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Theorem dflinc2 30956
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 30952 . 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 fvex 5713 . . . . . . . . . 10  |-  ( Base `  (Scalar `  m )
)  e.  _V
3 vex 2987 . . . . . . . . . 10  |-  v  e. 
_V
42, 3elmap 7253 . . . . . . . . 9  |-  ( s  e.  ( ( Base `  (Scalar `  m )
)  ^m  v )  <->  s : v --> ( Base `  (Scalar `  m )
) )
5 ffn 5571 . . . . . . . . 9  |-  ( s : v --> ( Base `  (Scalar `  m )
)  ->  s  Fn  v )
64, 5sylbi 195 . . . . . . . 8  |-  ( s  e.  ( ( Base `  (Scalar `  m )
)  ^m  v )  ->  s  Fn  v )
76adantr 465 . . . . . . 7  |-  ( ( s  e.  ( (
Base `  (Scalar `  m
) )  ^m  v
)  /\  v  e.  ~P ( Base `  m
) )  ->  s  Fn  v )
8 fnresi 5540 . . . . . . . 8  |-  (  _I  |`  v )  Fn  v
98a1i 11 . . . . . . 7  |-  ( ( s  e.  ( (
Base `  (Scalar `  m
) )  ^m  v
)  /\  v  e.  ~P ( Base `  m
) )  ->  (  _I  |`  v )  Fn  v )
103a1i 11 . . . . . . 7  |-  ( ( s  e.  ( (
Base `  (Scalar `  m
) )  ^m  v
)  /\  v  e.  ~P ( Base `  m
) )  ->  v  e.  _V )
11 inidm 3571 . . . . . . 7  |-  ( v  i^i  v )  =  v
12 eqidd 2444 . . . . . . 7  |-  ( ( ( s  e.  ( ( Base `  (Scalar `  m ) )  ^m  v )  /\  v  e.  ~P ( Base `  m
) )  /\  i  e.  v )  ->  (
s `  i )  =  ( s `  i ) )
13 fvresi 5916 . . . . . . . 8  |-  ( i  e.  v  ->  (
(  _I  |`  v
) `  i )  =  i )
1413adantl 466 . . . . . . 7  |-  ( ( ( s  e.  ( ( Base `  (Scalar `  m ) )  ^m  v )  /\  v  e.  ~P ( Base `  m
) )  /\  i  e.  v )  ->  (
(  _I  |`  v
) `  i )  =  i )
157, 9, 10, 10, 11, 12, 14offval 6339 . . . . . 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 ) ) )
1615eqcomd 2448 . . . . 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 )
) )
1716oveq2d 6119 . . . 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
) ) ) )
1817mpt2eq3ia 6163 . . 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
) ) ) )
1918mpteq2i 4387 . 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
) ) ) ) )
201, 19eqtri 2463 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 1369    e. wcel 1756   _Vcvv 2984   ~Pcpw 3872    e. cmpt 4362    _I cid 4643    |` cres 4854    Fn wfn 5425   -->wf 5426   ` cfv 5430  (class class class)co 6103    e. cmpt2 6105    oFcof 6330    ^m cmap 7226   Basecbs 14186  Scalarcsca 14253   .scvsca 14254    gsumg cgsu 14391   linC clinc 30950
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4415  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-ral 2732  df-rex 2733  df-reu 2734  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-op 3896  df-uni 4104  df-iun 4185  df-br 4305  df-opab 4363  df-mpt 4364  df-id 4648  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-of 6332  df-map 7228  df-linc 30952
This theorem is referenced by: (None)
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