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Theorem suppmptcfin 30936
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 2456 . . . . . 6  |-  ( x  =  v  ->  (
x  =  X  <->  v  =  X ) )
32ifbid 3914 . . . . 5  |-  ( x  =  v  ->  if ( x  =  X ,  .1.  ,  .0.  )  =  if ( v  =  X ,  .1.  ,  .0.  ) )
43cbvmptv 4486 . . . 4  |-  ( x  e.  V  |->  if ( x  =  X ,  .1.  ,  .0.  ) )  =  ( v  e.  V  |->  if ( v  =  X ,  .1.  ,  .0.  ) )
51, 4eqtri 2481 . . 3  |-  F  =  ( v  e.  V  |->  if ( v  =  X ,  .1.  ,  .0.  ) )
6 simp2 989 . . 3  |-  ( ( M  e.  LMod  /\  V  e.  ~P B  /\  X  e.  V )  ->  V  e.  ~P B )
7 suppmptcfin.0 . . . . 5  |-  .0.  =  ( 0g `  R )
8 fvex 5804 . . . . 5  |-  ( 0g
`  R )  e. 
_V
97, 8eqeltri 2536 . . . 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 5804 . . . . . 6  |-  ( 1r
`  R )  e. 
_V
1311, 12eqeltri 2536 . . . . 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 6817 . 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 7495 . . 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 3902 . . . . . . . . . 10  |-  ( -.  v  =  X  ->  if ( v  =  X ,  .1.  ,  .0.  )  =  .0.  )
2221adantr 465 . . . . . . . . 9  |-  ( ( -.  v  =  X  /\  ( ( M  e.  LMod  /\  V  e. 
~P B  /\  X  e.  V )  /\  v  e.  V ) )  ->  if ( v  =  X ,  .1.  ,  .0.  )  =  .0.  )
2322neeq1d 2726 . . . . . . . 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 2452 . . . . . . . . 9  |-  .0.  =  .0.
25 eqneqall 2658 . . . . . . . . 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 434 . . . . . 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 2827 . . . 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 30861 . . . 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 7639 . . 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 663 . 2  |-  ( ( M  e.  LMod  /\  V  e.  ~P B  /\  X  e.  V )  ->  { v  e.  V  |  if ( v  =  X ,  .1.  ,  .0.  )  =/=  .0.  }  e.  Fin )
3517, 34eqeltrd 2540 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 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2645   A.wral 2796   {crab 2800   _Vcvv 3072    C_ wss 3431   ifcif 3894   ~Pcpw 3963   {csn 3980    |-> cmpt 4453   ` cfv 5521  (class class class)co 6195   supp csupp 6795   Fincfn 7415   Basecbs 14287  Scalarcsca 14355   0gc0g 14492   1rcur 16720   LModclmod 17066
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 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477
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 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-reu 2803  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-pss 3447  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4195  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4489  df-eprel 4735  df-id 4739  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-ord 4825  df-on 4826  df-lim 4827  df-suc 4828  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-om 6582  df-supp 6796  df-1o 7025  df-er 7206  df-en 7416  df-fin 7419
This theorem is referenced by:  mptcfsupp  30937
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