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Theorem mptcfsupp 33246
Description: A mapping with value 0 except of one is finitely supported. (Contributed by AV, 9-Jun-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
mptcfsupp  |-  ( ( M  e.  LMod  /\  V  e.  ~P B  /\  X  e.  V )  ->  F finSupp  .0.  )
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 mptcfsupp
StepHypRef Expression
1 suppmptcfin.f . . . 4  |-  F  =  ( x  e.  V  |->  if ( x  =  X ,  .1.  ,  .0.  ) )
21funmpt2 5607 . . 3  |-  Fun  F
32a1i 11 . 2  |-  ( ( M  e.  LMod  /\  V  e.  ~P B  /\  X  e.  V )  ->  Fun  F )
4 suppmptcfin.b . . 3  |-  B  =  ( Base `  M
)
5 suppmptcfin.r . . 3  |-  R  =  (Scalar `  M )
6 suppmptcfin.0 . . 3  |-  .0.  =  ( 0g `  R )
7 suppmptcfin.1 . . 3  |-  .1.  =  ( 1r `  R )
84, 5, 6, 7, 1suppmptcfin 33245 . 2  |-  ( ( M  e.  LMod  /\  V  e.  ~P B  /\  X  e.  V )  ->  ( F supp  .0.  )  e.  Fin )
9 mptexg 6117 . . . . 5  |-  ( V  e.  ~P B  -> 
( x  e.  V  |->  if ( x  =  X ,  .1.  ,  .0.  ) )  e.  _V )
101, 9syl5eqel 2546 . . . 4  |-  ( V  e.  ~P B  ->  F  e.  _V )
11103ad2ant2 1016 . . 3  |-  ( ( M  e.  LMod  /\  V  e.  ~P B  /\  X  e.  V )  ->  F  e.  _V )
12 fvex 5858 . . . 4  |-  ( 0g
`  R )  e. 
_V
136, 12eqeltri 2538 . . 3  |-  .0.  e.  _V
14 isfsupp 7825 . . 3  |-  ( ( F  e.  _V  /\  .0.  e.  _V )  -> 
( F finSupp  .0.  <->  ( Fun  F  /\  ( F supp  .0.  )  e.  Fin )
) )
1511, 13, 14sylancl 660 . 2  |-  ( ( M  e.  LMod  /\  V  e.  ~P B  /\  X  e.  V )  ->  ( F finSupp  .0.  <->  ( Fun  F  /\  ( F supp  .0.  )  e.  Fin ) ) )
163, 8, 15mpbir2and 920 1  |-  ( ( M  e.  LMod  /\  V  e.  ~P B  /\  X  e.  V )  ->  F finSupp  .0.  )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823   _Vcvv 3106   ifcif 3929   ~Pcpw 3999   class class class wbr 4439    |-> cmpt 4497   Fun wfun 5564   ` cfv 5570  (class class class)co 6270   supp csupp 6891   Fincfn 7509   finSupp cfsupp 7821   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  df-fsupp 7822
This theorem is referenced by:  lcoss  33310  el0ldep  33340
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