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Theorem dflinc2 39391
Description: Alternative definition of linear combinations using the function operation. (Contributed by AV, 1-Apr-2019.)
Assertion
Ref Expression
dflinc2  |- linC  =  ( m  e.  _V  |->  ( s  e.  ( (
Base `  (Scalar `  m
) )  ^m  v
) ,  v  e. 
~P ( Base `  m
)  |->  ( m  gsumg  ( s  oF ( .s
`  m ) (  _I  |`  v )
) ) ) )
Distinct variable group:    m, s, v

Proof of Theorem dflinc2
Dummy variable  i is distinct from all other variables.
StepHypRef Expression
1 df-linc 39387 . 2  |- linC  =  ( m  e.  _V  |->  ( s  e.  ( (
Base `  (Scalar `  m
) )  ^m  v
) ,  v  e. 
~P ( Base `  m
)  |->  ( m  gsumg  ( i  e.  v  |->  ( ( s `  i ) ( .s `  m
) i ) ) ) ) )
2 elmapfn 7498 . . . . . . . 8  |-  ( s  e.  ( ( Base `  (Scalar `  m )
)  ^m  v )  ->  s  Fn  v )
32adantr 466 . . . . . . 7  |-  ( ( s  e.  ( (
Base `  (Scalar `  m
) )  ^m  v
)  /\  v  e.  ~P ( Base `  m
) )  ->  s  Fn  v )
4 fnresi 5707 . . . . . . . 8  |-  (  _I  |`  v )  Fn  v
54a1i 11 . . . . . . 7  |-  ( ( s  e.  ( (
Base `  (Scalar `  m
) )  ^m  v
)  /\  v  e.  ~P ( Base `  m
) )  ->  (  _I  |`  v )  Fn  v )
6 vex 3084 . . . . . . . 8  |-  v  e. 
_V
76a1i 11 . . . . . . 7  |-  ( ( s  e.  ( (
Base `  (Scalar `  m
) )  ^m  v
)  /\  v  e.  ~P ( Base `  m
) )  ->  v  e.  _V )
8 inidm 3671 . . . . . . 7  |-  ( v  i^i  v )  =  v
9 eqidd 2423 . . . . . . 7  |-  ( ( ( s  e.  ( ( Base `  (Scalar `  m ) )  ^m  v )  /\  v  e.  ~P ( Base `  m
) )  /\  i  e.  v )  ->  (
s `  i )  =  ( s `  i ) )
10 fvresi 6101 . . . . . . . 8  |-  ( i  e.  v  ->  (
(  _I  |`  v
) `  i )  =  i )
1110adantl 467 . . . . . . 7  |-  ( ( ( s  e.  ( ( Base `  (Scalar `  m ) )  ^m  v )  /\  v  e.  ~P ( Base `  m
) )  /\  i  e.  v )  ->  (
(  _I  |`  v
) `  i )  =  i )
123, 5, 7, 7, 8, 9, 11offval 6548 . . . . . 6  |-  ( ( s  e.  ( (
Base `  (Scalar `  m
) )  ^m  v
)  /\  v  e.  ~P ( Base `  m
) )  ->  (
s  oF ( .s `  m ) (  _I  |`  v
) )  =  ( i  e.  v  |->  ( ( s `  i
) ( .s `  m ) i ) ) )
1312eqcomd 2430 . . . . 5  |-  ( ( s  e.  ( (
Base `  (Scalar `  m
) )  ^m  v
)  /\  v  e.  ~P ( Base `  m
) )  ->  (
i  e.  v  |->  ( ( s `  i
) ( .s `  m ) i ) )  =  ( s  oF ( .s
`  m ) (  _I  |`  v )
) )
1413oveq2d 6317 . . . 4  |-  ( ( s  e.  ( (
Base `  (Scalar `  m
) )  ^m  v
)  /\  v  e.  ~P ( Base `  m
) )  ->  (
m  gsumg  ( i  e.  v 
|->  ( ( s `  i ) ( .s
`  m ) i ) ) )  =  ( m  gsumg  ( s  oF ( .s `  m
) (  _I  |`  v
) ) ) )
1514mpt2eq3ia 6366 . . 3  |-  ( s  e.  ( ( Base `  (Scalar `  m )
)  ^m  v ) ,  v  e.  ~P ( Base `  m )  |->  ( m  gsumg  ( i  e.  v 
|->  ( ( s `  i ) ( .s
`  m ) i ) ) ) )  =  ( s  e.  ( ( Base `  (Scalar `  m ) )  ^m  v ) ,  v  e.  ~P ( Base `  m )  |->  ( m 
gsumg  ( s  oF ( .s `  m
) (  _I  |`  v
) ) ) )
1615mpteq2i 4504 . 2  |-  ( m  e.  _V  |->  ( s  e.  ( ( Base `  (Scalar `  m )
)  ^m  v ) ,  v  e.  ~P ( Base `  m )  |->  ( m  gsumg  ( i  e.  v 
|->  ( ( s `  i ) ( .s
`  m ) i ) ) ) ) )  =  ( m  e.  _V  |->  ( s  e.  ( ( Base `  (Scalar `  m )
)  ^m  v ) ,  v  e.  ~P ( Base `  m )  |->  ( m  gsumg  ( s  oF ( .s `  m
) (  _I  |`  v
) ) ) ) )
171, 16eqtri 2451 1  |- linC  =  ( m  e.  _V  |->  ( s  e.  ( (
Base `  (Scalar `  m
) )  ^m  v
) ,  v  e. 
~P ( Base `  m
)  |->  ( m  gsumg  ( s  oF ( .s
`  m ) (  _I  |`  v )
) ) ) )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 370    = wceq 1437    e. wcel 1868   _Vcvv 3081   ~Pcpw 3979    |-> cmpt 4479    _I cid 4759    |` cres 4851    Fn wfn 5592   ` cfv 5597  (class class class)co 6301    |-> cmpt2 6303    oFcof 6539    ^m cmap 7476   Basecbs 15106  Scalarcsca 15178   .scvsca 15179    gsumg cgsu 15324   linC clinc 39385
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-rep 4533  ax-sep 4543  ax-nul 4551  ax-pow 4598  ax-pr 4656  ax-un 6593
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-ral 2780  df-rex 2781  df-reu 2782  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-op 4003  df-uni 4217  df-iun 4298  df-br 4421  df-opab 4480  df-mpt 4481  df-id 4764  df-xp 4855  df-rel 4856  df-cnv 4857  df-co 4858  df-dm 4859  df-rn 4860  df-res 4861  df-ima 4862  df-iota 5561  df-fun 5599  df-fn 5600  df-f 5601  df-f1 5602  df-fo 5603  df-f1o 5604  df-fv 5605  df-ov 6304  df-oprab 6305  df-mpt2 6306  df-of 6541  df-1st 6803  df-2nd 6804  df-map 7478  df-linc 39387
This theorem is referenced by: (None)
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