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Theorem mptscmfsupp0 17137
Description: A mapping to a scalar product is finitely supported if the mapping to the scalar is finitely supported. (Contributed by AV, 5-Oct-2019.)
Hypotheses
Ref Expression
mptscmfsupp0.d  |-  ( ph  ->  D  e.  V )
mptscmfsupp0.q  |-  ( ph  ->  Q  e.  LMod )
mptscmfsupp0.r  |-  ( ph  ->  R  =  (Scalar `  Q ) )
mptscmfsupp0.k  |-  K  =  ( Base `  Q
)
mptscmfsupp0.s  |-  ( (
ph  /\  k  e.  D )  ->  S  e.  B )
mptscmfsupp0.w  |-  ( (
ph  /\  k  e.  D )  ->  W  e.  K )
mptscmfsupp0.0  |-  .0.  =  ( 0g `  Q )
mptscmfsupp0.z  |-  Z  =  ( 0g `  R
)
mptscmfsupp0.m  |-  .*  =  ( .s `  Q )
mptscmfsupp0.f  |-  ( ph  ->  ( k  e.  D  |->  S ) finSupp  Z )
Assertion
Ref Expression
mptscmfsupp0  |-  ( ph  ->  ( k  e.  D  |->  ( S  .*  W
) ) finSupp  .0.  )
Distinct variable groups:    B, k    D, k    k, K    ph, k    .* , k
Allowed substitution hints:    Q( k)    R( k)    S( k)    V( k)    W( k)    .0. ( k)    Z( k)

Proof of Theorem mptscmfsupp0
Dummy variable  d is distinct from all other variables.
StepHypRef Expression
1 mptscmfsupp0.d . . 3  |-  ( ph  ->  D  e.  V )
2 mptexg 6059 . . 3  |-  ( D  e.  V  ->  (
k  e.  D  |->  ( S  .*  W ) )  e.  _V )
31, 2syl 16 . 2  |-  ( ph  ->  ( k  e.  D  |->  ( S  .*  W
) )  e.  _V )
4 funmpt 5565 . . 3  |-  Fun  (
k  e.  D  |->  ( S  .*  W ) )
54a1i 11 . 2  |-  ( ph  ->  Fun  ( k  e.  D  |->  ( S  .*  W ) ) )
6 mptscmfsupp0.0 . . . 4  |-  .0.  =  ( 0g `  Q )
7 fvex 5812 . . . 4  |-  ( 0g
`  Q )  e. 
_V
86, 7eqeltri 2538 . . 3  |-  .0.  e.  _V
98a1i 11 . 2  |-  ( ph  ->  .0.  e.  _V )
10 mptscmfsupp0.f . . 3  |-  ( ph  ->  ( k  e.  D  |->  S ) finSupp  Z )
1110fsuppimpd 7741 . 2  |-  ( ph  ->  ( ( k  e.  D  |->  S ) supp  Z
)  e.  Fin )
12 simpr 461 . . . . . . . 8  |-  ( (
ph  /\  d  e.  D )  ->  d  e.  D )
13 mptscmfsupp0.s . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  D )  ->  S  e.  B )
1413ralrimiva 2830 . . . . . . . . . 10  |-  ( ph  ->  A. k  e.  D  S  e.  B )
1514adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  d  e.  D )  ->  A. k  e.  D  S  e.  B )
16 rspcsbela 3816 . . . . . . . . 9  |-  ( ( d  e.  D  /\  A. k  e.  D  S  e.  B )  ->  [_ d  /  k ]_ S  e.  B )
1712, 15, 16syl2anc 661 . . . . . . . 8  |-  ( (
ph  /\  d  e.  D )  ->  [_ d  /  k ]_ S  e.  B )
18 eqid 2454 . . . . . . . . 9  |-  ( k  e.  D  |->  S )  =  ( k  e.  D  |->  S )
1918fvmpts 5888 . . . . . . . 8  |-  ( ( d  e.  D  /\  [_ d  /  k ]_ S  e.  B )  ->  ( ( k  e.  D  |->  S ) `  d )  =  [_ d  /  k ]_ S
)
2012, 17, 19syl2anc 661 . . . . . . 7  |-  ( (
ph  /\  d  e.  D )  ->  (
( k  e.  D  |->  S ) `  d
)  =  [_ d  /  k ]_ S
)
2120eqeq1d 2456 . . . . . 6  |-  ( (
ph  /\  d  e.  D )  ->  (
( ( k  e.  D  |->  S ) `  d )  =  Z  <->  [_ d  /  k ]_ S  =  Z
) )
