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Theorem mptscmfsuppd 17357
Description: A function mapping to a scalar product in which one factor is finitely supported is finitely supported. Formerly part of proof for ply1coe 18105. (Contributed by Stefan O'Rear, 21-Mar-2015.) (Revised by AV, 8-Aug-2019.) (Proof shortened by AV, 18-Oct-2019.)
Hypotheses
Ref Expression
mptscmfsuppd.b  |-  B  =  ( Base `  P
)
mptscmfsuppd.s  |-  S  =  (Scalar `  P )
mptscmfsuppd.n  |-  .x.  =  ( .s `  P )
mptscmfsuppd.p  |-  ( ph  ->  P  e.  LMod )
mptscmfsuppd.x  |-  ( ph  ->  X  e.  V )
mptscmfsuppd.z  |-  ( (
ph  /\  k  e.  X )  ->  Z  e.  B )
mptscmfsuppd.a  |-  ( ph  ->  A : X --> Y )
mptscmfsuppd.f  |-  ( ph  ->  A finSupp  ( 0g `  S ) )
Assertion
Ref Expression
mptscmfsuppd  |-  ( ph  ->  ( k  e.  X  |->  ( ( A `  k )  .x.  Z
) ) finSupp  ( 0g `  P ) )
Distinct variable groups:    A, k    B, k    P, k    S, k   
k, X    .x. , k    ph, k
Allowed substitution hints:    V( k)    Y( k)    Z( k)

Proof of Theorem mptscmfsuppd
StepHypRef Expression
1 mptscmfsuppd.x . 2  |-  ( ph  ->  X  e.  V )
2 mptscmfsuppd.p . 2  |-  ( ph  ->  P  e.  LMod )
3 mptscmfsuppd.s . . 3  |-  S  =  (Scalar `  P )
43a1i 11 . 2  |-  ( ph  ->  S  =  (Scalar `  P ) )
5 mptscmfsuppd.b . 2  |-  B  =  ( Base `  P
)
6 fvex 5874 . . 3  |-  ( A `
 k )  e. 
_V
76a1i 11 . 2  |-  ( (
ph  /\  k  e.  X )  ->  ( A `  k )  e.  _V )
8 mptscmfsuppd.z . 2  |-  ( (
ph  /\  k  e.  X )  ->  Z  e.  B )
9 eqid 2467 . 2  |-  ( 0g
`  P )  =  ( 0g `  P
)
10 eqid 2467 . 2  |-  ( 0g
`  S )  =  ( 0g `  S
)
11 mptscmfsuppd.n . 2  |-  .x.  =  ( .s `  P )
12 mptscmfsuppd.f . . 3  |-  ( ph  ->  A finSupp  ( 0g `  S ) )
13 mptscmfsuppd.a . . . . . . 7  |-  ( ph  ->  A : X --> Y )
14 ffn 5729 . . . . . . 7  |-  ( A : X --> Y  ->  A  Fn  X )
1513, 14syl 16 . . . . . 6  |-  ( ph  ->  A  Fn  X )
16 dffn5 5911 . . . . . 6  |-  ( A  Fn  X  <->  A  =  ( k  e.  X  |->  ( A `  k
) ) )
1715, 16sylib 196 . . . . 5  |-  ( ph  ->  A  =  ( k  e.  X  |->  ( A `
 k ) ) )
1817eqcomd 2475 . . . 4  |-  ( ph  ->  ( k  e.  X  |->  ( A `  k
) )  =  A )
1918breq1d 4457 . . 3  |-  ( ph  ->  ( ( k  e.  X  |->  ( A `  k ) ) finSupp  ( 0g `  S )  <->  A finSupp  ( 0g
`  S ) ) )
2012, 19mpbird 232 . 2  |-  ( ph  ->  ( k  e.  X  |->  ( A `  k
) ) finSupp  ( 0g `  S ) )
211, 2, 4, 5, 7, 8, 9, 10, 11, 20mptscmfsupp0 17356 1  |-  ( ph  ->  ( k  e.  X  |->  ( ( A `  k )  .x.  Z
) ) finSupp  ( 0g `  P ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   _Vcvv 3113   class class class wbr 4447    |-> cmpt 4505    Fn wfn 5581   -->wf 5582   ` cfv 5586  (class class class)co 6282   finSupp cfsupp 7825   Basecbs 14483  Scalarcsca 14551   .scvsca 14552   0gc0g 14688   LModclmod 17292
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 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  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-rmo 2822  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-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  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 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-supp 6899  df-er 7308  df-en 7514  df-fin 7517  df-fsupp 7826  df-0g 14690  df-mnd 15725  df-grp 15855  df-rng 16985  df-lmod 17294
This theorem is referenced by:  ply1coefsupp  18104
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