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Theorem lcoop 30945
Description: A linear combination as operation. (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
lcoop  |-  ( ( 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 ) ) } )
Distinct variable groups:    B, c    M, c, s    R, c, s    V, c, s
Allowed substitution hints:    B( s)    S( s, c)    X( s, c)

Proof of Theorem lcoop
Dummy variables  m  v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 2981 . . 3  |-  ( M  e.  X  ->  M  e.  _V )
21adantr 465 . 2  |-  ( ( M  e.  X  /\  V  e.  ~P B
)  ->  M  e.  _V )
3 lcoop.b . . . . . 6  |-  B  =  ( Base `  M
)
43pweqi 3864 . . . . 5  |-  ~P B  =  ~P ( Base `  M
)
54eleq2i 2507 . . . 4  |-  ( V  e.  ~P B  <->  V  e.  ~P ( Base `  M
) )
65biimpi 194 . . 3  |-  ( V  e.  ~P B  ->  V  e.  ~P ( Base `  M ) )
76adantl 466 . 2  |-  ( ( M  e.  X  /\  V  e.  ~P B
)  ->  V  e.  ~P ( Base `  M
) )
8 fvex 5701 . . . 4  |-  ( Base `  M )  e.  _V
93, 8eqeltri 2513 . . 3  |-  B  e. 
_V
10 rabexg 4442 . . 3  |-  ( B  e.  _V  ->  { c  e.  B  |  E. s  e.  ( R  ^m  V ) ( s finSupp 
( 0g `  S
)  /\  c  =  ( s ( linC  `  M ) V ) ) }  e.  _V )
119, 10mp1i 12 . 2  |-  ( ( M  e.  X  /\  V  e.  ~P B
)  ->  { c  e.  B  |  E. s  e.  ( R  ^m  V ) ( s finSupp 
( 0g `  S
)  /\  c  =  ( s ( linC  `  M ) V ) ) }  e.  _V )
12 fveq2 5691 . . . . . 6  |-  ( m  =  M  ->  ( Base `  m )  =  ( Base `  M
) )
1312, 3syl6eqr 2493 . . . . 5  |-  ( m  =  M  ->  ( Base `  m )  =  B )
1413adantr 465 . . . 4  |-  ( ( m  =  M  /\  v  =  V )  ->  ( Base `  m
)  =  B )
15 fveq2 5691 . . . . . . . . 9  |-  ( m  =  M  ->  (Scalar `  m )  =  (Scalar `  M ) )
1615fveq2d 5695 . . . . . . . 8  |-  ( m  =  M  ->  ( Base `  (Scalar `  m
) )  =  (
Base `  (Scalar `  M
) ) )
1716adantr 465 . . . . . . 7  |-  ( ( m  =  M  /\  v  =  V )  ->  ( Base `  (Scalar `  m ) )  =  ( Base `  (Scalar `  M ) ) )
18 lcoop.r . . . . . . . 8  |-  R  =  ( Base `  S
)
19 lcoop.s . . . . . . . . 9  |-  S  =  (Scalar `  M )
2019fveq2i 5694 . . . . . . . 8  |-  ( Base `  S )  =  (
Base `  (Scalar `  M
) )
2118, 20eqtri 2463 . . . . . . 7  |-  R  =  ( Base `  (Scalar `  M ) )
2217, 21syl6eqr 2493 . . . . . 6  |-  ( ( m  =  M  /\  v  =  V )  ->  ( Base `  (Scalar `  m ) )  =  R )
23 simpr 461 . . . . . 6  |-  ( ( m  =  M  /\  v  =  V )  ->  v  =  V )
2422, 23oveq12d 6109 . . . . 5  |-  ( ( m  =  M  /\  v  =  V )  ->  ( ( Base `  (Scalar `  m ) )  ^m  v )  =  ( R  ^m  V ) )
2515fveq2d 5695 . . . . . . . . 9  |-  ( m  =  M  ->  ( 0g `  (Scalar `  m
) )  =  ( 0g `  (Scalar `  M ) ) )
2619a1i 11 . . . . . . . . . . 11  |-  ( m  =  M  ->  S  =  (Scalar `  M )
)
2726eqcomd 2448 . . . . . . . . . 10  |-  ( m  =  M  ->  (Scalar `  M )  =  S )
2827fveq2d 5695 . . . . . . . . 9  |-  ( m  =  M  ->  ( 0g `  (Scalar `  M
) )  =  ( 0g `  S ) )
2925, 28eqtrd 2475 . . . . . . . 8  |-  ( m  =  M  ->  ( 0g `  (Scalar `  m
) )  =  ( 0g `  S ) )
3029adantr 465 . . . . . . 7  |-  ( ( m  =  M  /\  v  =  V )  ->  ( 0g `  (Scalar `  m ) )  =  ( 0g `  S
) )