22 oveq1 6210 . . . . . . . . 9  |-  ( [_ d  /  k ]_ S  =  Z  ->  ( [_ d  /  k ]_ S  .*  [_ d  /  k ]_ W )  =  ( Z  .*  [_ d  /  k ]_ W
) )
23 mptscmfsupp0.z . . . . . . . . . . . 12  |-  Z  =  ( 0g `  R
)
24 mptscmfsupp0.r . . . . . . . . . . . . . 14  |-  ( ph  ->  R  =  (Scalar `  Q ) )
2524adantr 465 . . . . . . . . . . . . 13  |-  ( (
ph  /\  d  e.  D )  ->  R  =  (Scalar `  Q )
)
2625fveq2d 5806 . . . . . . . . . . . 12  |-  ( (
ph  /\  d  e.  D )  ->  ( 0g `  R )  =  ( 0g `  (Scalar `  Q ) ) )
2723, 26syl5eq 2507 . . . . . . . . . . 11  |-  ( (
ph  /\  d  e.  D )  ->  Z  =  ( 0g `  (Scalar `  Q ) ) )
2827oveq1d 6218 . . . . . . . . . 10  |-  ( (
ph  /\  d  e.  D )  ->  ( Z  .*  [_ d  / 
k ]_ W )  =  ( ( 0g `  (Scalar `  Q ) )  .*  [_ d  / 
k ]_ W ) )
29 mptscmfsupp0.q . . . . . . . . . . . 12  |-  ( ph  ->  Q  e.  LMod )
3029adantr 465 . . . . . . . . . . 11  |-  ( (
ph  /\  d  e.  D )  ->  Q  e.  LMod )
31 mptscmfsupp0.w . . . . . . . . . . . . . 14  |-  ( (
ph  /\  k  e.  D )  ->  W  e.  K )
3231ralrimiva 2830 . . . . . . . . . . . . 13  |-  ( ph  ->  A. k  e.  D  W  e.  K )
3332adantr 465 . . . . . . . . . . . 12  |-  ( (
ph  /\  d  e.  D )  ->  A. k  e.  D  W  e.  K )
34 rspcsbela 3816 . . . . . . . . . . . 12  |-  ( ( d  e.  D  /\  A. k  e.  D  W  e.  K )  ->  [_ d  /  k ]_ W  e.  K )
3512, 33, 34syl2anc 661 . . . . . . . . . . 11  |-  ( (
ph  /\  d  e.  D )  ->  [_ d  /  k ]_ W  e.  K )
36 mptscmfsupp0.k . . . . . . . . . . . 12  |-  K  =  ( Base `  Q
)
37 eqid 2454 . . . . . . . . . . . 12  |-  (Scalar `  Q )  =  (Scalar `  Q )
38 mptscmfsupp0.m . . . . . . . . . . . 12  |-  .*  =  ( .s `  Q )
39 eqid 2454 . . . . . . . . . . . 12  |-  ( 0g
`  (Scalar `  Q )
)  =  ( 0g
`  (Scalar `  Q )
)
4036, 37, 38, 39, 6lmod0vs 17107 . . . . . . . . . . 11  |-  ( ( Q  e.  LMod  /\  [_ d  /  k ]_ W  e.  K )  ->  (
( 0g `  (Scalar `  Q ) )  .* 
[_ d  /  k ]_ W )  =  .0.  )
4130, 35, 40syl2anc 661 . . . . . . . . . 10  |-  ( (
ph  /\  d  e.  D )  ->  (
( 0g `  (Scalar `  Q ) )  .* 
[_ d  /  k ]_ W )  =  .0.  )
4228, 41eqtrd 2495 . . . . . . . . 9  |-  ( (
ph  /\  d  e.  D )  ->  ( Z  .*  [_ d  / 
k ]_ W )  =  .0.  )
4322, 42sylan9eqr 2517 . . . . . . . 8  |-  ( ( ( ph  /\  d  e.  D )  /\  [_ d  /  k ]_ S  =  Z )  ->  ( [_ d  /  k ]_ S  .*  [_ d  /  k ]_ W
)  =  .0.  )
44 csbov12g 6237 . . . . . . . . . . . . . 14  |-  ( d  e.  D  ->  [_ d  /  k ]_ ( S  .*  W )  =  ( [_ d  / 
k ]_ S  .*  [_ d  /  k ]_ W
) )
4544adantl 466 . . . . . . . . . . . . 13  |-  ( (
ph  /\  d  e.  D )  ->  [_ d  /  k ]_ ( S  .*  W )  =  ( [_ d  / 
k ]_ S  .*  [_ d  /  k ]_ W
) )
46 ovex 6228 . . . . . . . . . . . . 13  |-  ( [_ d  /  k ]_ S  .*  [_ d  /  k ]_ W )  e.  _V
4745, 46syl6eqel 2550 . . . . . . . . . . . 12  |-  ( (
ph  /\  d  e.  D )  ->  [_ d  /  k ]_ ( S  .*  W )  e. 
