : I --> B )
65ffvelrnda 5833 . . . . . . 7  |-  ( (
ph  /\  x  e.  I )  ->  ( Y `  x )  e.  B )
7 rrgval.e . . . . . . . 8  |-  E  =  (RLReg `  R )
8 rrgval.b . . . . . . . 8  |-  B  =  ( Base `  R
)
9 rrgval.t . . . . . . . 8  |-  .x.  =  ( .r `  R )
10 rrgval.z . . . . . . . 8  |-  .0.  =  ( 0g `  R )
117, 8, 9, 10rrgeq0 16309 . . . . . . 7  |-  ( ( R  e.  Ring  /\  X  e.  E  /\  ( Y `  x )  e.  B )  ->  (
( X  .x.  ( Y `  x )
)  =  .0.  <->  ( Y `  x )  =  .0.  ) )
122, 4, 6, 11syl3anc 1184 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  (
( X  .x.  ( Y `  x )
)  =  .0.  <->  ( Y `  x )  =  .0.  ) )
1312necon3bid 2606 . . . . 5  |-  ( (
ph  /\  x  e.  I )  ->  (
( X  .x.  ( Y `  x )
)  =/=  .0.  <->  ( Y `  x )  =/=  .0.  ) )
14 ovex 6069 . . . . . 6  |-  ( X 
.x.  ( Y `  x ) )  e. 
_V
15 eldifsn 3891 . . . . . 6  |-  ( ( X  .x.  ( Y `
 x ) )  e.  ( _V  \  {  .0.  } )  <->  ( ( X  .x.  ( Y `  x ) )  e. 
_V  /\  ( X  .x.  ( Y `  x
) )  =/=  .0.  ) )
1614, 15mpbiran 885 . . . . 5  |-  ( ( X  .x.  ( Y `
 x ) )  e.  ( _V  \  {  .0.  } )  <->  ( X  .x.  ( Y `  x
) )  =/=  .0.  )
17 fvex 5705 . . . . . 6  |-  ( Y `
 x )  e. 
_V
18 eldifsn 3891 . . . . . 6  |-  ( ( Y `  x )  e.  ( _V  \  {  .0.  } )  <->  ( ( Y `  x )  e.  _V  /\  ( Y `
 x )  =/= 
.0.  ) )
1917, 18mpbiran 885 . . . . 5  |-  ( ( Y `  x )  e.  ( _V  \  {  .0.  } )  <->  ( Y `  x )  =/=  .0.  )
2013, 16, 193bitr4g 280 . . . 4  |-  ( (
ph  /\  x  e.  I )  ->  (
( X  .x.  ( Y `  x )
)  e.  ( _V 
\  {  .0.  }
)  <->  ( Y `  x )  e.  ( _V  \  {  .0.  } ) ) )
2120rabbidva 2911 . . 3  |-  ( ph  ->  { x  e.  I  |  ( X  .x.  ( Y `  x ) )  e.  ( _V 
\  {  .0.  }
) }  =  {
x  e.  I  |  ( Y `  x
)  e.  ( _V 
\  {  .0.  }
) } )
22 eqid 2408 . . . 4  |-  ( x  e.  I  |->  ( X 
.x.  ( Y `  x ) ) )  =  ( x  e.  I  |->  ( X  .x.  ( Y `  x ) ) )
2322mptpreima 5326 . . 3  |-  ( `' ( x  e.  I  |->  ( X  .x.  ( Y `  x )
) ) " ( _V  \  {  .0.  }
) )  =  {
x  e.  I  |  ( X  .x.  ( Y `  x )
)  e.  ( _V 
\  {  .0.  }
) }
24 eqid 2408 . . . 4  |-  ( x  e.  I  |->  ( Y `
 x ) )  =  ( x  e.  I  |->  ( Y `  x ) )
2524mptpreima 5326 . . 3  |-  ( `' ( x  e.  I  |->  ( Y `  x
) ) " ( _V  \  {  .0.  }
) )  =  {
x  e.  I  |  ( Y `  x
)  e.  ( _V 
\  {  .0.  }
) }
2621, 23, 253eqtr4g 2465 . 2  |-  ( ph  ->  ( `' ( x  e.  I  |->  ( X 
.x.  ( Y `  x ) ) )
" ( _V  \  {  .0.  } ) )  =  ( `' ( x  e.  I  |->  ( Y `  x ) ) " ( _V 
\  {  .0.  }
) ) )
27 rrgsupp.i . . . . 5  |-  ( ph  ->  I  e.  V )
2817a1i 11 . . . . 5  |-  ( (
ph  /\  x  e.  I )  ->  ( Y `  x )  e.  _V )
29 fconstmpt 4884 . . . . . 6  |-  ( I  X.  { X }
)  =  ( x  e.  I  |->  X )
3029a1i 11 . . . . 5  |-  ( ph  ->  ( I  X.  { X } )  =  ( x  e.  I  |->  X ) )
