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Theorem rrgsupp 16356
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 453 . . . . . . 7  |-  ( (
ph  /\  x  e.  I )  ->  R  e.  Ring )
3 rrgsupp.x . . . . . . . 8  |-  ( ph  ->  X  e.  E )
43adantr 453 . . . . . . 7  |-  ( (
ph  /\  x  e.  I )  ->  X  e.  E )
5 rrgsupp.y . . . . . . . 8  |-  ( ph  ->  Y : I --> B )
65ffvelrnda 5873 . . . . . . 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 16355 . . . . . . 7  |-  ( ( R  e.  Ring  /\  X  e.  E  /\  ( Y `  x )  e.  B )  ->  (
( X  .x.  ( Y `  x )
)  =  .0.  <->  ( Y `  x )  =  .0.  ) )
122, 4, 6, 11syl3anc 1185 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  (
( X  .x.  ( Y `  x )
)  =  .0.  <->  ( Y `  x )  =  .0.  ) )
1312necon3bid 2638 . . . . 5  |-  ( (
ph  /\  x  e.  I )  ->  (
( X  .x.  ( Y `  x )
)  =/=  .0.  <->  ( Y `  x )  =/=  .0.  ) )
14 ovex 6109 . . . . . 6  |-  ( X 
.x.  ( Y `  x ) )  e. 
_V
15 eldifsn 3929 . . . . . 6  |-  ( ( X  .x.  ( Y `
 x ) )  e.  ( _V  \  {  .0.  } )  <->  ( ( X  .x.  ( Y `  x ) )  e. 
_V  /\  ( X  .x.  ( Y `  x
) )  =/=  .0.  ) )
1614, 15mpbiran 886 . . . . 5  |-  ( ( X  .x.  ( Y `
 x ) )  e.  ( _V  \  {  .0.  } )  <->  ( X  .x.  ( Y `  x
) )  =/=  .0.  )
17 fvex 5745 . . . . . 6  |-  ( Y `
 x )  e. 
_V
18 eldifsn 3929 . . . . . 6  |-  ( ( Y `  x )  e.  ( _V  \  {  .0.  } )  <->  ( ( Y `  x )  e.  _V  /\  ( Y `
 x )  =/= 
.0.  ) )
1917, 18mpbiran 886 . . . . 5  |-  ( ( Y `  x )  e.  ( _V  \  {  .0.  } )  <->  ( Y `  x )  =/=  .0.  )
2013, 16, 193bitr4g 281 . . . 4  |-  ( (
ph  /\  x  e.  I )  ->  (
( X  .x.  ( Y `  x )
)  e.  ( _V 
\  {  .0.  }
)  <->  ( Y `  x )  e.  ( _V  \  {  .0.  } ) ) )
2120rabbidva 2949 . . 3  |-  ( ph  ->  { x  e.  I  |  ( X  .x.  ( Y `  x ) )  e.  ( _V 
\  {  .0.  }
) }  =  {
x  e.  I  |  ( Y `  x
)  e.  ( _V 
\  {  .0.  }
) } )
22 eqid 2438 . . . 4  |-  ( x  e.  I  |->  ( X 
.x.  ( Y `  x ) ) )  =  ( x  e.  I  |->  ( X  .x.  ( Y `  x ) ) )
2322mptpreima 5366 . . 3  |-  ( `' ( x  e.  I  |->  ( X  .x.  ( Y `  x )
) ) " ( _V  \  {  .0.  }
) )  =  {
x  e.  I  |  ( X  .x.  ( Y `  x )
)  e.  ( _V 
\  {  .0.  }
) }
24 eqid 2438 . . . 4  |-  ( x  e.  I  |->  ( Y `
 x ) )  =  ( x  e.  I  |->  ( Y `  x ) )
2524mptpreima 5366 . . 3  |-  ( `' ( x  e.  I  |->  ( Y `  x
) ) " ( _V  \  {  .0.  }
) )  =  {
x  e.  I  |  ( Y `  x
)  e.  ( _V 
\  {  .0.  }
) }
2621, 23, 253eqtr4g 2495 . 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 4924 . . . . . 6  |-  ( I  X.  { X }
)  =  ( x  e.  I  |->  X )
3029a1i 11 . . . . 5  |-  ( ph  ->  ( I  X.  { X } )  =  ( x  e.  I  |->  X ) )
315feqmptd 5782 . . . . 5  |-  ( ph  ->  Y  =  ( x  e.  I  |->  ( Y `
 x ) ) )
3227, 4, 28, 30, 31offval2 6325 . . . 4  |-  ( ph  ->  ( ( I  X.  { X } )  o F  .x.  Y )  =  ( x  e.  I  |->  ( X  .x.  ( Y `  x ) ) ) )
3332cnveqd 5051 . . 3  |-  ( ph  ->  `' ( ( I  X.  { X }
)  o F  .x.  Y )  =  `' ( x  e.  I  |->  ( X  .x.  ( Y `  x )
) ) )
3433imaeq1d 5205 . 2  |-  ( ph  ->  ( `' ( ( I  X.  { X } )  o F 
.x.  Y ) "
( _V  \  {  .0.  } ) )  =  ( `' ( x  e.  I  |->  ( X 
.x.  ( Y `  x ) ) )
" ( _V  \  {  .0.  } ) ) )
3531cnveqd 5051 . . 3  |-  ( ph  ->  `' Y  =  `' ( x  e.  I  |->  ( Y `  x
) ) )
3635imaeq1d 5205 . 2  |-  ( ph  ->  ( `' Y "
( _V  \  {  .0.  } ) )  =  ( `' ( x  e.  I  |->  ( Y `
 x ) )
" ( _V  \  {  .0.  } ) ) )
3726, 34, 363eqtr4d 2480 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 178    /\ wa 360    = wceq 1653    e. wcel 1726    =/= wne 2601   {crab 2711   _Vcvv 2958    \ cdif 3319   {csn 3816    e. cmpt 4269    X. cxp 4879   `'ccnv 4880   "cima 4884   -->wf 5453   ` cfv 5457  (class class class)co 6084    o Fcof 6306   Basecbs 13474   .rcmulr 13535   0gc0g 13728   Ringcrg 15665  RLRegcrlreg 16344
This theorem is referenced by:  mdegvsca  20004  deg1mul3  20043
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-of 6308  df-riota 6552  df-recs 6636  df-rdg 6671  df-er 6908  df-en 7113  df-dom 7114  df-sdom 7115  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-nn 10006  df-2 10063  df-ndx 13477  df-slot 13478  df-base 13479  df-sets 13480  df-plusg 13547  df-0g 13732  df-mnd 14695  df-grp 14817  df-mgp 15654  df-rng 15668  df-rlreg 16348
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