_V )
48 eqid 2454 . . . . . . . . . . . . 13  |-  ( k  e.  D  |->  ( S  .*  W ) )  =  ( k  e.  D  |->  ( S  .*  W ) )
4948fvmpts 5888 . . . . . . . . . . . 12  |-  ( ( d  e.  D  /\  [_ d  /  k ]_ ( S  .*  W
)  e.  _V )  ->  ( ( k  e.  D  |->  ( S  .*  W ) ) `  d )  =  [_ d  /  k ]_ ( S  .*  W ) )
5012, 47, 49syl2anc 661 . . . . . . . . . . 11  |-  ( (
ph  /\  d  e.  D )  ->  (
( k  e.  D  |->  ( S  .*  W
) ) `  d
)  =  [_ d  /  k ]_ ( S  .*  W ) )
5150, 45eqtrd 2495 . . . . . . . . . 10  |-  ( (
ph  /\  d  e.  D )  ->  (
( k  e.  D  |->  ( S  .*  W
) ) `  d
)  =  ( [_ d  /  k ]_ S  .*  [_ d  /  k ]_ W ) )
5251eqeq1d 2456 . . . . . . . . 9  |-  ( (
ph  /\  d  e.  D )  ->  (
( ( k  e.  D  |->  ( S  .*  W ) ) `  d )  =  .0.  <->  (
[_ d  /  k ]_ S  .*  [_ d  /  k ]_ W
)  =  .0.  )
)
5352adantr 465 . . . . . . . 8  |-  ( ( ( ph  /\  d  e.  D )  /\  [_ d  /  k ]_ S  =  Z )  ->  (
( ( k  e.  D  |->  ( S  .*  W ) ) `  d )  =  .0.  <->  (
[_ d  /  k ]_ S  .*  [_ d  /  k ]_ W
)  =  .0.  )
)
5443, 53mpbird 232 . . . . . . 7  |-  ( ( ( ph  /\  d  e.  D )  /\  [_ d  /  k ]_ S  =  Z )  ->  (
( k  e.  D  |->  ( S  .*  W
) ) `  d
)  =  .0.  )
5554ex 434 . . . . . 6  |-  ( (
ph  /\  d  e.  D )  ->  ( [_ d  /  k ]_ S  =  Z  ->  ( ( k  e.  D  |->  ( S  .*  W ) ) `  d )  =  .0.  ) )
5621, 55sylbid 215 . . . . 5  |-  ( (
ph  /\  d  e.  D )  ->  (
( ( k  e.  D  |->  S ) `  d )  =  Z  ->  ( ( k  e.  D  |->  ( S  .*  W ) ) `
 d )  =  .0.  ) )
5756necon3d 2676 . . . 4  |-  ( (
ph  /\  d  e.  D )  ->  (
( ( k  e.  D  |->  ( S  .*  W ) ) `  d )  =/=  .0.  ->  ( ( k  e.  D  |->  S ) `  d )  =/=  Z
) )
5857ss2rabdv 3544 . . 3  |-  ( ph  ->  { d  e.  D  |  ( ( k  e.  D  |->  ( S  .*  W ) ) `
 d )  =/= 
.0.  }  C_  { d  e.  D  |  ( ( k  e.  D  |->  S ) `  d
)  =/=  Z }
)
59 ovex 6228 . . . . . 6  |-  ( S  .*  W )  e. 
