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Theorem rrgsupp 16032
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 451 . . . . . . 7  |-  ( (
ph  /\  x  e.  I )  ->  R  e.  Ring )
3 rrgsupp.x . . . . . . . 8  |-  ( ph  ->  X  e.  E )
43adantr 451 . . . . . . 7  |-  ( (
ph  /\  x  e.  I )  ->  X  e.  E )
5 rrgsupp.y . . . . . . . 8  |-  ( ph  ->  Y : I --> B )
6 ffvelrn 5663 . . . . . . . 8  |-  ( ( Y : I --> B  /\  x  e.  I )  ->  ( Y `  x
)  e.  B )
75, 6sylan 457 . . . . . . 7  |-  ( (
ph  /\  x  e.  I )  ->  ( Y `  x )  e.  B )
8 rrgval.e . . . . . . . 8  |-  E  =  (RLReg `  R )
9 rrgval.b . . . . . . . 8  |-  B  =  ( Base `  R
)
10 rrgval.t . . . . . . . 8  |-  .x.  =  ( .r `  R )
11 rrgval.z . . . . . . . 8  |-  .0.  =  ( 0g `  R )
128, 9, 10, 11rrgeq0 16031 . . . . . . 7  |-  ( ( R  e.  Ring  /\  X  e.  E  /\  ( Y `  x )  e.  B )  ->  (
( X  .x.  ( Y `  x )
)  =  .0.  <->  ( Y `  x )  =  .0.  ) )
132, 4, 7, 12syl3anc 1182 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  (
( X  .x.  ( Y `  x )
)  =  .0.  <->  ( Y `  x )  =  .0.  ) )
1413necon3bid 2481 . . . . 5  |-  ( (
ph  /\  x  e.  I )  ->  (
( X  .x.  ( Y `  x )
)  =/=  .0.  <->  ( Y `  x )  =/=  .0.  ) )
15 ovex 5883 . . . . . 6  |-  ( X 
.x.  ( Y `  x ) )  e. 
_V
16 eldifsn 3749 . . . . . 6  |-  ( ( X  .x.  ( Y `
 x ) )  e.  ( _V  \  {  .0.  } )  <->  ( ( X  .x.  ( Y `  x ) )  e. 
_V  /\  ( X  .x.  ( Y `  x
) )  =/=  .0.  ) )
1715, 16mpbiran 884 . . . . 5  |-  ( ( X  .x.  ( Y `
 x ) )  e.  ( _V  \  {  .0.  } )  <->  ( X  .x.  ( Y `  x
) )  =/=  .0.  )
18 fvex 5539 . . . . . 6  |-  ( Y `
 x )  e. 
_V
19 eldifsn 3749 . . . . . 6  |-  ( ( Y `  x )  e.  ( _V  \  {  .0.  } )  <->  ( ( Y `  x )  e.  _V  /\  ( Y `
 x )  =/= 
.0.  ) )
2018, 19mpbiran 884 . . . . 5  |-  ( ( Y `  x )  e.  ( _V  \  {  .0.  } )  <->  ( Y `  x )  =/=  .0.  )
2114, 17, 203bitr4g 279 . . . 4  |-  ( (
ph  /\  x  e.  I )  ->  (
( X  .x.  ( Y `  x )
)  e.  ( _V 
\  {  .0.  }
)  <->  ( Y `  x )  e.  ( _V  \  {  .0.  } ) ) )
2221rabbidva 2779 . . 3  |-  ( ph  ->  { x  e.  I  |  ( X  .x.  ( Y `  x ) )  e.  ( _V 
\  {  .0.  }
) }  =  {
x  e.  I  |  ( Y `  x
)  e.  ( _V 
\  {  .0.  }
) } )
23 eqid 2283 . . . 4  |-  ( x  e.  I  |->  ( X 
.x.  ( Y `  x ) ) )  =  ( x  e.  I  |->  ( X  .x.  ( Y `  x ) ) )
2423mptpreima 5166 . . 3  |-  ( `' ( x  e.  I  |->  ( X  .x.  ( Y `  x )
) ) " ( _V  \  {  .0.  }
) )  =  {
x  e.  I  |  ( X  .x.  ( Y `  x )
)  e.  ( _V 
\  {  .0.  }
) }
25 eqid 2283 . . . 4  |-  ( x  e.  I  |->  ( Y `
 x ) )  =  ( x  e.  I  |->  ( Y `  x ) )
2625mptpreima 5166 . . 3  |-  ( `' ( x  e.  I  |->  ( Y `  x
) ) " ( _V  \  {  .0.  }
) )  =  {
x  e.  I  |  ( Y `  x
)  e.  ( _V 
\  {  .0.  }
) }
2722, 24, 263eqtr4g 2340 . 2  |-  ( ph  ->  ( `' ( x  e.  I  |->  ( X 
.x.  ( Y `  x ) ) )
" ( _V  \  {  .0.  } ) )  =  ( `' ( x  e.  I  |->  ( Y `  x ) ) " ( _V 
\  {  .0.  }
) ) )
28 rrgsupp.i . . . . 5  |-  ( ph  ->  I  e.  V )
2918a1i 10 . . . . 5  |-  ( (
ph  /\  x  e.  I )  ->  ( Y `  x )  e.  _V )
30 fconstmpt 4732 . . . . . 6  |-  ( I  X.  { X }
)  =  ( x  e.  I  |->  X )
3130a1i 10 . . . . 5  |-  ( ph  ->  ( I  X.  { X } )  =  ( x  e.  I  |->  X ) )
325feqmptd 5575 . . . . 5  |-  ( ph  ->  Y  =  ( x  e.  I  |->  ( Y `
 x ) ) )
3328, 4, 29, 31, 32offval2 6095 . . . 4  |-  ( ph  ->  ( ( I  X.  { X } )  o F  .x.  Y )  =  ( x  e.  I  |->  ( X  .x.  ( Y `  x ) ) ) )
3433cnveqd 4857 . . 3  |-  ( ph  ->  `' ( ( I  X.  { X }
)  o F  .x.  Y )  =  `' ( x  e.  I  |->  ( X  .x.  ( Y `  x )
) ) )
3534imaeq1d 5011 . 2  |-  ( ph  ->  ( `' ( ( I  X.  { X } )  o F 
.x.  Y ) "
( _V  \  {  .0.  } ) )  =  ( `' ( x  e.  I  |->  ( X 
.x.  ( Y `  x ) ) )
" ( _V  \  {  .0.  } ) ) )
3632cnveqd 4857 . . 3  |-  ( ph  ->  `' Y  =  `' ( x  e.  I  |->  ( Y `  x
) ) )
3736imaeq1d 5011 . 2  |-  ( ph  ->  ( `' Y "
( _V  \  {  .0.  } ) )  =  ( `' ( x  e.  I  |->  ( Y `
 x ) )
" ( _V  \  {  .0.  } ) ) )
3827, 35, 373eqtr4d 2325 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 176    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   {crab 2547   _Vcvv 2788    \ cdif 3149   {csn 3640    e. cmpt 4077    X. cxp 4687   `'ccnv 4688   "cima 4692   -->wf 5251   ` cfv 5255  (class class class)co 5858    o Fcof 6076   Basecbs 13148   .rcmulr 13209   0gc0g 13400   Ringcrg 15337  RLRegcrlreg 16020
This theorem is referenced by:  mdegvsca  19462  deg1mul3  19501
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-plusg 13221  df-0g 13404  df-mnd 14367  df-grp 14489  df-mgp 15326  df-rng 15340  df-rlreg 16024
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