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Theorem suppmptcfin 33245
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 2458 . . . . . 6  |-  ( x  =  v  ->  (
x  =  X  <->  v  =  X ) )
32ifbid 3951 . . . . 5  |-  ( x  =  v  ->  if ( x  =  X ,  .1.  ,  .0.  )  =  if ( v  =  X ,  .1.  ,  .0.  ) )
43cbvmptv 4530 . . . 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 995 . . 3  |-  ( ( M  e.  LMod  /\  V  e.  ~P B  /\  X  e.  V )  ->  V  e.  ~P B )
7 suppmptcfin.0 . . . . 5  |-  .0.  =  ( 0g `  R )
8 fvex 5858 . . . . 5  |-  ( 0g
`  R )  e. 
_V
97, 8eqeltri 2538 . . . 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 5858 . . . . . 6  |-  ( 1r
`  R )  e. 
_V
1311, 12eqeltri 2538 . . . . 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 3972 . . 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 6915 . 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 7589 . . 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 3938 . . . . . . . . . 10  |-  ( -.  v  =  X  ->  if ( v  =  X ,  .1.  ,  .0.  )  =  .0.  )
2221adantr 463 . . . . . . . . 9  |-  ( ( -.  v  =  X  /\  ( ( M  e.  LMod  /\  V  e. 
~P B  /\  X  e.  V )  /\  v  e.  V ) )  ->  if ( v  =  X ,  .1.  ,  .0.  )  =  .0.  )
2322neeq1d 2731 . . . . . . . 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 2454 . . . . . . . . 9  |-  .0.  =  .0.
25 eqneqall 2661 . . . . . . . . 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 432 . . . . . 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 2868 . . . 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 33193 . . . 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 7733 . . 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 661 . 2  |-  ( ( M  e.  LMod  /\  V  e.  ~P B  /\  X  e.  V )  ->  { v  e.  V  |  if ( v  =  X ,  .1.  ,  .0.  )  =/=  .0.  }  e.  Fin )
3517, 34eqeltrd 2542 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 367    /\ w3a 971    = wceq 1398    e. wcel 1823    =/= wne 2649   A.wral 2804   {crab 2808   _Vcvv 3106    C_ wss 3461   ifcif 3929   ~Pcpw 3999   {csn 4016    |-> cmpt 4497   ` cfv 5570  (class class class)co 6270   supp csupp 6891   Fincfn 7509   Basecbs 14719  Scalarcsca 14790   0gc0g 14932   1rcur 17351   LModclmod 17710
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-supp 6892  df-1o 7122  df-er 7303  df-en 7510  df-fin 7513
This theorem is referenced by:  mptcfsupp  33246
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