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Theorem mndpfsupp 39754
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 7444 . . . . . 6  |-  ( A  e.  ( R  ^m  V )  ->  A  Fn  V )
21adantr 466 . . . . 5  |-  ( ( A  e.  ( R  ^m  V )  /\  B  e.  ( R  ^m  V ) )  ->  A  Fn  V )
323ad2ant2 1027 . . . 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 7444 . . . . . 6  |-  ( B  e.  ( R  ^m  V )  ->  B  Fn  V )
54adantl 467 . . . . 5  |-  ( ( A  e.  ( R  ^m  V )  /\  B  e.  ( R  ^m  V ) )  ->  B  Fn  V )
653ad2ant2 1027 . . . 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 1030 . . . 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 3609 . . . 4  |-  ( V  i^i  V )  =  V
93, 6, 7, 7, 8offn 6495 . . 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 5629 . . 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 7838 . . . 4  |-  ( A finSupp 
( 0g `  M
)  ->  ( A supp  ( 0g `  M ) )  e.  Fin )
14 id 22 . . . . 5  |-  ( B finSupp 
( 0g `  M
)  ->  B finSupp  ( 0g
`  M ) )
1514fsuppimpd 7838 . . . 4  |-  ( B finSupp 
( 0g `  M
)  ->  ( B supp  ( 0g `  M ) )  e.  Fin )
1613, 15anim12i 568 . . 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 39753 . . 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 1299 . 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 6272 . . 3  |-  ( A  oF ( +g  `  M ) B )  e.  _V
21 fvex 5830 . . . 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 7835 . . 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 667 . 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 930 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 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1872   _Vcvv 3017   class class class wbr 4361   Fun wfun 5533    Fn wfn 5534   ` cfv 5539  (class class class)co 6244    oFcof 6482   supp csupp 6864    ^m cmap 7422   Fincfn 7519   finSupp cfsupp 7831   Basecbs 15059   +g cplusg 15128   0gc0g 15276   Mndcmnd 16473
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2058  ax-ext 2403  ax-rep 4474  ax-sep 4484  ax-nul 4493  ax-pow 4540  ax-pr 4598  ax-un 6536
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2275  df-mo 2276  df-clab 2410  df-cleq 2416  df-clel 2419  df-nfc 2553  df-ne 2596  df-ral 2714  df-rex 2715  df-reu 2716  df-rmo 2717  df-rab 2718  df-v 3019  df-sbc 3238  df-csb 3334  df-dif 3377  df-un 3379  df-in 3381  df-ss 3388  df-pss 3390  df-nul 3700  df-if 3850  df-pw 3921  df-sn 3937  df-pr 3939  df-tp 3941  df-op 3943  df-uni 4158  df-int 4194  df-iun 4239  df-br 4362  df-opab 4421  df-mpt 4422  df-tr 4457  df-eprel 4702  df-id 4706  df-po 4712  df-so 4713  df-fr 4750  df-we 4752  df-xp 4797  df-rel 4798  df-cnv 4799  df-co 4800  df-dm 4801  df-rn 4802  df-res 4803  df-ima 4804  df-pred 5337  df-ord 5383  df-on 5384  df-lim 5385  df-suc 5386  df-iota 5503  df-fun 5541  df-fn 5542  df-f 5543  df-f1 5544  df-fo 5545  df-f1o 5546  df-fv 5547  df-riota 6206  df-ov 6247  df-oprab 6248  df-mpt2 6249  df-of 6484  df-om 6646  df-1st 6746  df-2nd 6747  df-supp 6865  df-wrecs 6978  df-recs 7040  df-rdg 7078  df-oadd 7136  df-er 7313  df-map 7424  df-en 7520  df-fin 7523  df-fsupp 7832  df-0g 15278  df-mgm 16426  df-sgrp 16465  df-mnd 16475
This theorem is referenced by:  lincsumcl  39817
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