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Theorem mndpfsupp 38480
Description: A mapping of a scalar multiplication with a function of scalars is finitely supported if the function of scalars is finitely supported. (Contributed by AV, 9-Jun-2019.)
Hypothesis
Ref Expression
mndpsuppfi.r  |-  R  =  ( Base `  M
)
Assertion
Ref Expression
mndpfsupp  |-  ( ( ( M  e.  Mnd  /\  V  e.  X )  /\  ( A  e.  ( R  ^m  V
)  /\  B  e.  ( R  ^m  V ) )  /\  ( A finSupp 
( 0g `  M
)  /\  B finSupp  ( 0g
`  M ) ) )  ->  ( A  oF ( +g  `  M ) B ) finSupp 
( 0g `  M
) )

Proof of Theorem mndpfsupp
StepHypRef Expression
1 elmapfn 7479 . . . . . 6  |-  ( A  e.  ( R  ^m  V )  ->  A  Fn  V )
21adantr 463 . . . . 5  |-  ( ( A  e.  ( R  ^m  V )  /\  B  e.  ( R  ^m  V ) )  ->  A  Fn  V )
323ad2ant2 1019 . . . 4  |-  ( ( ( M  e.  Mnd  /\  V  e.  X )  /\  ( A  e.  ( R  ^m  V
)  /\  B  e.  ( R  ^m  V ) )  /\  ( A finSupp 
( 0g `  M
)  /\  B finSupp  ( 0g
`  M ) ) )  ->  A  Fn  V )
4 elmapfn 7479 . . . . . 6  |-  ( B  e.  ( R  ^m  V )  ->  B  Fn  V )
54adantl 464 . . . . 5  |-  ( ( A  e.  ( R  ^m  V )  /\  B  e.  ( R  ^m  V ) )  ->  B  Fn  V )
653ad2ant2 1019 . . . 4  |-  ( ( ( M  e.  Mnd  /\  V  e.  X )  /\  ( A  e.  ( R  ^m  V
)  /\  B  e.  ( R  ^m  V ) )  /\  ( A finSupp 
( 0g `  M
)  /\  B finSupp  ( 0g
`  M ) ) )  ->  B  Fn  V )
7 simp1r 1022 . . . 4  |-  ( ( ( M  e.  Mnd  /\  V  e.  X )  /\  ( A  e.  ( R  ^m  V
)  /\  B  e.  ( R  ^m  V ) )  /\  ( A finSupp 
( 0g `  M
)  /\  B finSupp  ( 0g
`  M ) ) )  ->  V  e.  X )
8 inidm 3648 . . . 4  |-  ( V  i^i  V )  =  V
93, 6, 7, 7, 8offn 6532 . . 3  |-  ( ( ( M  e.  Mnd  /\  V  e.  X )  /\  ( A  e.  ( R  ^m  V
)  /\  B  e.  ( R  ^m  V ) )  /\  ( A finSupp 
( 0g `  M
)  /\  B finSupp  ( 0g
`  M ) ) )  ->  ( A  oF ( +g  `  M ) B )  Fn  V )
10 fnfun 5659 . . 3  |-  ( ( A  oF ( +g  `  M ) B )  Fn  V  ->  Fun  ( A  oF ( +g  `  M
) B ) )
119, 10syl 17 . 2  |-  ( ( ( M  e.  Mnd  /\  V  e.  X )  /\  ( A  e.  ( R  ^m  V
)  /\  B  e.  ( R  ^m  V ) )  /\  ( A finSupp 
( 0g `  M
)  /\  B finSupp  ( 0g
`  M ) ) )  ->  Fun  ( A  oF ( +g  `  M ) B ) )
12 id 22 . . . . 5  |-  ( A finSupp 
( 0g `  M
)  ->  A finSupp  ( 0g
`  M ) )
1312fsuppimpd 7870 . . . 4  |-  ( A finSupp 
( 0g `  M
)  ->  ( A supp  ( 0g `  M ) )  e.  Fin )
14 id 22 . . . . 5  |-  ( B finSupp 
( 0g `  M
)  ->  B finSupp  ( 0g
`  M ) )
1514fsuppimpd 7870 . . . 4  |-  ( B finSupp 
( 0g `  M
)  ->  ( B supp  ( 0g `  M ) )  e.  Fin )
1613, 15anim12i 564 . . 3  |-  ( ( A finSupp  ( 0g `  M )  /\  B finSupp  ( 0g `  M ) )  ->  ( ( A supp  ( 0g `  M
) )  e.  Fin  /\  ( B supp  ( 0g
`  M ) )  e.  Fin ) )
17 mndpsuppfi.r . . . 4  |-  R  =  ( Base `  M
)
1817mndpsuppfi 38479 . . 3  |-  ( ( ( M  e.  Mnd  /\  V  e.  X )  /\  ( A  e.  ( R  ^m  V
)  /\  B  e.  ( R  ^m  V ) )  /\  ( ( A supp  ( 0g `  M ) )  e. 
