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Theorem lincop 31051
Description: A linear combination as operation. (Contributed by AV, 30-Mar-2019.)
Assertion
Ref Expression
lincop  |-  ( M  e.  X  ->  ( linC  `  M )  =  ( s  e.  ( (
Base `  (Scalar `  M
) )  ^m  v
) ,  v  e. 
~P ( Base `  M
)  |->  ( M  gsumg  ( x  e.  v  |->  ( ( s `  x ) ( .s `  M
) x ) ) ) ) )
Distinct variable groups:    M, s,
v, x    v, X
Allowed substitution hints:    X( x, s)

Proof of Theorem lincop
Dummy variable  m is distinct from all other variables.
StepHypRef Expression
1 df-linc 31049 . . 3  |- linC  =  ( m  e.  _V  |->  ( s  e.  ( (
Base `  (Scalar `  m
) )  ^m  v
) ,  v  e. 
~P ( Base `  m
)  |->  ( m  gsumg  ( x  e.  v  |->  ( ( s `  x ) ( .s `  m
) x ) ) ) ) )
21a1i 11 . 2  |-  ( M  e.  X  -> linC  =  ( m  e.  _V  |->  ( s  e.  ( (
Base `  (Scalar `  m
) )  ^m  v
) ,  v  e. 
~P ( Base `  m
)  |->  ( m  gsumg  ( x  e.  v  |->  ( ( s `  x ) ( .s `  m
) x ) ) ) ) ) )
3 fveq2 5791 . . . . . 6  |-  ( m  =  M  ->  (Scalar `  m )  =  (Scalar `  M ) )
43fveq2d 5795 . . . . 5  |-  ( m  =  M  ->  ( Base `  (Scalar `  m
) )  =  (
Base `  (Scalar `  M
) ) )
54oveq1d 6207 . . . 4  |-  ( m  =  M  ->  (
( Base `  (Scalar `  m
) )  ^m  v
)  =  ( (
Base `  (Scalar `  M
) )  ^m  v
) )
6 fveq2 5791 . . . . 5  |-  ( m  =  M  ->  ( Base `  m )  =  ( Base `  M
) )
76pweqd 3965 . . . 4  |-  ( m  =  M  ->  ~P ( Base `  m )  =  ~P ( Base `  M
) )
8 id 22 . . . . 5  |-  ( m  =  M  ->  m  =  M )
9 fveq2 5791 . . . . . . 7  |-  ( m  =  M  ->  ( .s `  m )  =  ( .s `  M
) )
109oveqd 6209 . . . . . 6  |-  ( m  =  M  ->  (
( s `  x
) ( .s `  m ) x )  =  ( ( s `
 x ) ( .s `  M ) x ) )
1110mpteq2dv 4479 . . . . 5  |-  ( m  =  M  ->  (
x  e.  v  |->  ( ( s `  x
) ( .s `  m ) x ) )  =  ( x  e.  v  |->  ( ( s `  x ) ( .s `  M
) x ) ) )
128, 11oveq12d 6210 . . . 4  |-  ( m  =  M  ->  (
m  gsumg  ( x  e.  v 
|->  ( ( s `  x ) ( .s
`  m ) x ) ) )  =  ( M  gsumg  ( x  e.  v 
|->  ( ( s `  x ) ( .s
`  M ) x ) ) ) )
135, 7, 12mpt2eq123dv 6249 . . 3  |-  ( m  =  M  ->  (
s  e.  ( (
Base `  (Scalar `  m
) )  ^m  v
) ,  v  e. 
~P ( Base `  m
)  |->  ( m  gsumg  ( x  e.  v  |->  ( ( s `  x ) ( .s `  m
) x ) ) ) )  =  ( s  e.  ( (
Base `  (Scalar `  M
) )  ^m  v
) ,  v  e. 
