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Theorem mptcfsupp 30918
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 5539 . . 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 30917 . 2  |-  ( ( M  e.  LMod  /\  V  e.  ~P B  /\  X  e.  V )  ->  ( F supp  .0.  )  e.  Fin )
9 mptexg 6032 . . . . 5  |-  ( V  e.  ~P B  -> 
( x  e.  V  |->  if ( x  =  X ,  .1.  ,  .0.  ) )  e.  _V )
101, 9syl5eqel 2540 . . . 4  |-  ( V  e.  ~P B  ->  F  e.  _V )
11103ad2ant2 1010 . . 3  |-  ( ( M  e.  LMod  /\  V  e.  ~P B  /\  X  e.  V )  ->  F  e.  _V )
12 fvex 5785 . . . 4  |-  ( 0g
`  R )  e. 
_V
136, 12eqeltri 2532 . . 3  |-  .0.  e.  _V
14 isfsupp 7711 . . 3  |-  ( ( F  e.  _V  /\  .0.  e.  _V )  -> 
( F finSupp  .0.  <->  ( Fun  F  /\  ( F supp  .0.  )  e.  Fin )
) )
1511, 13, 14sylancl 662 . 2  |-  ( ( M  e.  LMod  /\  V  e.  ~P B  /\  X  e.  V )  ->  ( F finSupp  .0.  <->  ( Fun  F  /\  ( F supp  .0.  )  e.  Fin ) ) )
163, 8, 15mpbir2and 913 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 369    /\ w3a 965    = wceq 1370    e. wcel 1757   _Vcvv 3054   ifcif 3875   ~Pcpw 3944   class class class wbr 4376    |-> cmpt 4434   Fun wfun 5496   ` cfv 5502  (class class class)co 6176   supp csupp 6776   Fincfn 7396   finSupp cfsupp 7707   Basecbs 14262  Scalarcsca 14329   0gc0g 14466   1rcur 16694   LModclmod 17040
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1709  ax-7 1729  ax-8 1759  ax-9 1761  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429  ax-rep 4487  ax-sep 4497  ax-nul 4505  ax-pow 4554  ax-pr 4615  ax-un 6458
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1702  df-eu 2263  df-mo 2264  df-clab 2436  df-cleq 2442  df-clel 2445  df-nfc 2598  df-ne 2643  df-ral 2797  df-rex 2798  df-reu 2799  df-rab 2801  df-v 3056  df-sbc 3271  df-csb 3373  df-dif 3415  df-un 3417  df-in 3419  df-ss 3426  df-pss 3428  df-nul 3722  df-if 3876  df-pw 3946  df-sn 3962  df-pr 3964  df-tp 3966  df-op 3968  df-uni 4176  df-iun 4257  df-br 4377  df-opab 4435  df-mpt 4436  df-tr 4470  df-eprel 4716  df-id 4720  df-po 4725  df-so 4726  df-fr 4763  df-we 4765  df-ord 4806  df-on 4807  df-lim 4808  df-suc 4809  df-xp 4930  df-rel 4931  df-cnv 4932  df-co 4933  df-dm 4934  df-rn 4935  df-res 4936  df-ima 4937  df-iota 5465  df-fun 5504  df-fn 5505  df-f 5506  df-f1 5507  df-fo 5508  df-f1o 5509  df-fv 5510  df-ov 6179  df-oprab 6180  df-mpt2 6181  df-om 6563  df-supp 6777  df-1o 7006  df-er 7187  df-en 7397  df-fin 7400  df-fsupp 7708
This theorem is referenced by:  lcoss  31063  el0ldep  31093
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