_V
6059rgenw 2901 . . . . 5  |-  A. k  e.  D  ( S  .*  W )  e.  _V
6148fnmpt 5648 . . . . 5  |-  ( A. k  e.  D  ( S  .*  W )  e. 
_V  ->  ( k  e.  D  |->  ( S  .*  W ) )  Fn  D )
6260, 61mp1i 12 . . . 4  |-  ( ph  ->  ( k  e.  D  |->  ( S  .*  W
) )  Fn  D
)
63 suppvalfn 6810 . . . 4  |-  ( ( ( k  e.  D  |->  ( S  .*  W
) )  Fn  D  /\  D  e.  V  /\  .0.  e.  _V )  ->  ( ( k  e.  D  |->  ( S  .*  W ) ) supp  .0.  )  =  { d  e.  D  |  (
( k  e.  D  |->  ( S  .*  W
) ) `  d
)  =/=  .0.  }
)
6462, 1, 9, 63syl3anc 1219 . . 3  |-  ( ph  ->  ( ( k  e.  D  |->  ( S  .*  W ) ) supp  .0.  )  =  { d  e.  D  |  (
( k  e.  D  |->  ( S  .*  W
) ) `  d
)  =/=  .0.  }
)
6518fnmpt 5648 . . . . 5  |-  ( A. k  e.  D  S  e.  B  ->  ( k  e.  D  |->  S )  Fn  D )
6614, 65syl 16 . . . 4  |-  ( ph  ->  ( k  e.  D  |->  S )  Fn  D
)
67 fvex 5812 . . . . . 6  |-  ( 0g
`  R )  e. 
_V
6823, 67eqeltri 2538 . . . . 5  |-  Z  e. 
_V
6968a1i 11 . . . 4  |-  ( ph  ->  Z  e.  _V )
70 suppvalfn 6810 . . . 4  |-  ( ( ( k  e.  D  |->  S )  Fn  D  /\  D  e.  V  /\  Z  e.  _V )  ->  ( ( k  e.  D  |->  S ) supp 
Z )  =  {
d  e.  D  | 
( ( k  e.  D  |->  S ) `  d )  =/=  Z } )
7166, 1, 69, 70syl3anc 1219 . . 3  |-  ( ph  ->  ( ( k  e.  D  |->  S ) supp  Z
)  =  { d  e.  D  |  ( ( k  e.  D  |->  S ) `  d
)  =/=  Z }
)
7258, 64, 713sstr4d 3510 . 2  |-  ( ph  ->  ( ( k  e.  D  |->  ( S  .*  W ) ) supp  .0.  )  C_  ( ( k  e.  D  |->  S ) supp 
Z ) )
73 suppssfifsupp 7749 . 2  |-  ( ( ( ( k  e.  D  |->  ( S  .*  W ) )  e. 
_V  /\  Fun  ( k  e.  D  |->  ( S  .*  W ) )  /\  .0.  e.  _V )  /\  ( ( ( k  e.  D  |->  S ) supp  Z )  e. 
Fin  /\  ( (
k  e.  D  |->  ( S  .*  W ) ) supp  .0.  )  C_  ( ( k  e.  D  |->  S ) supp  Z
) ) )  -> 
( k  e.  D  |->  ( S  .*  W
) ) finSupp  .0.  )
743, 5, 9, 11, 72, 73syl32anc 1227 1  |-  ( ph  ->  ( k  e.  D  |->  ( S  .*  W
) ) finSupp  .0.  )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758    =/= wne 2648   A.wral 2799   {crab 2803   _Vcvv 3078   [_csb 3398    C_ wss 3439   class class class wbr 4403    |-> cmpt 4461   Fun wfun 5523    Fn wfn 5524   ` cfv 5529  (class class class)co 6203   supp csupp 6803   Fincfn 7423   finSupp cfsupp 7734   Basecbs 14295  Scalarcsca 14363   .scvsca 14364   0gc0g 14500   LModclmod 17074
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 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485
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 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-supp 6804  df-er 7214  df-en 7424  df-fin 7427  df-fsupp 7735  df-0g 14502  df-mnd 15537  df-grp 15667  df-rng 16773  df-lmod 17076
This theorem is referenced by:  mptscmfsuppd  17138  gsumsmonply1  31015  pm2mpcl  31304  mply1topmatcllem  31310  mp2pm2mplem2  31314  mp2pm2mplem5  31317  pm2mpghmlem2  31319  chcoeffeqlem  31392
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