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Theorem mptscmfsuppd 17136
Description: A function mapping to a scalar product in which one factor is finitely supported is finitely supported. Formerly part of proof for ply1coe 17872. (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 5810 . . 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 2454 . 2  |-  ( 0g
`  P )  =  ( 0g `  P
)
10 eqid 2454 . 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 5668 . . . . . . 7  |-  ( A : X --> Y  ->  A  Fn  X )
1513, 14syl 16 . . . . . 6  |-  ( ph  ->  A  Fn  X )
16 dffn5 5847 . . . . . 6  |-  ( A  Fn  X  <->  A  =  ( k  e.  X  |->  ( A `  k
) ) )
1715, 16sylib 196 . . . . 5  |-  ( ph  ->  A  =  ( k  e.  X  |->  ( A `
 k ) ) )
1817eqcomd 2462 . . . 4  |-  ( ph  ->  ( k  e.  X  |->  ( A `  k
) )  =  A )
1918breq1d 4411 . . 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 17135 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 1370    e. wcel 1758   _Vcvv 3078   class class class wbr 4401    |-> cmpt 4459    Fn wfn 5522   -->wf 5523   ` cfv 5527  (class class class)co 6201   finSupp cfsupp 7732   Basecbs 14293  Scalarcsca 14361   .scvsca 14362   0gc0g 14498   LModclmod 17072
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-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-tp 3991  df-op 3993  df-uni 4201  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-tr 4495  df-eprel 4741  df-id 4745  df-po 4750  df-so 4751  df-fr 4788  df-we 4790  df-ord 4831  df-on 4832  df-lim 4833  df-suc 4834  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-riota 6162  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-om 6588  df-supp 6802  df-er 7212  df-en 7422  df-fin 7425  df-fsupp 7733  df-0g 14500  df-mnd 15535  df-grp 15665  df-rng 16771  df-lmod 17074
This theorem is referenced by:  ply1coefsupp  17871
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