3130breq2d 4304 . . . . . 6  |-  ( ( m  =  M  /\  v  =  V )  ->  ( s finSupp  ( 0g
`  (Scalar `  m )
)  <->  s finSupp  ( 0g `  S ) ) )
32 fveq2 5691 . . . . . . . . 9  |-  ( m  =  M  ->  ( linC  `  m )  =  ( linC  `  M ) )
3332adantr 465 . . . . . . . 8  |-  ( ( m  =  M  /\  v  =  V )  ->  ( linC  `  m )  =  ( linC  `  M ) )
34 eqidd 2444 . . . . . . . 8  |-  ( ( m  =  M  /\  v  =  V )  ->  s  =  s )
3533, 34, 23oveq123d 6112 . . . . . . 7  |-  ( ( m  =  M  /\  v  =  V )  ->  ( s ( linC  `  m ) v )  =  ( s ( linC  `  M ) V ) )
3635eqeq2d 2454 . . . . . 6  |-  ( ( m  =  M  /\  v  =  V )  ->  ( c  =  ( s ( linC  `  m
) v )  <->  c  =  ( s ( linC  `  M ) V ) ) )
3731, 36anbi12d 710 . . . . 5  |-  ( ( m  =  M  /\  v  =  V )  ->  ( ( s finSupp  ( 0g `  (Scalar `  m
) )  /\  c  =  ( s ( linC  `  m ) v ) )  <->  ( s finSupp  ( 0g `  S )  /\  c  =  ( s
( linC  `  M ) V ) ) ) )
3824, 37rexeqbidv 2932 . . . 4  |-  ( ( m  =  M  /\  v  =  V )  ->  ( E. s  e.  ( ( Base `  (Scalar `  m ) )  ^m  v ) ( s finSupp 
( 0g `  (Scalar `  m ) )  /\  c  =  ( s
( linC  `  m )
v ) )  <->  E. s  e.  ( R  ^m  V
) ( s finSupp  ( 0g `  S )  /\  c  =  ( s
( linC  `  M ) V ) ) ) )
3914, 38rabeqbidv 2967 . . 3  |-  ( ( m  =  M  /\  v  =  V )  ->  { c  e.  (
Base `  m )  |  E. s  e.  ( ( Base `  (Scalar `  m ) )  ^m  v ) ( s finSupp 
( 0g `  (Scalar `  m ) )  /\  c  =  ( s
( linC  `  m )
v ) ) }  =  { c  e.  B  |  E. s  e.  ( R  ^m  V
) ( s finSupp  ( 0g `  S )  /\  c  =  ( s
( linC  `  M ) V ) ) } )
4012pweqd 3865 . . 3  |-  ( m  =  M  ->  ~P ( Base `  m )  =  ~P ( Base `  M
) )
41 df-lco 30941 . . 3  |- LinCo  =  ( m  e.  _V , 
v  e.  ~P ( Base `  m )  |->  { c  e.  ( Base `  m )  |  E. s  e.  ( ( Base `  (Scalar `  m
) )  ^m  v
) ( s finSupp  ( 0g `  (Scalar `  m
) )  /\  c  =  ( s ( linC  `  m ) v ) ) } )
4239, 40, 41ovmpt2x 6219 . 2  |-  ( ( M  e.  _V  /\  V  e.  ~P ( Base `  M )  /\  { c  e.  B  |  E. s  e.  ( R  ^m  V ) ( s finSupp  ( 0g `  S )  /\  c  =  ( s ( linC  `  M ) V ) ) }  e.  _V )  ->  ( M LinCo  V
)  =  { c  e.  B  |  E. s  e.  ( R  ^m  V ) ( s finSupp 
( 0g `  S
)  /\  c  =  ( s ( linC  `  M ) V ) ) } )
432, 7, 11, 42syl3anc 1218 1  |-  ( ( 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 ) ) } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   E.wrex 2716   {crab 2719   _Vcvv 2972   ~Pcpw 3860   class class class wbr 4292   ` cfv 5418  (class class class)co 6091    ^m cmap 7214   finSupp cfsupp 7620   Basecbs 14174  Scalarcsca 14241   0gc0g 14378   linC clinc 30938   LinCo clinco 30939
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-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pr 4531
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 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-opab 4351  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-iota 5381  df-fun 5420  df-fv 5426  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-lco 30941
This theorem is referenced by:  lcoval  30946  lco0  30961
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