Fin  /\  ( B supp  ( 0g `  M ) )  e.  Fin )
)  ->  ( ( A  oF ( +g  `  M ) B ) supp  ( 0g `  M
) )  e.  Fin )
1916, 18syl3an3 1265 . 2  |-  ( ( ( M  e.  Mnd  /\  V  e.  X )  /\  ( A  e.  ( R  ^m  V
)  /\  B  e.  ( R  ^m  V ) )  /\  ( A finSupp 
( 0g `  M
)  /\  B finSupp  ( 0g
`  M ) ) )  ->  ( ( A  oF ( +g  `  M ) B ) supp  ( 0g `  M
) )  e.  Fin )
20 ovex 6306 . . 3  |-  ( A  oF ( +g  `  M ) B )  e.  _V
21 fvex 5859 . . . 4  |-  ( 0g
`  M )  e. 
_V
2221a1i 11 . . 3  |-  ( ( ( M  e.  Mnd  /\  V  e.  X )  /\  ( A  e.  ( R  ^m  V
)  /\  B  e.  ( R  ^m  V ) )  /\  ( A finSupp 
( 0g `  M
)  /\  B finSupp  ( 0g
`  M ) ) )  ->  ( 0g `  M )  e.  _V )
23 isfsupp 7867 . . 3  |-  ( ( ( A  oF ( +g  `  M
) B )  e. 
_V  /\  ( 0g `  M )  e.  _V )  ->  ( ( A  oF ( +g  `  M ) B ) finSupp 
( 0g `  M
)  <->  ( Fun  ( A  oF ( +g  `  M ) B )  /\  ( ( A  oF ( +g  `  M ) B ) supp  ( 0g `  M
) )  e.  Fin ) ) )
2420, 22, 23sylancr 661 . 2  |-  ( ( ( M  e.  Mnd  /\  V  e.  X )  /\  ( A  e.  ( R  ^m  V
)  /\  B  e.  ( R  ^m  V ) )  /\  ( A finSupp 
( 0g `  M
)  /\  B finSupp  ( 0g
`  M ) ) )  ->  ( ( A  oF ( +g  `  M ) B ) finSupp 
( 0g `  M
)  <->  ( Fun  ( A  oF ( +g  `  M ) B )  /\  ( ( A  oF ( +g  `  M ) B ) supp  ( 0g `  M
) )  e.  Fin ) ) )
2511, 19, 24mpbir2and 923 1  |-  ( ( ( M  e.  Mnd  /\  V  e.  X )  /\  ( A  e.  ( R  ^m  V
)  /\  B  e.  ( R  ^m  V ) )  /\  ( A finSupp 
( 0g `  M
)  /\  B finSupp  ( 0g
`  M ) ) )  ->  ( A  oF ( +g  `  M ) B ) finSupp 
( 0g `  M
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842   _Vcvv 3059   class class class wbr 4395   Fun wfun 5563    Fn wfn 5564   ` cfv 5569  (class class class)co 6278    oFcof 6519   supp csupp 6902    ^m cmap 7457   Fincfn 7554   finSupp cfsupp 7863   Basecbs 14841   +g cplusg 14909   0gc0g 15054   Mndcmnd 16243
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 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  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 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-int 4228  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-of 6521  df-om 6684  df-1st 6784  df-2nd 6785  df-supp 6903  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-oadd 7171  df-er 7348  df-map 7459  df-en 7555  df-fin 7558  df-fsupp 7864  df-0g 15056  df-mgm 16196  df-sgrp 16235  df-mnd 16245
This theorem is referenced by:  lincsumcl  38543
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