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Theorem dflinc2 32498
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 32494 . 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 7453 . . . . . . . 8  |-  ( s  e.  ( ( Base `  (Scalar `  m )
)  ^m  v )  ->  s  Fn  v )
32adantr 465 . . . . . . 7  |-  ( ( s  e.  ( (
Base `  (Scalar `  m
) )  ^m  v
)  /\  v  e.  ~P ( Base `  m
) )  ->  s  Fn  v )
4 fnresi 5704 . . . . . . . 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 3121 . . . . . . . 8  |-  v  e. 
_V
76a1i 11 . . . . . . 7  |-  ( ( s  e.  ( (
Base `  (Scalar `  m
) )  ^m  v
)  /\  v  e.  ~P ( Base `  m
) )  ->  v  e.  _V )
8 inidm 3712 . . . . . . 7  |-  ( v  i^i  v )  =  v
9 eqidd 2468 . . . . . . 7  |-  ( ( ( s  e.  ( ( Base `  (Scalar `  m ) )  ^m  v )  /\  v  e.  ~P ( Base `  m
) )  /\  i  e.  v )  ->  (
s `  i )  =  ( s `  i ) )
10 fvresi 6098 . . . . . . . 8  |-  ( i  e.  v  ->  (
(  _I  |`  v
) `  i )  =  i )
1110adantl 466 . . . . . . 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 6542 . . . . . 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 2475 . . . . 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 6311 . . . 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 6357 . . 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 4536 . 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 2496 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 369    = wceq 1379    e. wcel 1767   _Vcvv 3118   ~Pcpw 4016    |-> cmpt 4511    _I cid 4796    |` cres 5007    Fn wfn 5589   ` cfv 5594  (class class class)co 6295    |-> cmpt2 6297    oFcof 6533    ^m cmap 7432   Basecbs 14506  Scalarcsca 14574   .scvsca 14575    gsumg cgsu 14712   linC clinc 32492
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  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 6298  df-oprab 6299  df-mpt2 6300  df-of 6535  df-1st 6795  df-2nd 6796  df-map 7434  df-linc 32494
This theorem is referenced by: (None)
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