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Theorem suppmptcfin 40436
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 2465 . . . . . 6  |-  ( x  =  v  ->  (
x  =  X  <->  v  =  X ) )
32ifbid 3914 . . . . 5  |-  ( x  =  v  ->  if ( x  =  X ,  .1.  ,  .0.  )  =  if ( v  =  X ,  .1.  ,  .0.  ) )
43cbvmptv 4508 . . . 4  |-  ( x  e.  V  |->  if ( x  =  X ,  .1.  ,  .0.  ) )  =  ( v  e.  V  |->  if ( v  =  X ,  .1.  ,  .0.  ) )
51, 4eqtri 2483 . . 3  |-  F  =  ( v  e.  V  |->  if ( v  =  X ,  .1.  ,  .0.  ) )
6 simp2 1015 . . 3  |-  ( ( M  e.  LMod  /\  V  e.  ~P B  /\  X  e.  V )  ->  V  e.  ~P B )
7 suppmptcfin.0 . . . . 5  |-  .0.  =  ( 0g `  R )
8 fvex 5897 . . . . 5  |-  ( 0g
`  R )  e. 
_V
97, 8eqeltri 2535 . . . 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 5897 . . . . . 6  |-  ( 1r
`  R )  e. 
_V
1311, 12eqeltri 2535 . . . . 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 3935 . . 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 6964 . 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 7675 . . 3  |-  { X }  e.  Fin
19 2a1 28 . . . . . 6  |-  ( v  =  X  ->  (
( ( M  e. 
LMod  /\  V  e.  ~P B  /\  X  e.  V
)  /\  v  e.  V )  ->  ( if ( v  =  X ,  .1.  ,  .0.  )  =/=  .0.  ->  v  =  X ) ) )
20 iffalse 3901 . . . . . . . . . 10  |-  ( -.  v  =  X  ->  if ( v  =  X ,  .1.  ,  .0.  )  =  .0.  )
2120adantr 471 . . . . . . . . 9  |-  ( ( -.  v  =  X  /\  ( ( M  e.  LMod  /\  V  e. 
~P B  /\  X  e.  V )  /\  v  e.  V ) )  ->  if ( v  =  X ,  .1.  ,  .0.  )  =  .0.  )
2221neeq1d 2694 . . . . . . . 8  |-  ( ( -.  v  =  X  /\  ( ( M  e.  LMod  /\  V  e. 
~P B  /\  X  e.  V )  /\  v  e.  V ) )  -> 
( if ( v  =  X ,  .1.  ,  .0.  )  =/=  .0.  <->  .0.  =/=  .0.  ) )
23 eqid 2461 . . . . . . . . 9  |-  .0.  =  .0.
24 eqneqall 2645 . . . . . . . . 9  |-  (  .0.  =  .0.  ->  (  .0.  =/=  .0.  ->  v  =  X ) )
2523, 24ax-mp 5 . . . . . . . 8  |-  (  .0. 
=/=  .0.  ->  v  =  X )
2622, 25syl6bi 236 . . . . . . 7  |-  ( ( -.  v  =  X  /\  ( ( M  e.  LMod  /\  V  e. 
~P B  /\  X  e.  V )  /\  v  e.  V ) )  -> 
( if ( v  =  X ,  .1.  ,  .0.  )  =/=  .0.  ->  v  =  X ) )
2726ex 440 . . . . . 6  |-  ( -.  v  =  X  -> 
( ( ( M  e.  LMod  /\  V  e. 
~P B  /\  X  e.  V )  /\  v  e.  V )  ->  ( if ( v  =  X ,  .1.  ,  .0.  )  =/=  .0.  ->  v  =  X ) ) )
2819, 27pm2.61i 169 . . . . 5  |-  ( ( ( M  e.  LMod  /\  V  e.  ~P B  /\  X  e.  V
)  /\  v  e.  V )  ->  ( if ( v  =  X ,  .1.  ,  .0.  )  =/=  .0.  ->  v  =  X ) )
2928ralrimiva 2813 . . . 4  |-  ( ( M  e.  LMod  /\  V  e.  ~P B  /\  X  e.  V )  ->  A. v  e.  V  ( if ( v  =  X ,  .1.  ,  .0.  )  =/=  .0.  ->  v  =  X ) )
30 rabsssn 4013 . . . 4  |-  ( { v  e.  V  |  if ( v  =  X ,  .1.  ,  .0.  )  =/=  .0.  }  C_  { X }  <->  A. v  e.  V  ( if ( v  =  X ,  .1.  ,  .0.  )  =/=  .0.  ->  v  =  X ) )
3129, 30sylibr 217 . . 3  |-  ( ( M  e.  LMod  /\  V  e.  ~P B  /\  X  e.  V )  ->  { v  e.  V  |  if ( v  =  X ,  .1.  ,  .0.  )  =/=  .0.  }  C_  { X } )
32 ssfi 7817 . . 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 )
3318, 31, 32sylancr 674 . 2  |-  ( ( M  e.  LMod  /\  V  e.  ~P B  /\  X  e.  V )  ->  { v  e.  V  |  if ( v  =  X ,  .1.  ,  .0.  )  =/=  .0.  }  e.  Fin )
3417, 33eqeltrd 2539 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 375    /\ w3a 991    = wceq 1454    e. wcel 1897    =/= wne 2632   A.wral 2748   {crab 2752   _Vcvv 3056    C_ wss 3415   ifcif 3892   ~Pcpw 3962   {csn 3979    |-> cmpt 4474   ` cfv 5600  (class class class)co 6314   supp csupp 6940   Fincfn 7594   Basecbs 15169  Scalarcsca 15241   0gc0g 15386   1rcur 17783   LModclmod 18139
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-8 1899  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-rep 4528  ax-sep 4538  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6609
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-ral 2753  df-rex 2754  df-reu 2755  df-rab 2757  df-v 3058  df-sbc 3279  df-csb 3375  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-pss 3431  df-nul 3743  df-if 3893  df-pw 3964  df-sn 3980  df-pr 3982  df-tp 3984  df-op 3986  df-uni 4212  df-iun 4293  df-br 4416  df-opab 4475  df-mpt 4476  df-tr 4511  df-eprel 4763  df-id 4767  df-po 4773  df-so 4774  df-fr 4811  df-we 4813  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-ord 5444  df-on 5445  df-lim 5446  df-suc 5447  df-iota 5564  df-fun 5602  df-fn 5603  df-f 5604  df-f1 5605  df-fo 5606  df-f1o 5607  df-fv 5608  df-ov 6317  df-oprab 6318  df-mpt2 6319  df-om 6719  df-supp 6941  df-1o 7207  df-er 7388  df-en 7595  df-fin 7598
This theorem is referenced by:  mptcfsupp  40437
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