~P ( Base `  M
)  |->  ( M  gsumg  ( x  e.  v  |->  ( ( s `  x ) ( .s `  M
) x ) ) ) ) )
1413adantl 466 . 2  |-  ( ( M  e.  X  /\  m  =  M )  ->  ( s  e.  ( ( Base `  (Scalar `  m ) )  ^m  v ) ,  v  e.  ~P ( Base `  m )  |->  ( m 
gsumg  ( x  e.  v  |->  ( ( s `  x ) ( .s
`  m ) x ) ) ) )  =  ( s  e.  ( ( Base `  (Scalar `  M ) )  ^m  v ) ,  v  e.  ~P ( Base `  M )  |->  ( M 
gsumg  ( x  e.  v  |->  ( ( s `  x ) ( .s
`  M ) x ) ) ) ) )
15 elex 3079 . 2  |-  ( M  e.  X  ->  M  e.  _V )
16 fvex 5801 . . . 4  |-  ( Base `  M )  e.  _V
1716pwex 4575 . . 3  |-  ~P ( Base `  M )  e. 
_V
18 ovex 6217 . . . . 5  |-  ( (
Base `  (Scalar `  M
) )  ^m  v
)  e.  _V
1918a1i 11 . . . 4  |-  ( M  e.  X  ->  (
( Base `  (Scalar `  M
) )  ^m  v
)  e.  _V )
2019ralrimivw 2823 . . 3  |-  ( M  e.  X  ->  A. v  e.  ~P  ( Base `  M
) ( ( Base `  (Scalar `  M )
)  ^m  v )  e.  _V )
21 eqid 2451 . . . 4  |-  ( s  e.  ( ( Base `  (Scalar `  M )
)  ^m  v ) ,  v  e.  ~P ( Base `  M )  |->  ( M  gsumg  ( x  e.  v 
|->  ( ( s `  x ) ( .s
`  M ) x ) ) ) )  =  ( s  e.  ( ( Base `  (Scalar `  M ) )  ^m  v ) ,  v  e.  ~P ( Base `  M )  |->  ( M 
gsumg  ( x  e.  v  |->  ( ( s `  x ) ( .s
`  M ) x ) ) ) )
2221mpt2exxg2 30868 . . 3  |-  ( ( ~P ( Base `  M
)  e.  _V  /\  A. v  e.  ~P  ( Base `  M ) ( ( Base `  (Scalar `  M ) )  ^m  v )  e.  _V )  ->  ( s  e.  ( ( Base `  (Scalar `  M ) )  ^m  v ) ,  v  e.  ~P ( Base `  M )  |->  ( M 
gsumg  ( x  e.  v  |->  ( ( s `  x ) ( .s
`  M ) x ) ) ) )  e.  _V )
2317, 20, 22sylancr 663 . 2  |-  ( M  e.  X  ->  (
s  e.  ( (
Base `  (Scalar `  M
) )  ^m  v
) ,  v  e. 
~P ( Base `  M
)  |->  ( M  gsumg  ( x  e.  v  |->  ( ( s `  x ) ( .s `  M
) x ) ) ) )  e.  _V )
242, 14, 15, 23fvmptd 5880 1  |-  ( M  e.  X  ->  ( linC  `  M )  =  ( s  e.  ( (
Base `  (Scalar `  M
) )  ^m  v
) ,  v  e. 
~P ( Base `  M
)  |->  ( M  gsumg  ( x  e.  v  |->  ( ( s `  x ) ( .s `  M
) x ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    e. wcel 1758   A.wral 2795   _Vcvv 3070   ~Pcpw 3960    |-> cmpt 4450   ` cfv 5518  (class class class)co 6192    |-> cmpt2 6194    ^m cmap 7316   Basecbs 14278  Scalarcsca 14345   .scvsca 14346    gsumg cgsu 14483   linC clinc 31047
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4503  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4736  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-1st 6679  df-2nd 6680  df-linc 31049
This theorem is referenced by:  lincval  31052
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