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Theorem lincop 33209
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 33207 . . 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 5774 . . . . . 6  |-  ( m  =  M  ->  (Scalar `  m )  =  (Scalar `  M ) )
43fveq2d 5778 . . . . 5  |-  ( m  =  M  ->  ( Base `  (Scalar `  m
) )  =  (
Base `  (Scalar `  M
) ) )
54oveq1d 6211 . . . 4  |-  ( m  =  M  ->  (
( Base `  (Scalar `  m
) )  ^m  v
)  =  ( (
Base `  (Scalar `  M
) )  ^m  v
) )
6 fveq2 5774 . . . . 5  |-  ( m  =  M  ->  ( Base `  m )  =  ( Base `  M
) )
76pweqd 3932 . . . 4  |-  ( m  =  M  ->  ~P ( Base `  m )  =  ~P ( Base `  M
) )
8 id 22 . . . . 5  |-  ( m  =  M  ->  m  =  M )
9 fveq2 5774 . . . . . . 7  |-  ( m  =  M  ->  ( .s `  m )  =  ( .s `  M
) )
109oveqd 6213 . . . . . 6  |-  ( m  =  M  ->  (
( s `  x
) ( .s `  m ) x )  =  ( ( s `
 x ) ( .s `  M ) x ) )
1110mpteq2dv 4454 . . . . 5  |-  ( m  =  M  ->  (
x  e.  v  |->  ( ( s `  x
) ( .s `  m ) x ) )  =  ( x  e.  v  |->  ( ( s `  x ) ( .s `  M
) x ) ) )
128, 11oveq12d 6214 . . . 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 6258 . . 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 464 . 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 3043 . 2  |-  ( M  e.  X  ->  M  e.  _V )
16 fvex 5784 . . . 4  |-  ( Base `  M )  e.  _V
1716pwex 4548 . . 3  |-  ~P ( Base `  M )  e. 
_V
18 ovex 6224 . . . . 5  |-  ( (
Base `  (Scalar `  M
) )  ^m  v
)  e.  _V
1918a1i 11 . . . 4  |-  ( M  e.  X  ->  (
( Base `  (Scalar `  M
) )  ^m  v
)  e.  _V )
2019ralrimivw 2797 . . 3  |-  ( M  e.  X  ->  A. v  e.  ~P  ( Base `  M
) ( ( Base `  (Scalar `  M )
)  ^m  v )  e.  _V )
21 eqid 2382 . . . 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 33127 . . 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 661 . 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 5862 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 1399    e. wcel 1826   A.wral 2732   _Vcvv 3034   ~Pcpw 3927    |-> cmpt 4425   ` cfv 5496  (class class class)co 6196    |-> cmpt2 6198    ^m cmap 7338   Basecbs 14634  Scalarcsca 14705   .scvsca 14706    gsumg cgsu 14848   linC clinc 33205
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-reu 2739  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-1st 6699  df-2nd 6700  df-linc 33207
This theorem is referenced by:  lincval  33210
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