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Theorem mptscmfsuppd 17895
Description: A function mapping to a scalar product in which one factor is finitely supported is finitely supported. Formerly part of proof for ply1coe 18655. (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 5858 . . 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 2402 . 2  |-  ( 0g
`  P )  =  ( 0g `  P
)
10 eqid 2402 . 2  |-  ( 0g
`  S )  =  ( 0g `  S
)
11 mptscmfsuppd.n . 2  |-  .x.  =  ( .s `  P )
12 mptscmfsuppd.a . . . 4  |-  ( ph  ->  A : X --> Y )
1312feqmptd 5901 . . 3  |-  ( ph  ->  A  =  ( k  e.  X  |->  ( A `
 k ) ) )
14 mptscmfsuppd.f . . 3  |-  ( ph  ->  A finSupp  ( 0g `  S ) )
1513, 14eqbrtrrd 4416 . 2  |-  ( ph  ->  ( k  e.  X  |->  ( A `  k
) ) finSupp  ( 0g `  S ) )
161, 2, 4, 5, 7, 8, 9, 10, 11, 15mptscmfsupp0 17894 1  |-  ( ph  ->  ( k  e.  X  |->  ( ( A `  k )  .x.  Z
) ) finSupp  ( 0g `  P ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1405    e. wcel 1842   _Vcvv 3058   class class class wbr 4394    |-> cmpt 4452   -->wf 5564   ` cfv 5568  (class class class)co 6277   finSupp cfsupp 7862   Basecbs 14839  Scalarcsca 14910   .scvsca 14911   0gc0g 15052   LModclmod 17830
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-fal 1411  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-we 4783  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-ord 5412  df-on 5413  df-lim 5414  df-suc 5415  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6683  df-supp 6902  df-er 7347  df-en 7554  df-fin 7557  df-fsupp 7863  df-0g 15054  df-mgm 16194  df-sgrp 16233  df-mnd 16243  df-grp 16379  df-ring 17518  df-lmod 17832
This theorem is referenced by:  ply1coefsupp  18654
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