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Theorem mptscmfsupp0 18164
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 6121 . . 3  |-  ( D  e.  V  ->  (
k  e.  D  |->  ( S  .*  W ) )  e.  _V )
31, 2syl 17 . 2  |-  ( ph  ->  ( k  e.  D  |->  ( S  .*  W
) )  e.  _V )
4 funmpt 5597 . . 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 5858 . . . 4  |-  ( 0g
`  Q )  e. 
_V
86, 7eqeltri 2526 . . 3  |-  .0.  e.  _V
98a1i 11 . 2  |-  ( ph  ->  .0.  e.  _V )
10 mptscmfsupp0.f . . 3  |-  ( ph  ->  ( k  e.  D  |->  S ) finSupp  Z )
1110fsuppimpd 7877 . 2  |-  ( ph  ->  ( ( k  e.  D  |->  S ) supp  Z
)  e.  Fin )
12 simpr 467 . . . . . . . 8  |-  ( (
ph  /\  d  e.  D )  ->  d  e.  D )
13 mptscmfsupp0.s . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  D )  ->  S  e.  B )
1413ralrimiva 2790 . . . . . . . . . 10  |-  ( ph  ->  A. k  e.  D  S  e.  B )
1514adantr 471 . . . . . . . . 9  |-  ( (
ph  /\  d  e.  D )  ->  A. k  e.  D  S  e.  B )
16 rspcsbela 3763 . . . . . . . . 9  |-  ( ( d  e.  D  /\  A. k  e.  D  S  e.  B )  ->  [_ d  /  k ]_ S  e.  B )
1712, 15, 16syl2anc 671 . . . . . . . 8  |-  ( (
ph  /\  d  e.  D )  ->  [_ d  /  k ]_ S  e.  B )
18 eqid 2452 . . . . . . . . 9  |-  ( k  e.  D  |->  S )  =  ( k  e.  D  |->  S )
1918fvmpts 5935 . . . . . . . 8  |-  ( ( d  e.  D  /\  [_ d  /  k ]_ S  e.  B )  ->  ( ( k  e.  D  |->  S ) `  d )  =  [_ d  /  k ]_ S
)
2012, 17, 19syl2anc 671 . . . . . . 7  |-  ( (
ph  /\  d  e.  D )  ->  (
( k  e.  D  |->  S ) `  d
)  =  [_ d  /  k ]_ S
)
2120eqeq1d 2454 . . . . . 6  |-  ( (
ph  /\  d  e.  D )  ->  (
( ( k  e.  D  |->  S ) `  d )  =  Z  <->  [_ d  /  k ]_ S  =  Z
) )
22 oveq1 6283 . . . . . . . . 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 471 . . . . . . . . . . . . 13  |-  ( (
ph  /\  d  e.  D )  ->  R  =  (Scalar `  Q )
)
2625fveq2d 5852 . . . . . . . . . . . 12  |-  ( (
ph  /\  d  e.  D )  ->  ( 0g `  R )  =  ( 0g `  (Scalar `  Q ) ) )
2723, 26syl5eq 2498 . . . . . . . . . . 11  |-  ( (
ph  /\  d  e.  D )  ->  Z  =  ( 0g `  (Scalar `  Q ) ) )
2827oveq1d 6291 . . . . . . . . . 10  |-  ( (
ph  /\  d  e.  D )  ->  ( Z  .*  [_ d  / 
k ]_ W )  =  ( ( 0g `  (Scalar `  Q ) )  .*  [_ d  / 
k ]_ W ) )
29 mptscmfsupp0.q . . . . . . . . . . . 12  |-  ( ph  ->  Q  e.  LMod )
3029adantr 471 . . . . . . . . . . 11  |-  ( (
ph  /\  d  e.  D )  ->  Q  e.  LMod )
31 mptscmfsupp0.w . . . . . . . . . . . . . 14  |-  ( (
ph  /\  k  e.  D )  ->  W  e.  K )
3231ralrimiva 2790 . . . . . . . . . . . . 13  |-  ( ph  ->  A. k  e.  D  W  e.  K )
3332adantr 471 . . . . . . . . . . . 12  |-  ( (
ph  /\  d  e.  D )  ->  A. k  e.  D  W  e.  K )
34 rspcsbela 3763 . . . . . . . . . . . 12  |-  ( ( d  e.  D  /\  A. k  e.  D  W  e.  K )  ->  [_ d  /  k ]_ W  e.  K )
3512, 33, 34syl2anc 671 . . . . . . . . . . 11  |-  ( (
ph  /\  d  e.  D )  ->  [_ d  /  k ]_ W  e.  K )
36 mptscmfsupp0.k . . . . . . . . . . . 12  |-  K  =  ( Base `  Q
)
