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Theorem rmsuppss 33107
Description: The support of a mapping of a multiplication of a constant with a function into a ring is a subset of the support of the function. (Contributed by AV, 11-Apr-2019.)
Hypothesis
Ref Expression
rmsuppss.r  |-  R  =  ( Base `  M
)
Assertion
Ref Expression
rmsuppss  |-  ( ( ( M  e.  Ring  /\  V  e.  X  /\  C  e.  R )  /\  A  e.  ( R  ^m  V ) )  ->  ( ( v  e.  V  |->  ( C ( .r `  M
) ( A `  v ) ) ) supp  ( 0g `  M
) )  C_  ( A supp  ( 0g `  M
) ) )
Distinct variable groups:    v, A    v, C    v, M    v, R    v, X    v, V

Proof of Theorem rmsuppss
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 oveq2 6304 . . . . . . 7  |-  ( ( A `  w )  =  ( 0g `  M )  ->  ( C ( .r `  M ) ( A `
 w ) )  =  ( C ( .r `  M ) ( 0g `  M
) ) )
2 simpll1 1035 . . . . . . . 8  |-  ( ( ( ( M  e. 
Ring  /\  V  e.  X  /\  C  e.  R
)  /\  A  e.  ( R  ^m  V ) )  /\  w  e.  V )  ->  M  e.  Ring )
3 simpll3 1037 . . . . . . . 8  |-  ( ( ( ( M  e. 
Ring  /\  V  e.  X  /\  C  e.  R
)  /\  A  e.  ( R  ^m  V ) )  /\  w  e.  V )  ->  C  e.  R )
4 rmsuppss.r . . . . . . . . 9  |-  R  =  ( Base `  M
)
5 eqid 2457 . . . . . . . . 9  |-  ( .r
`  M )  =  ( .r `  M
)
6 eqid 2457 . . . . . . . . 9  |-  ( 0g
`  M )  =  ( 0g `  M
)
74, 5, 6ringrz 17363 . . . . . . . 8  |-  ( ( M  e.  Ring  /\  C  e.  R )  ->  ( C ( .r `  M ) ( 0g
`  M ) )  =  ( 0g `  M ) )
82, 3, 7syl2anc 661 . . . . . . 7  |-  ( ( ( ( M  e. 
Ring  /\  V  e.  X  /\  C  e.  R
)  /\  A  e.  ( R  ^m  V ) )  /\  w  e.  V )  ->  ( C ( .r `  M ) ( 0g
`  M ) )  =  ( 0g `  M ) )
91, 8sylan9eqr 2520 . . . . . 6  |-  ( ( ( ( ( M  e.  Ring  /\  V  e.  X  /\  C  e.  R )  /\  A  e.  ( R  ^m  V
) )  /\  w  e.  V )  /\  ( A `  w )  =  ( 0g `  M ) )  -> 
( C ( .r
`  M ) ( A `  w ) )  =  ( 0g
`  M ) )
109ex 434 . . . . 5  |-  ( ( ( ( M  e. 
Ring  /\  V  e.  X  /\  C  e.  R
)  /\  A  e.  ( R  ^m  V ) )  /\  w  e.  V )  ->  (
( A `  w
)  =  ( 0g
`  M )  -> 
( C ( .r
`  M ) ( A `  w ) )  =  ( 0g
`  M ) ) )
1110necon3d 2681 . . . 4  |-  ( ( ( ( M  e. 
Ring  /\  V  e.  X  /\  C  e.  R
)  /\  A  e.  ( R  ^m  V ) )  /\  w  e.  V )  ->  (
( C ( .r
`  M ) ( A `  w ) )  =/=  ( 0g
`  M )  -> 
( A `  w
)  =/=  ( 0g
`  M ) ) )
1211ss2rabdv 3577 . . 3  |-  ( ( ( M  e.  Ring  /\  V  e.  X  /\  C  e.  R )  /\  A  e.  ( R  ^m  V ) )  ->  { w  e.  V  |  ( C ( .r `  M
) ( A `  w ) )  =/=  ( 0g `  M
) }  C_  { w  e.  V  |  ( A `  w )  =/=  ( 0g `  M
) } )
13 elmapi 7459 . . . . . 6  |-  ( A  e.  ( R  ^m  V )  ->  A : V --> R )
14 fdm 5741 . . . . . 6  |-  ( A : V --> R  ->  dom  A  =  V )
1513, 14syl 16 . . . . 5  |-  ( A  e.  ( R  ^m  V )  ->  dom  A  =  V )
1615adantl 466 . . . 4  |-  ( ( ( M  e.  Ring  /\  V  e.  X  /\  C  e.  R )  /\  A  e.  ( R  ^m  V ) )  ->  dom  A  =  V )
17 rabeq 3103 . . . 4  |-  ( dom 
A  =  V  ->  { w  e.  dom  A  |  ( A `  w )  =/=  ( 0g `  M ) }  =  { w  e.  V  |  ( A `
 w )  =/=  ( 0g `  M
) } )
1816, 17syl 16 . . 3  |-  ( ( ( M  e.  Ring  /\  V  e.  X  /\  C  e.  R )  /\  A  e.  ( R  ^m  V ) )  ->  { w  e. 
