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Theorem lincval0 39811
Description: The value of an empty linear combination. (Contributed by AV, 12-Apr-2019.)
Assertion
Ref Expression
lincval0  |-  ( M  e.  X  ->  ( (/) ( linC  `  M ) (/) )  =  ( 0g
`  M ) )

Proof of Theorem lincval0
Dummy variable  v is distinct from all other variables.
StepHypRef Expression
1 0ex 4499 . . . . 5  |-  (/)  e.  _V
21snid 3969 . . . 4  |-  (/)  e.  { (/)
}
3 fvex 5835 . . . . . 6  |-  ( Base `  (Scalar `  M )
)  e.  _V
4 map0e 7464 . . . . . 6  |-  ( (
Base `  (Scalar `  M
) )  e.  _V  ->  ( ( Base `  (Scalar `  M ) )  ^m  (/) )  =  1o )
53, 4mp1i 13 . . . . 5  |-  ( M  e.  X  ->  (
( Base `  (Scalar `  M
) )  ^m  (/) )  =  1o )
6 df1o2 7149 . . . . 5  |-  1o  =  { (/) }
75, 6syl6eq 2478 . . . 4  |-  ( M  e.  X  ->  (
( Base `  (Scalar `  M
) )  ^m  (/) )  =  { (/) } )
82, 7syl5eleqr 2513 . . 3  |-  ( M  e.  X  ->  (/)  e.  ( ( Base `  (Scalar `  M ) )  ^m  (/) ) )
9 0elpw 4536 . . . 4  |-  (/)  e.  ~P ( Base `  M )
109a1i 11 . . 3  |-  ( M  e.  X  ->  (/)  e.  ~P ( Base `  M )
)
11 lincval 39805 . . 3  |-  ( ( M  e.  X  /\  (/) 
e.  ( ( Base `  (Scalar `  M )
)  ^m  (/) )  /\  (/) 
e.  ~P ( Base `  M
) )  ->  ( (/) ( linC  `  M ) (/) )  =  ( M 
gsumg  ( v  e.  (/)  |->  ( ( (/) `  v
) ( .s `  M ) v ) ) ) )
128, 10, 11mpd3an23 1362 . 2  |-  ( M  e.  X  ->  ( (/) ( linC  `  M ) (/) )  =  ( M 
gsumg  ( v  e.  (/)  |->  ( ( (/) `  v
) ( .s `  M ) v ) ) ) )
13 mpt0 5666 . . . . 5  |-  ( v  e.  (/)  |->  ( ( (/) `  v ) ( .s
`  M ) v ) )  =  (/)
1413a1i 11 . . . 4  |-  ( M  e.  X  ->  (
v  e.  (/)  |->  ( (
(/) `  v )
( .s `  M
) v ) )  =  (/) )
1514oveq2d 6265 . . 3  |-  ( M  e.  X  ->  ( M  gsumg  ( v  e.  (/)  |->  ( ( (/) `  v
) ( .s `  M ) v ) ) )  =  ( M  gsumg  (/) ) )
16 eqid 2428 . . . 4  |-  ( 0g
`  M )  =  ( 0g `  M
)
1716gsum0 16464 . . 3  |-  ( M 
gsumg  (/) )  =  ( 0g
`  M )
1815, 17syl6eq 2478 . 2  |-  ( M  e.  X  ->  ( M  gsumg  ( v  e.  (/)  |->  ( ( (/) `  v
) ( .s `  M ) v ) ) )  =  ( 0g `  M ) )
1912, 18eqtrd 2462 1  |-  ( M  e.  X  ->  ( (/) ( linC  `  M ) (/) )  =  ( 0g
`  M ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1437    e. wcel 1872   _Vcvv 3022   (/)c0 3704   ~Pcpw 3924   {csn 3941    |-> cmpt 4425   ` cfv 5544  (class class class)co 6249   1oc1o 7130    ^m cmap 7427   Basecbs 15064  Scalarcsca 15136   .scvsca 15137   0gc0g 15281    gsumg cgsu 15282   linC clinc 39800
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-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-nul 3705  df-if 3855  df-pw 3926  df-sn 3942  df-pr 3944  df-op 3948  df-uni 4163  df-iun 4244  df-br 4367  df-opab 4426  df-mpt 4427  df-id 4711  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-pred 5342  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-1st 6751  df-2nd 6752  df-wrecs 6983  df-recs 7045  df-rdg 7083  df-1o 7137  df-map 7429  df-seq 12164  df-gsum 15284  df-linc 39802
This theorem is referenced by:  lco0  39823
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