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Theorem rmsuppss 39748
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 6257 . . . . . . 7  |-  ( ( A `  w )  =  ( 0g `  M )  ->  ( C ( .r `  M ) ( A `
 w ) )  =  ( C ( .r `  M ) ( 0g `  M
) ) )
2 simpll1 1044 . . . . . . . 8  |-  ( ( ( ( M  e. 
Ring  /\  V  e.  X  /\  C  e.  R
)  /\  A  e.  ( R  ^m  V ) )  /\  w  e.  V )  ->  M  e.  Ring )
3 simpll3 1046 . . . . . . . 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 2428 . . . . . . . . 9  |-  ( .r
`  M )  =  ( .r `  M
)
6 eqid 2428 . . . . . . . . 9  |-  ( 0g
`  M )  =  ( 0g `  M
)
74, 5, 6ringrz 17761 . . . . . . . 8  |-  ( ( M  e.  Ring  /\  C  e.  R )  ->  ( C ( .r `  M ) ( 0g
`  M ) )  =  ( 0g `  M ) )
82, 3, 7syl2anc 665 . . . . . . 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 2484 . . . . . 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 435 . . . . 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 2622 . . . 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 3485 . . 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 7448 . . . . . 6  |-  ( A  e.  ( R  ^m  V )  ->  A : V --> R )
14 fdm 5693 . . . . . 6  |-  ( A : V --> R  ->  dom  A  =  V )
1513, 14syl 17 . . . . 5  |-  ( A  e.  ( R  ^m  V )  ->  dom  A  =  V )
1615adantl 467 . . . 4  |-  ( ( ( M  e.  Ring  /\  V  e.  X  /\  C  e.  R )  /\  A  e.  ( R  ^m  V ) )  ->  dom  A  =  V )
17 rabeq 3015 . . . 4  |-  ( dom 
A  =  V  ->  { w  e.  dom  A  |  ( A `  w )  =/=  ( 0g `  M ) }  =  { w  e.  V  |  ( A `
 w )  =/=  ( 0g `  M
) } )
1816, 17syl 17 . . 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 3444 . 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 5825 . . . . 5  |-  ( v  =  w  ->  ( A `  v )  =  ( A `  w ) )
2120oveq2d 6265 . . . 4  |-  ( v  =  w  ->  ( C ( .r `  M ) ( A `
 v ) )  =  ( C ( .r `  M ) ( A `  w
) ) )
2221cbvmptv 4459 . . 3  |-  ( v  e.  V  |->  ( C ( .r `  M
) ( A `  v ) ) )  =  ( w  e.  V  |->  ( C ( .r `  M ) ( A `  w
) ) )
23 simpl2 1009 . . 3  |-  ( ( ( M  e.  Ring  /\  V  e.  X  /\  C  e.  R )  /\  A  e.  ( R  ^m  V ) )  ->  V  e.  X
)
24 fvex 5835 . . . 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 6277 . . . 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 6893 . 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 7450 . . . 4  |-  ( A  e.  ( R  ^m  V )  ->  Fun  A )
3029adantl 467 . . 3  |-  ( ( ( M  e.  Ring  /\  V  e.  X  /\  C  e.  R )  /\  A  e.  ( R  ^m  V ) )  ->  Fun  A )
31 simpr 462 . . 3  |-  ( ( ( M  e.  Ring  /\  V  e.  X  /\  C  e.  R )  /\  A  e.  ( R  ^m  V ) )  ->  A  e.  ( R  ^m  V ) )
32 suppval1 6875 . . 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 1264 . 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 3450 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 370    /\ w3a 982    = wceq 1437    e. wcel 1872    =/= wne 2599   {crab 2718   _Vcvv 3022    C_ wss 3379    |-> cmpt 4425   dom cdm 4796   Fun wfun 5538   -->wf 5540   ` cfv 5544  (class class class)co 6249   supp csupp 6869    ^m cmap 7427   Basecbs 15064   .rcmulr 15134   0gc0g 15281   Ringcrg 17723
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 2063  ax-ext 2408  ax-rep 4479  ax-sep 4489  ax-nul 4498  ax-pow 4545  ax-pr 4603  ax-un 6541  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567
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 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-nel 2602  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 3024  df-sbc 3243  df-csb 3339  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-pss 3395  df-nul 3705  df-if 3855  df-pw 3926  df-sn 3942  df-pr 3944  df-tp 3946  df-op 3948  df-uni 4163  df-iun 4244  df-br 4367  df-opab 4426  df-mpt 4427  df-tr 4462  df-eprel 4707  df-id 4711  df-po 4717  df-so 4718  df-fr 4755  df-we 4757  df-xp 4802  df-rel 4803  df-cnv 4804  df-co 4805  df-dm 4806  df-rn 4807  df-res 4808  df-ima 4809  df-pred 5342  df-ord 5388  df-on 5389  df-lim 5390  df-suc 5391  df-iota 5508  df-fun 5546  df-fn 5547  df-f 5548  df-f1 5549  df-fo 5550  df-f1o 5551  df-fv 5552  df-riota 6211  df-ov 6252  df-oprab 6253  df-mpt2 6254  df-om 6651  df-1st 6751  df-2nd 6752  df-supp 6870  df-wrecs 6983  df-recs 7045  df-rdg 7083  df-er 7318  df-map 7429  df-en 7525  df-dom 7526  df-sdom 7527  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9813  df-neg 9814  df-nn 10561  df-2 10619  df-ndx 15067  df-slot 15068  df-base 15069  df-sets 15070  df-plusg 15146  df-0g 15283  df-mgm 16431  df-sgrp 16470  df-mnd 16480  df-grp 16616  df-mgp 17667  df-ring 17725
This theorem is referenced by:  rmsuppfi  39751
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