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Theorem lincval 32109
Description: The value of a linear combination. (Contributed by AV, 30-Mar-2019.)
Assertion
Ref Expression
lincval  |-  ( ( M  e.  X  /\  S  e.  ( ( Base `  (Scalar `  M
) )  ^m  V
)  /\  V  e.  ~P ( Base `  M
) )  ->  ( S ( linC  `  M ) V )  =  ( M  gsumg  ( x  e.  V  |->  ( ( S `  x ) ( .s
`  M ) x ) ) ) )
Distinct variable groups:    x, M    x, S    x, V
Allowed substitution hint:    X( x)

Proof of Theorem lincval
Dummy variables  s 
v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lincop 32108 . . . 4  |-  ( 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 ) ) ) ) )
213ad2ant1 1017 . . 3  |-  ( ( M  e.  X  /\  S  e.  ( ( Base `  (Scalar `  M
) )  ^m  V
)  /\  V  e.  ~P ( Base `  M
) )  ->  ( linC  `  M )  =  ( s  e.  ( (
Base `  (Scalar `  M
) )  ^m  v
) ,  v  e. 
~P ( Base `  M
)  |->  ( M  gsumg  ( x  e.  v  |->  ( ( s `  x ) ( .s `  M
) x ) ) ) ) )
32oveqd 6301 . 2  |-  ( ( M  e.  X  /\  S  e.  ( ( Base `  (Scalar `  M
) )  ^m  V
)  /\  V  e.  ~P ( Base `  M
) )  ->  ( S ( linC  `  M ) V )  =  ( S ( s  e.  ( ( Base `  (Scalar `  M ) )  ^m  v ) ,  v  e.  ~P ( Base `  M )  |->  ( M 
gsumg  ( x  e.  v  |->  ( ( s `  x ) ( .s
`  M ) x ) ) ) ) V ) )
4 simp2 997 . . 3  |-  ( ( M  e.  X  /\  S  e.  ( ( Base `  (Scalar `  M
) )  ^m  V
)  /\  V  e.  ~P ( Base `  M
) )  ->  S  e.  ( ( Base `  (Scalar `  M ) )  ^m  V ) )
5 simp3 998 . . 3  |-  ( ( M  e.  X  /\  S  e.  ( ( Base `  (Scalar `  M
) )  ^m  V
)  /\  V  e.  ~P ( Base `  M
) )  ->  V  e.  ~P ( Base `  M
) )
6 ovex 6309 . . . 4  |-  ( M 
gsumg  ( x  e.  V  |->  ( ( S `  x ) ( .s
`  M ) x ) ) )  e. 
_V
76a1i 11 . . 3  |-  ( ( 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 )
8 simpr 461 . . . . . 6  |-  ( ( s  =  S  /\  v  =  V )  ->  v  =  V )
9 fveq1 5865 . . . . . . . 8  |-  ( s  =  S  ->  (
s `  x )  =  ( S `  x ) )
109oveq1d 6299 . . . . . . 7  |-  ( s  =  S  ->  (
( s `  x
) ( .s `  M ) x )  =  ( ( S `
 x ) ( .s `  M ) x ) )
1110adantr 465 . . . . . 6  |-  ( ( s  =  S  /\  v  =  V )  ->  ( ( s `  x ) ( .s
`  M ) x )  =  ( ( S `  x ) ( .s `  M
) x ) )
128, 11mpteq12dv 4525 . . . . 5  |-  ( ( s  =  S  /\  v  =  V )  ->  ( x  e.  v 
|->  ( ( s `  x ) ( .s
`  M ) x ) )  =  ( x  e.  V  |->  ( ( S `  x
) ( .s `  M ) x ) ) )
1312oveq2d 6300 . . . 4  |-  ( ( s  =  S  /\  v  =  V )  ->  ( M  gsumg  ( x  e.  v 
|->  ( ( s `  x ) ( .s
`  M ) x ) ) )  =  ( M  gsumg  ( x  e.  V  |->  ( ( S `  x ) ( .s
`  M ) x ) ) ) )
14 oveq2 6292 . . . 4  |-  ( v  =  V  ->  (
( Base `  (Scalar `  M
) )  ^m  v
)  =  ( (
Base `  (Scalar `  M
) )  ^m  V
) )
15 eqid 2467 . . . 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 ) ) ) )
1613, 14, 15ovmpt2x2 32026 . . 3  |-  ( ( S  e.  ( (
Base `  (Scalar `  M
) )  ^m  V
)  /\  V  e.  ~P ( Base `  M
)  /\  ( M  gsumg  ( x  e.  V  |->  ( ( S `  x
) ( .s `  M ) x ) ) )  e.  _V )  ->  ( S ( s  e.  ( (
Base `  (Scalar `  M
) )  ^m  v
) ,  v  e. 
~P ( Base `  M
)  |->  ( M  gsumg  ( x  e.  v  |->  ( ( s `  x ) ( .s `  M
) x ) ) ) ) V )  =  ( M  gsumg  ( x  e.  V  |->  ( ( S `  x ) ( .s `  M
) x ) ) ) )
174, 5, 7, 16syl3anc 1228 . 2  |-  ( ( M  e.  X  /\  S  e.  ( ( Base `  (Scalar `  M
) )  ^m  V
)  /\  V  e.  ~P ( Base `  M
) )  ->  ( S ( s  e.  ( ( Base `  (Scalar `  M ) )  ^m  v ) ,  v  e.  ~P ( Base `  M )  |->  ( M 
gsumg  ( x  e.  v  |->  ( ( s `  x ) ( .s
`  M ) x ) ) ) ) V )  =  ( M  gsumg  ( x  e.  V  |->  ( ( S `  x ) ( .s
`  M ) x ) ) ) )
183, 17eqtrd 2508 1  |-  ( ( M  e.  X  /\  S  e.  ( ( Base `  (Scalar `  M
) )  ^m  V
)  /\  V  e.  ~P ( Base `  M
) )  ->  ( S ( linC  `  M ) V )  =  ( M  gsumg  ( x  e.  V  |->  ( ( S `  x ) ( .s
`  M ) x ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   _Vcvv 3113   ~Pcpw 4010    |-> cmpt 4505   ` cfv 5588  (class class class)co 6284    |-> cmpt2 6286    ^m cmap 7420   Basecbs 14490  Scalarcsca 14558   .scvsca 14559    gsumg cgsu 14696   linC clinc 32104
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 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576
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 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-1st 6784  df-2nd 6785  df-linc 32106
This theorem is referenced by:  lincfsuppcl  32113  linccl  32114  lincval0  32115  lincvalsng  32116  lincvalpr  32118  lincvalsc0  32121  linc0scn0  32123  lincdifsn  32124  linc1  32125  lincellss  32126  lincsum  32129  lincscm  32130  lindslinindimp2lem4  32161  lindslinindsimp2lem5  32162  snlindsntor  32171  lincresunit3lem2  32180  lincresunit3  32181  zlmodzxzldeplem3  32202
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