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Theorem suppmptcfin 30709
Description: The support of a mapping with value 0 except of one is finite. (Contributed by AV, 27-Apr-2019.)
Hypotheses
Ref Expression
suppmptcfin.b  |-  B  =  ( Base `  M
)
suppmptcfin.r  |-  R  =  (Scalar `  M )
suppmptcfin.0  |-  .0.  =  ( 0g `  R )
suppmptcfin.1  |-  .1.  =  ( 1r `  R )
suppmptcfin.f  |-  F  =  ( x  e.  V  |->  if ( x  =  X ,  .1.  ,  .0.  ) )
Assertion
Ref Expression
suppmptcfin  |-  ( ( M  e.  LMod  /\  V  e.  ~P B  /\  X  e.  V )  ->  ( F supp  .0.  )  e.  Fin )
Distinct variable groups:    x, B    x, F    x, M    x, V    x, X    x,  .1.    x,  .0.
Allowed substitution hint:    R( x)

Proof of Theorem suppmptcfin
Dummy variable  v is distinct from all other variables.
StepHypRef Expression
1 suppmptcfin.f . . . 4  |-  F  =  ( x  e.  V  |->  if ( x  =  X ,  .1.  ,  .0.  ) )
2 eqeq1 2447 . . . . . 6  |-  ( x  =  v  ->  (
x  =  X  <->  v  =  X ) )
32ifbid 3808 . . . . 5  |-  ( x  =  v  ->  if ( x  =  X ,  .1.  ,  .0.  )  =  if ( v  =  X ,  .1.  ,  .0.  ) )
43cbvmptv 4380 . . . 4  |-  ( x  e.  V  |->  if ( x  =  X ,  .1.  ,  .0.  ) )  =  ( v  e.  V  |->  if ( v  =  X ,  .1.  ,  .0.  ) )
51, 4eqtri 2461 . . 3  |-  F  =  ( v  e.  V  |->  if ( v  =  X ,  .1.  ,  .0.  ) )
6 simp2 984 . . 3  |-  ( ( M  e.  LMod  /\  V  e.  ~P B  /\  X  e.  V )  ->  V  e.  ~P B )
7 suppmptcfin.0 . . . . 5  |-  .0.  =  ( 0g `  R )
8 fvex 5698 . . . . 5  |-  ( 0g
`  R )  e. 
_V
97, 8eqeltri 2511 . . . 4  |-  .0.  e.  _V
109a1i 11 . . 3  |-  ( ( M  e.  LMod  /\  V  e.  ~P B  /\  X  e.  V )  ->  .0.  e.  _V )
11 suppmptcfin.1 . . . . . 6  |-  .1.  =  ( 1r `  R )
12 fvex 5698 . . . . . 6  |-  ( 1r
`  R )  e. 
_V
1311, 12eqeltri 2511 . . . . 5  |-  .1.  e.  _V
1413a1i 11 . . . 4  |-  ( ( ( M  e.  LMod  /\  V  e.  ~P B  /\  X  e.  V
)  /\  v  e.  V )  ->  .1.  e.  _V )
159a1i 11 . . . 4  |-  ( ( ( M  e.  LMod  /\  V  e.  ~P B  /\  X  e.  V
)  /\  v  e.  V )  ->  .0.  e.  _V )
1614, 15ifcld 3829 . . 3  |-  ( ( ( M  e.  LMod  /\  V  e.  ~P B  /\  X  e.  V
)  /\  v  e.  V )  ->  if ( v  =  X ,  .1.  ,  .0.  )  e.  _V )
175, 6, 10, 16mptsuppd 6711 . 2  |-  ( ( M  e.  LMod  /\  V  e.  ~P B  /\  X  e.  V )  ->  ( F supp  .0.  )  =  {
v  e.  V  |  if ( v  =  X ,  .1.  ,  .0.  )  =/=  .0.  } )
18 snfi 7386 . . 3  |-  { X }  e.  Fin
19 ax-1 6 . . . . . . 7  |-  ( v  =  X  ->  ( if ( v  =  X ,  .1.  ,  .0.  )  =/=  .0.  ->  v  =  X ) )
2019a1d 25 . . . . . 6  |-  ( v  =  X  ->  (
( ( M  e. 
LMod  /\  V  e.  ~P B  /\  X  e.  V
)  /\  v  e.  V )  ->  ( if ( v  =  X ,  .1.  ,  .0.  )  =/=  .0.  ->  v  =  X ) ) )
21 iffalse 3796 . . . . . . . . . 10  |-  ( -.  v  =  X  ->  if ( v  =  X ,  .1.  ,  .0.  )  =  .0.  )
2221adantr 462 . . . . . . . . 9  |-  ( ( -.  v  =  X  /\  ( ( M  e.  LMod  /\  V  e. 