dom  A  |  ( A `  w )  =/=  ( 0g `  M
) }  =  {
w  e.  V  | 
( A `  w
)  =/=  ( 0g
`  M ) } )
1912, 18sseqtr4d 3536 . 2  |-  ( ( ( M  e.  Ring  /\  V  e.  X  /\  C  e.  R )  /\  A  e.  ( R  ^m  V ) )  ->  { w  e.  V  |  ( C ( .r `  M
) ( A `  w ) )  =/=  ( 0g `  M
) }  C_  { w  e.  dom  A  |  ( A `  w )  =/=  ( 0g `  M ) } )
20 fveq2 5872 . . . . 5  |-  ( v  =  w  ->  ( A `  v )  =  ( A `  w ) )
2120oveq2d 6312 . . . 4  |-  ( v  =  w  ->  ( C ( .r `  M ) ( A `
 v ) )  =  ( C ( .r `  M ) ( A `  w
) ) )
2221cbvmptv 4548 . . 3  |-  ( v  e.  V  |->  ( C ( .r `  M
) ( A `  v ) ) )  =  ( w  e.  V  |->  ( C ( .r `  M ) ( A `  w
) ) )
23 simpl2 1000 . . 3  |-  ( ( ( M  e.  Ring  /\  V  e.  X  /\  C  e.  R )  /\  A  e.  ( R  ^m  V ) )  ->  V  e.  X
)
24 fvex 5882 . . . 4  |-  ( 0g
`  M )  e. 
_V
2524a1i 11 . . 3  |-  ( ( ( M  e.  Ring  /\  V  e.  X  /\  C  e.  R )  /\  A  e.  ( R  ^m  V ) )  ->  ( 0g `  M )  e.  _V )
26 ovex 6324 . . . 4  |-  ( C ( .r `  M
) ( A `  w ) )  e. 
_V
2726a1i 11 . . 3  |-  ( ( ( ( M  e. 
Ring  /\  V  e.  X  /\  C  e.  R
)  /\  A  e.  ( R  ^m  V ) )  /\  w  e.  V )  ->  ( C ( .r `  M ) ( A `
 w ) )  e.  _V )
2822, 23, 25, 27mptsuppd 6941 . 2  |-  ( ( ( M  e.  Ring  /\  V  e.  X  /\  C  e.  R )  /\  A  e.  ( R  ^m  V ) )  ->  ( ( v  e.  V  |->  ( C ( .r `  M
) ( A `  v ) ) ) supp  ( 0g `  M
) )  =  {
w  e.  V  | 
( C ( .r
`  M ) ( A `  w ) )  =/=  ( 0g
`  M ) } )
29 elmapfun 7461 . . . 4  |-  ( A  e.  ( R  ^m  V )  ->  Fun  A )
3029adantl 466 . . 3  |-  ( ( ( M  e.  Ring  /\  V  e.  X  /\  C  e.  R )  /\  A  e.  ( R  ^m  V ) )  ->  Fun  A )
31 simpr 461 . . 3  |-  ( ( ( M  e.  Ring  /\  V  e.  X  /\  C  e.  R )  /\  A  e.  ( R  ^m  V ) )  ->  A  e.  ( R  ^m  V ) )
32 suppval1 6923 . . 3  |-  ( ( Fun  A  /\  A  e.  ( R  ^m  V
)  /\  ( 0g `  M )  e.  _V )  ->  ( A supp  ( 0g `  M ) )  =  { w  e. 
dom  A  |  ( A `  w )  =/=  ( 0g `  M
) } )
3330, 31, 25, 32syl3anc 1228 . 2  |-  ( ( ( M  e.  Ring  /\  V  e.  X  /\  C  e.  R )  /\  A  e.  ( R  ^m  V ) )  ->  ( A supp  ( 0g `  M ) )  =  { w  e. 
dom  A  |  ( A `  w )  =/=  ( 0g `  M
) } )
3419, 28, 333sstr4d 3542 1  |-  ( ( ( M  e.  Ring  /\  V  e.  X  /\  C  e.  R )  /\  A  e.  ( R  ^m  V ) )  ->  ( ( v  e.  V  |->  ( C ( .r `  M
) ( A `  v ) ) ) supp  ( 0g `  M
) )  C_  ( A supp  ( 0g `  M
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819    =/= wne 2652   {crab 2811   _Vcvv 3109    C_ wss 3471    |-> cmpt 4515   dom cdm 5008   Fun wfun 5588   -->wf 5590   ` cfv 5594  (class class class)co 6296   supp csupp 6917    ^m cmap 7438   Basecbs 14644   .rcmulr 14713   0gc0g 14857   Ringcrg 17325
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-supp 6918  df-recs 7060  df-rdg 7094  df-er 7329  df-map 7440  df-en 7536  df-dom 7537  df-sdom 7538  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-ndx 14647  df-slot 14648  df-base 14649  df-sets 14650  df-plusg 14725  df-0g 14859  df-mgm 15999  df-sgrp 16038  df-mnd 16048  df-grp 16184  df-mgp 17269  df-ring 17327
This theorem is referenced by:  rmsuppfi  33110
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