37 eqid 2452 . . . . . . . . . . . 12  |-  (Scalar `  Q )  =  (Scalar `  Q )
38 mptscmfsupp0.m . . . . . . . . . . . 12  |-  .*  =  ( .s `  Q )
39 eqid 2452 . . . . . . . . . . . 12  |-  ( 0g
`  (Scalar `  Q )
)  =  ( 0g
`  (Scalar `  Q )
)
4036, 37, 38, 39, 6lmod0vs 18135 . . . . . . . . . . 11  |-  ( ( Q  e.  LMod  /\  [_ d  /  k ]_ W  e.  K )  ->  (
( 0g `  (Scalar `  Q ) )  .* 
[_ d  /  k ]_ W )  =  .0.  )
4130, 35, 40syl2anc 671 . . . . . . . . . 10  |-  ( (
ph  /\  d  e.  D )  ->  (
( 0g `  (Scalar `  Q ) )  .* 
[_ d  /  k ]_ W )  =  .0.  )
4228, 41eqtrd 2486 . . . . . . . . 9  |-  ( (
ph  /\  d  e.  D )  ->  ( Z  .*  [_ d  / 
k ]_ W )  =  .0.  )
4322, 42sylan9eqr 2508 . . . . . . . 8  |-  ( ( ( ph  /\  d  e.  D )  /\  [_ d  /  k ]_ S  =  Z )  ->  ( [_ d  /  k ]_ S  .*  [_ d  /  k ]_ W
)  =  .0.  )
44 csbov12g 6312 . . . . . . . . . . . . . 14  |-  ( d  e.  D  ->  [_ d  /  k ]_ ( S  .*  W )  =  ( [_ d  / 
k ]_ S  .*  [_ d  /  k ]_ W
) )
4544adantl 472 . . . . . . . . . . . . 13  |-  ( (
ph  /\  d  e.  D )  ->  [_ d  /  k ]_ ( S  .*  W )  =  ( [_ d  / 
k ]_ S  .*  [_ d  /  k ]_ W
) )
46 ovex 6304 . . . . . . . . . . . . 13  |-  ( [_ d  /  k ]_ S  .*  [_ d  /  k ]_ W )  e.  _V
4745, 46syl6eqel 2538 . . . . . . . . . . . 12  |-  ( (
ph  /\  d  e.  D )  ->  [_ d  /  k ]_ ( S  .*  W )  e. 
_V )
48 eqid 2452 . . . . . . . . . . . . 13  |-  ( k  e.  D  |->  ( S  .*  W ) )  =  ( k  e.  D  |->  ( S  .*  W ) )
4948fvmpts 5935 . . . . . . . . . . . 12  |-  ( ( d  e.  D  /\  [_ d  /  k ]_ ( S  .*  W
)  e.  _V )  ->  ( ( k  e.  D  |->  ( S  .*  W ) ) `  d )  =  [_ d  /  k ]_ ( S  .*  W ) )
5012, 47, 49syl2anc 671 . . . . . . . . . . 11  |-  ( (
ph  /\  d  e.  D )  ->  (
( k  e.  D  |->  ( S  .*  W
) ) `  d
)  =  [_ d  /  k ]_ ( S  .*  W ) )
5150, 45eqtrd 2486 . . . . . . . . . 10  |-  ( (
ph  /\  d  e.  D )  ->  (
( k  e.  D  |->  ( S  .*  W
) ) `  d
)  =  ( [_ d  /  k ]_ S  .*  [_ d  /  k ]_ W ) )
5251eqeq1d 2454 . . . . . . . . 9  |-  ( (
ph  /\  d  e.  D )  ->  (
( ( k  e.  D  |->  ( S  .*  W ) ) `  d )  =  .0.  <->  (
[_ d  /  k ]_ S  .*  [_ d  /  k ]_ W
)  =  .0.  )
)
5352adantr 471 . . . . . . . 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 240 . . . . . . 7  |-  ( ( ( ph  /\  d  e.  D )  /\  [_ d  /  k ]_ S  =  Z )  ->  (
( k  e.  D  |->  ( S  .*  W
) ) `  d
)  =  .0.  )
5554ex 440 . . . . . 6  |-  ( (
ph  /\  d  e.  D )  ->  ( [_ d  /  k ]_ S  =  Z  ->  ( ( k  e.  D  |->  ( S  .*  W ) ) `  d )  =  .0.  ) )
5621, 55sylbid 223 . . . . 5  |-  ( (
ph  /\  d  e.  D )  ->  (
( ( k  e.  D  |->  S ) `  d )  =  Z  ->  ( ( k  e.  D  |->  ( S  .*  W ) ) `
 d )  =  .0.  ) )
5756necon3d 2645 . . . 4  |-  ( (
ph  /\  d  e.  D )  ->  (
( ( k  e.  D  |->  ( S  .*  W ) ) `  d )  =/=  .0.  ->  ( ( k  e.  D  |->  S ) `  d )  =/=  Z
) )
5857ss2rabdv 3478 . . 3  |-  ( ph  ->  { d  e.  D  |  ( ( k  e.  D  |->  ( S  .*  W ) ) `
 d )  =/= 
.0.  }  C_  { d  e.  D  |  ( ( k  e.  D  |->  S ) `  d
)  =/=  Z }
)
59 ovex 6304 . . . . . 6  |-  ( S  .*  W )  e. 