~P B  /\  X  e.  V )  /\  v  e.  V ) )  ->  if ( v  =  X ,  .1.  ,  .0.  )  =  .0.  )
2322neeq1d 2619 . . . . . . . 8  |-  ( ( -.  v  =  X  /\  ( ( M  e.  LMod  /\  V  e. 
~P B  /\  X  e.  V )  /\  v  e.  V ) )  -> 
( if ( v  =  X ,  .1.  ,  .0.  )  =/=  .0.  <->  .0.  =/=  .0.  ) )
24 eqid 2441 . . . . . . . . 9  |-  .0.  =  .0.
25 eqneqall 2703 . . . . . . . . 9  |-  (  .0.  =  .0.  ->  (  .0.  =/=  .0.  ->  v  =  X ) )
2624, 25ax-mp 5 . . . . . . . 8  |-  (  .0. 
=/=  .0.  ->  v  =  X )
2723, 26syl6bi 228 . . . . . . 7  |-  ( ( -.  v  =  X  /\  ( ( M  e.  LMod  /\  V  e. 
~P B  /\  X  e.  V )  /\  v  e.  V ) )  -> 
( if ( v  =  X ,  .1.  ,  .0.  )  =/=  .0.  ->  v  =  X ) )
2827ex 434 . . . . . 6  |-  ( -.  v  =  X  -> 
( ( ( M  e.  LMod  /\  V  e. 
~P B  /\  X  e.  V )  /\  v  e.  V )  ->  ( if ( v  =  X ,  .1.  ,  .0.  )  =/=  .0.  ->  v  =  X ) ) )
2920, 28pm2.61i 164 . . . . 5  |-  ( ( ( M  e.  LMod  /\  V  e.  ~P B  /\  X  e.  V
)  /\  v  e.  V )  ->  ( if ( v  =  X ,  .1.  ,  .0.  )  =/=  .0.  ->  v  =  X ) )
3029ralrimiva 2797 . . . 4  |-  ( ( M  e.  LMod  /\  V  e.  ~P B  /\  X  e.  V )  ->  A. v  e.  V  ( if ( v  =  X ,  .1.  ,  .0.  )  =/=  .0.  ->  v  =  X ) )
31 rabsssn 30642 . . . 4  |-  ( { v  e.  V  |  if ( v  =  X ,  .1.  ,  .0.  )  =/=  .0.  }  C_  { X }  <->  A. v  e.  V  ( if ( v  =  X ,  .1.  ,  .0.  )  =/=  .0.  ->  v  =  X ) )
3230, 31sylibr 212 . . 3  |-  ( ( M  e.  LMod  /\  V  e.  ~P B  /\  X  e.  V )  ->  { v  e.  V  |  if ( v  =  X ,  .1.  ,  .0.  )  =/=  .0.  }  C_  { X } )
33 ssfi 7529 . . 3  |-  ( ( { X }  e.  Fin  /\  { v  e.  V  |  if ( v  =  X ,  .1.  ,  .0.  )  =/= 
.0.  }  C_  { X } )  ->  { v  e.  V  |  if ( v  =  X ,  .1.  ,  .0.  )  =/=  .0.  }  e.  Fin )
3418, 32, 33sylancr 658 . 2  |-  ( ( M  e.  LMod  /\  V  e.  ~P B  /\  X  e.  V )  ->  { v  e.  V  |  if ( v  =  X ,  .1.  ,  .0.  )  =/=  .0.  }  e.  Fin )
3517, 34eqeltrd 2515 1  |-  ( ( M  e.  LMod  /\  V  e.  ~P B  /\  X  e.  V )  ->  ( F supp  .0.  )  e.  Fin )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 960    = wceq 1364    e. wcel 1761    =/= wne 2604   A.wral 2713   {crab 2717   _Vcvv 2970    C_ wss 3325   ifcif 3788   ~Pcpw 3857   {csn 3874    e. cmpt 4347   ` cfv 5415  (class class class)co 6090   supp csupp 6689   Fincfn 7306   Basecbs 14170  Scalarcsca 14237   0gc0g 14374   1rcur 16593   LModclmod 16928
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-supp 6690  df-1o 6916  df-er 7097  df-en 7307  df-fin 7310
This theorem is referenced by:  mptcfsupp  30710
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