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Theorem lcoval 31055
Description: The value of a linear combination. (Contributed by AV, 5-Apr-2019.) (Revised by AV, 28-Jul-2019.)
Hypotheses
Ref Expression
lcoop.b  |-  B  =  ( Base `  M
)
lcoop.s  |-  S  =  (Scalar `  M )
lcoop.r  |-  R  =  ( Base `  S
)
Assertion
Ref Expression
lcoval  |-  ( ( M  e.  X  /\  V  e.  ~P B
)  ->  ( C  e.  ( M LinCo  V )  <-> 
( C  e.  B  /\  E. s  e.  ( R  ^m  V ) ( s finSupp  ( 0g
`  S )  /\  C  =  ( s
( linC  `  M ) V ) ) ) ) )
Distinct variable groups:    M, s    R, s    V, s    C, s
Allowed substitution hints:    B( s)    S( s)    X( s)

Proof of Theorem lcoval
Dummy variable  c is distinct from all other variables.
StepHypRef Expression
1 lcoop.b . . . 4  |-  B  =  ( Base `  M
)
2 lcoop.s . . . 4  |-  S  =  (Scalar `  M )
3 lcoop.r . . . 4  |-  R  =  ( Base `  S
)
41, 2, 3lcoop 31054 . . 3  |-  ( ( M  e.  X  /\  V  e.  ~P B
)  ->  ( M LinCo  V )  =  { c  e.  B  |  E. s  e.  ( R  ^m  V ) ( s finSupp 
( 0g `  S
)  /\  c  =  ( s ( linC  `  M ) V ) ) } )
54eleq2d 2521 . 2  |-  ( ( M  e.  X  /\  V  e.  ~P B
)  ->  ( C  e.  ( M LinCo  V )  <-> 
C  e.  { c  e.  B  |  E. s  e.  ( R  ^m  V ) ( s finSupp 
( 0g `  S
)  /\  c  =  ( s ( linC  `  M ) V ) ) } ) )
6 eqeq1 2455 . . . . 5  |-  ( c  =  C  ->  (
c  =  ( s ( linC  `  M ) V )  <->  C  =  ( s ( linC  `  M ) V ) ) )
76anbi2d 703 . . . 4  |-  ( c  =  C  ->  (
( s finSupp  ( 0g `  S )  /\  c  =  ( s ( linC  `  M ) V ) )  <->  ( s finSupp  ( 0g `  S )  /\  C  =  ( s
( linC  `  M ) V ) ) ) )
87rexbidv 2848 . . 3  |-  ( c  =  C  ->  ( E. s  e.  ( R  ^m  V ) ( s finSupp  ( 0g `  S )  /\  c  =  ( s ( linC  `  M ) V ) )  <->  E. s  e.  ( R  ^m  V ) ( s finSupp  ( 0g
`  S )  /\  C  =  ( s
( linC  `  M ) V ) ) ) )
98elrab 3216 . 2  |-  ( C  e.  { c  e.  B  |  E. s  e.  ( R  ^m  V
) ( s finSupp  ( 0g `  S )  /\  c  =  ( s
( linC  `  M ) V ) ) }  <-> 
( C  e.  B  /\  E. s  e.  ( R  ^m  V ) ( s finSupp  ( 0g
`  S )  /\  C  =  ( s
( linC  `  M ) V ) ) ) )
105, 9syl6bb 261 1  |-  ( ( M  e.  X  /\  V  e.  ~P B
)  ->  ( C  e.  ( M LinCo  V )  <-> 
( C  e.  B  /\  E. s  e.  ( R  ^m  V ) ( s finSupp  ( 0g
`  S )  /\  C  =  ( s
( linC  `  M ) V ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758   E.wrex 2796   {crab 2799   ~Pcpw 3960   class class class wbr 4392   ` cfv 5518  (class class class)co 6192    ^m cmap 7316   finSupp cfsupp 7723   Basecbs 14278  Scalarcsca 14345   0gc0g 14482   linC clinc 31047   LinCo clinco 31048
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-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4513  ax-nul 4521  ax-pr 4631
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-rab 2804  df-v 3072  df-sbc 3287  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-br 4393  df-opab 4451  df-id 4736  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-iota 5481  df-fun 5520  df-fv 5526  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-lco 31050
This theorem is referenced by:  lcoel0  31071  lincsumcl  31074  lincscmcl  31075  lincolss  31077  ellcoellss  31078  lcoss  31079  lindslinindsimp1  31100  lindslinindsimp2  31106
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