_V
6059rgenw 2749 . . . . 5  |-  A. k  e.  D  ( S  .*  W )  e.  _V
6148fnmpt 5686 . . . . 5  |-  ( A. k  e.  D  ( S  .*  W )  e. 
_V  ->  ( k  e.  D  |->  ( S  .*  W ) )  Fn  D )
6260, 61mp1i 13 . . . 4  |-  ( ph  ->  ( k  e.  D  |->  ( S  .*  W
) )  Fn  D
)
63 suppvalfn 6909 . . . 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 1271 . . 3  |-  ( ph  ->  ( ( k  e.  D  |->  ( S  .*  W ) ) supp  .0.  )  =  { d  e.  D  |  (
( k  e.  D  |->  ( S  .*  W
) ) `  d
)  =/=  .0.  }
)
6518fnmpt 5686 . . . . 5  |-  ( A. k  e.  D  S  e.  B  ->  ( k  e.  D  |->  S )  Fn  D )
6614, 65syl 17 . . . 4  |-  ( ph  ->  ( k  e.  D  |->  S )  Fn  D
)
67 fvex 5858 . . . . . 6  |-  ( 0g
`  R )  e. 
_V
6823, 67eqeltri 2526 . . . . 5  |-  Z  e. 
_V
6968a1i 11 . . . 4  |-  ( ph  ->  Z  e.  _V )
70 suppvalfn 6909 . . . 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 1271 . . 3  |-  ( ph  ->  ( ( k  e.  D  |->  S ) supp  Z
)  =  { d  e.  D  |  ( ( k  e.  D  |->  S ) `  d
)  =/=  Z }
)
7258, 64, 713sstr4d 3443 . 2  |-  ( ph  ->  ( ( k  e.  D  |->  ( S  .*  W ) ) supp  .0.  )  C_  ( ( k  e.  D  |->  S ) supp 
Z ) )
73 suppssfifsupp 7885 . 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 1279 1  |-  ( ph  ->  ( k  e.  D  |->  ( S  .*  W
) ) finSupp  .0.  )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 375    = wceq 1448    e. wcel 1891    =/= wne 2622   A.wral 2737   {crab 2741   _Vcvv 3013   [_csb 3331    C_ wss 3372   class class class wbr 4374    |-> cmpt 4433   Fun wfun 5555    Fn wfn 5556   ` cfv 5561  (class class class)co 6276   supp csupp 6902   Fincfn 7556   finSupp cfsupp 7870   Basecbs 15132  Scalarcsca 15204   .scvsca 15205   0gc0g 15349   LModclmod 18102
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1673  ax-4 1686  ax-5 1762  ax-6 1809  ax-7 1855  ax-8 1893  ax-9 1900  ax-10 1919  ax-11 1924  ax-12 1937  ax-13 2092  ax-ext 2432  ax-rep 4487  ax-sep 4497  ax-nul 4506  ax-pow 4554  ax-pr 4612  ax-un 6571
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 987  df-3an 988  df-tru 1451  df-fal 1454  df-ex 1668  df-nf 1672  df-sb 1802  df-eu 2304  df-mo 2305  df-clab 2439  df-cleq 2445  df-clel 2448  df-nfc 2582  df-ne 2624  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3015  df-sbc 3236  df-csb 3332  df-dif 3375  df-un 3377  df-in 3379  df-ss 3386  df-pss 3388  df-nul 3700  df-if 3850  df-pw 3921  df-sn 3937  df-pr 3939  df-tp 3941  df-op 3943  df-uni 4169  df-iun 4250  df-br 4375  df-opab 4434  df-mpt 4435  df-tr 4470  df-eprel 4723  df-id 4727  df-po 4733  df-so 4734  df-fr 4771  df-we 4773  df-xp 4818  df-rel 4819  df-cnv 4820  df-co 4821  df-dm 4822  df-rn 4823  df-res 4824  df-ima 4825  df-ord 5405  df-on 5406  df-lim 5407  df-suc 5408  df-iota 5525  df-fun 5563  df-fn 5564  df-f 5565  df-f1 5566  df-fo 5567  df-f1o 5568  df-fv 5569  df-riota 6238  df-ov 6279  df-oprab 6280  df-mpt2 6281  df-om 6681  df-supp 6903  df-er 7350  df-en 7557  df-fin 7560  df-fsupp 7871  df-0g 15351  df-mgm 16499  df-sgrp 16538  df-mnd 16548  df-grp 16684  df-ring 17793  df-lmod 18104
This theorem is referenced by:  mptscmfsuppd  18165  gsumsmonply1  18908  pm2mpcl  19832  mply1topmatcllem  19838  mp2pm2mplem5  19845  pm2mpghmlem2  19847  chcoeffeqlem  19920
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