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Theorem mndpfsupp 31919
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 7433 . . . . . 6  |-  ( A  e.  ( R  ^m  V )  ->  A  Fn  V )
21adantr 465 . . . . 5  |-  ( ( A  e.  ( R  ^m  V )  /\  B  e.  ( R  ^m  V ) )  ->  A  Fn  V )
323ad2ant2 1013 . . . 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 7433 . . . . . 6  |-  ( B  e.  ( R  ^m  V )  ->  B  Fn  V )
54adantl 466 . . . . 5  |-  ( ( A  e.  ( R  ^m  V )  /\  B  e.  ( R  ^m  V ) )  ->  B  Fn  V )
653ad2ant2 1013 . . . 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 1016 . . . 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 3702 . . . 4  |-  ( V  i^i  V )  =  V
93, 6, 7, 7, 8offn 6528 . . 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 5671 . . 3  |-  ( ( A  oF ( +g  `  M ) B )  Fn  V  ->  Fun  ( A  oF ( +g  `  M
) B ) )
119, 10syl 16 . 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 7827 . . . 4  |-  ( A finSupp 
( 0g `  M
)  ->  ( A supp  ( 0g `  M ) )  e.  Fin )
14 id 22 . . . . 5  |-  ( B finSupp 
( 0g `  M
)  ->  B finSupp  ( 0g
`  M ) )
1514fsuppimpd 7827 . . . 4  |-  ( B finSupp 
( 0g `  M
)  ->  ( B supp  ( 0g `  M ) )  e.  Fin )
1613, 15anim12i 566 . . 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 31918 . . 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 1258 . 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 6302 . . 3  |-  ( A  oF ( +g  `  M ) B )  e.  _V
21 fvex 5869 . . . 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 7824 . . 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 663 . 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 915 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 369    /\ w3a 968    = wceq 1374    e. wcel 1762   _Vcvv 3108   class class class wbr 4442   Fun wfun 5575    Fn wfn 5576   ` cfv 5581  (class class class)co 6277    oFcof 6515   supp csupp 6893    ^m cmap 7412   Fincfn 7508   finSupp cfsupp 7820   Basecbs 14481   +g cplusg 14546   0gc0g 14686   Mndcmnd 15717
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-reu 2816  df-rmo 2817  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-uni 4241  df-int 4278  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-tr 4536  df-eprel 4786  df-id 4790  df-po 4795  df-so 4796  df-fr 4833  df-we 4835  df-ord 4876  df-on 4877  df-lim 4878  df-suc 4879  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-of 6517  df-om 6674  df-1st 6776  df-2nd 6777  df-supp 6894  df-recs 7034  df-rdg 7068  df-oadd 7126  df-er 7303  df-map 7414  df-en 7509  df-fin 7512  df-fsupp 7821  df-0g 14688  df-mnd 15723
This theorem is referenced by:  lincsumcl  31982
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