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Theorem lincval0 33160
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 4587 . . . . 5  |-  (/)  e.  _V
21snid 4060 . . . 4  |-  (/)  e.  { (/)
}
3 fvex 5882 . . . . . 6  |-  ( Base `  (Scalar `  M )
)  e.  _V
4 map0e 7475 . . . . . 6  |-  ( (
Base `  (Scalar `  M
) )  e.  _V  ->  ( ( Base `  (Scalar `  M ) )  ^m  (/) )  =  1o )
53, 4mp1i 12 . . . . 5  |-  ( M  e.  X  ->  (
( Base `  (Scalar `  M
) )  ^m  (/) )  =  1o )
6 df1o2 7160 . . . . 5  |-  1o  =  { (/) }
75, 6syl6eq 2514 . . . 4  |-  ( M  e.  X  ->  (
( Base `  (Scalar `  M
) )  ^m  (/) )  =  { (/) } )
82, 7syl5eleqr 2552 . . 3  |-  ( M  e.  X  ->  (/)  e.  ( ( Base `  (Scalar `  M ) )  ^m  (/) ) )
9 0elpw 4625 . . . 4  |-  (/)  e.  ~P ( Base `  M )
109a1i 11 . . 3  |-  ( M  e.  X  ->  (/)  e.  ~P ( Base `  M )
)
11 lincval 33154 . . 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 1326 . 2  |-  ( M  e.  X  ->  ( (/) ( linC  `  M ) (/) )  =  ( M 
gsumg  ( v  e.  (/)  |->  ( ( (/) `  v
) ( .s `  M ) v ) ) ) )
13 mpt0 5714 . . . . 5  |-  ( v  e.  (/)  |->  ( ( (/) `  v ) ( .s
`  M ) v ) )  =  (/)
1413a1i 11 . . . 4  |-  ( M  e.  X  ->  (
v  e.  (/)  |->  ( (
(/) `  v )
( .s `  M
) v ) )  =  (/) )
1514oveq2d 6312 . . 3  |-  ( M  e.  X  ->  ( M  gsumg  ( v  e.  (/)  |->  ( ( (/) `  v
) ( .s `  M ) v ) ) )  =  ( M  gsumg  (/) ) )
16 eqid 2457 . . . 4  |-  ( 0g
`  M )  =  ( 0g `  M
)
1716gsum0 16032 . . 3  |-  ( M 
gsumg  (/) )  =  ( 0g
`  M )
1815, 17syl6eq 2514 . 2  |-  ( M  e.  X  ->  ( M  gsumg  ( v  e.  (/)  |->  ( ( (/) `  v
) ( .s `  M ) v ) ) )  =  ( 0g `  M ) )
1912, 18eqtrd 2498 1  |-  ( M  e.  X  ->  ( (/) ( linC  `  M ) (/) )  =  ( 0g
`  M ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1395    e. wcel 1819   _Vcvv 3109   (/)c0 3793   ~Pcpw 4015   {csn 4032    |-> cmpt 4515   ` cfv 5594  (class class class)co 6296   1oc1o 7141    ^m cmap 7438   Basecbs 14644  Scalarcsca 14715   .scvsca 14716   0gc0g 14857    gsumg cgsu 14858   linC clinc 33149
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-map 7440  df-seq 12111  df-gsum 14860  df-linc 33151
This theorem is referenced by:  lco0  33172
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