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Theorem dflinc2 40256
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 40252 . 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 7494 . . . . . . . 8  |-  ( s  e.  ( ( Base `  (Scalar `  m )
)  ^m  v )  ->  s  Fn  v )
32adantr 467 . . . . . . 7  |-  ( ( s  e.  ( (
Base `  (Scalar `  m
) )  ^m  v
)  /\  v  e.  ~P ( Base `  m
) )  ->  s  Fn  v )
4 fnresi 5693 . . . . . . . 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 3048 . . . . . . . 8  |-  v  e. 
_V
76a1i 11 . . . . . . 7  |-  ( ( s  e.  ( (
Base `  (Scalar `  m
) )  ^m  v
)  /\  v  e.  ~P ( Base `  m
) )  ->  v  e.  _V )
8 inidm 3641 . . . . . . 7  |-  ( v  i^i  v )  =  v
9 eqidd 2452 . . . . . . 7  |-  ( ( ( s  e.  ( ( Base `  (Scalar `  m ) )  ^m  v )  /\  v  e.  ~P ( Base `  m
) )  /\  i  e.  v )  ->  (
s `  i )  =  ( s `  i ) )
10 fvresi 6090 . . . . . . . 8  |-  ( i  e.  v  ->  (
(  _I  |`  v
) `  i )  =  i )
1110adantl 468 . . . . . . 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 6538 . . . . . 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 2457 . . . . 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 6306 . . . 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 6356 . . 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 4486 . 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 2473 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 371    = wceq 1444    e. wcel 1887   _Vcvv 3045   ~Pcpw 3951    |-> cmpt 4461    _I cid 4744    |` cres 4836    Fn wfn 5577   ` cfv 5582  (class class class)co 6290    |-> cmpt2 6292    oFcof 6529    ^m cmap 7472   Basecbs 15121  Scalarcsca 15193   .scvsca 15194    gsumg cgsu 15339   linC clinc 40250
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-reu 2744  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-of 6531  df-1st 6793  df-2nd 6794  df-map 7474  df-linc 40252
This theorem is referenced by: (None)
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