315feqmptd 5742 . . . . 5  |-  ( ph  ->  Y  =  ( x  e.  I  |->  ( Y `
 x ) ) )
3227, 4, 28, 30, 31offval2 6285 . . . 4  |-  ( ph  ->  ( ( I  X.  { X } )  o F  .x.  Y )  =  ( x  e.  I  |->  ( X  .x.  ( Y `  x ) ) ) )
3332cnveqd 5011 . . 3  |-  ( ph  ->  `' ( ( I  X.  { X }
)  o F  .x.  Y )  =  `' ( x  e.  I  |->  ( X  .x.  ( Y `  x )
) ) )
3433imaeq1d 5165 . 2  |-  ( ph  ->  ( `' ( ( I  X.  { X } )  o F 
.x.  Y ) "
( _V  \  {  .0.  } ) )  =  ( `' ( x  e.  I  |->  ( X 
.x.  ( Y `  x ) ) )
" ( _V  \  {  .0.  } ) ) )
3531cnveqd 5011 . . 3  |-  ( ph  ->  `' Y  =  `' ( x  e.  I  |->  ( Y `  x
) ) )
3635imaeq1d 5165 . 2  |-  ( ph  ->  ( `' Y "
( _V  \  {  .0.  } ) )  =  ( `' ( x  e.  I  |->  ( Y `
 x ) )
" ( _V  \  {  .0.  } ) ) )
3726, 34, 363eqtr4d 2450 1  |-  ( ph  ->  ( `' ( ( I  X.  { X } )  o F 
.x.  Y ) "
( _V  \  {  .0.  } ) )  =  ( `' Y "
( _V  \  {  .0.  } ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721    =/= wne 2571   {crab 2674   _Vcvv 2920    \ cdif 3281   {csn 3778    e. cmpt 4230    X. cxp 4839   `'ccnv 4840   "cima 4844   -->wf 5413   ` cfv 5417  (class class class)co 6044    o Fcof 6266   Basecbs 13428   .rcmulr 13489   0gc0g 13682   Ringcrg 15619  RLRegcrlreg 16298
This theorem is referenced by:  mdegvsca  19956  deg1mul3  19995
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664  ax-cnex 9006  ax-resscn 9007  ax-1cn 9008  ax-icn 9009  ax-addcl 9010  ax-addrcl 9011  ax-mulcl 9012  ax-mulrcl 9013  ax-mulcom 9014  ax-addass 9015  ax-mulass 9016  ax-distr 9017  ax-i2m1 9018  ax-1ne0 9019  ax-1rid 9020  ax-rnegex 9021  ax-rrecex 9022  ax-cnre 9023  ax-pre-lttri 9024  ax-pre-lttrn 9025  ax-pre-ltadd 9026  ax-pre-mulgt0 9027
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-nel 2574  df-ral 2675  df-rex 2676  df-reu 2677  df-rmo 2678  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-pss 3300  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-tp 3786  df-op 3787  df-uni 3980  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-tr 4267  df-eprel 4458  df-id 4462  df-po 4467  df-so 4468  df-fr 4505  df-we 4507  df-ord 4548  df-on 4549  df-lim 4550  df-suc 4551  df-om 4809  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-of 6268  df-riota 6512  df-recs 6596  df-rdg 6631  df-er 6868  df-en 7073  df-dom 7074  df-sdom 7075  df-pnf 9082  df-mnf 9083  df-xr 9084  df-ltxr 9085  df-le 9086  df-sub 9253  df-neg 9254  df-nn 9961  df-2 10018  df-ndx 13431  df-slot 13432  df-base 13433  df-sets 13434  df-plusg 13501  df-0g 13686  df-mnd 14649  df-grp 14771  df-mgp 15608  df-rng 15622  df-rlreg 16302
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