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Theorem mptcfsupp 39768
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 5581 . . 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 39767 . 2  |-  ( ( M  e.  LMod  /\  V  e.  ~P B  /\  X  e.  V )  ->  ( F supp  .0.  )  e.  Fin )
9 mptexg 6094 . . . . 5  |-  ( V  e.  ~P B  -> 
( x  e.  V  |->  if ( x  =  X ,  .1.  ,  .0.  ) )  e.  _V )
101, 9syl5eqel 2510 . . . 4  |-  ( V  e.  ~P B  ->  F  e.  _V )
11103ad2ant2 1027 . . 3  |-  ( ( M  e.  LMod  /\  V  e.  ~P B  /\  X  e.  V )  ->  F  e.  _V )
12 fvex 5835 . . . 4  |-  ( 0g
`  R )  e. 
_V
136, 12eqeltri 2502 . . 3  |-  .0.  e.  _V
14 isfsupp 7840 . . 3  |-  ( ( F  e.  _V  /\  .0.  e.  _V )  -> 
( F finSupp  .0.  <->  ( Fun  F  /\  ( F supp  .0.  )  e.  Fin )
) )
1511, 13, 14sylancl 666 . 2  |-  ( ( M  e.  LMod  /\  V  e.  ~P B  /\  X  e.  V )  ->  ( F finSupp  .0.  <->  ( Fun  F  /\  ( F supp  .0.  )  e.  Fin ) ) )
163, 8, 15mpbir2and 930 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 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1872   _Vcvv 3022   ifcif 3854   ~Pcpw 3924   class class class wbr 4366    |-> cmpt 4425   Fun wfun 5538   ` cfv 5544  (class class class)co 6249   supp csupp 6869   Fincfn 7524   finSupp cfsupp 7836   Basecbs 15064  Scalarcsca 15136   0gc0g 15281   1rcur 17678   LModclmod 18034
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-rep 4479  ax-sep 4489  ax-nul 4498  ax-pow 4545  ax-pr 4603  ax-un 6541
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-ral 2719  df-rex 2720  df-reu 2721  df-rab 2723  df-v 3024  df-sbc 3243  df-csb 3339  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-pss 3395  df-nul 3705  df-if 3855  df-pw 3926  df-sn 3942  df-pr 3944  df-tp 3946  df-op 3948  df-uni 4163  df-iun 4244  df-br 4367  df-opab 4426  df-mpt 4427  df-tr 4462  df-eprel 4707  df-id 4711  df-po 4717  df-so 4718  df-fr 4755  df-we 4757  df-xp 4802  df-rel 4803  df-cnv 4804  df-co 4805  df-dm 4806  df-rn 4807  df-res 4808  df-ima 4809  df-ord 5388  df-on 5389  df-lim 5390  df-suc 5391  df-iota 5508  df-fun 5546  df-fn 5547  df-f 5548  df-f1 5549  df-fo 5550  df-f1o 5551  df-fv 5552  df-ov 6252  df-oprab 6253  df-mpt2 6254  df-om 6651  df-supp 6870  df-1o 7137  df-er 7318  df-en 7525  df-fin 7528  df-fsupp 7837
This theorem is referenced by:  lcoss  39832  el0ldep  39862
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