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Theorem rmsuppss 31911
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 6283 . . . . . . 7  |-  ( ( A `  w )  =  ( 0g `  M )  ->  ( C ( .r `  M ) ( A `
 w ) )  =  ( C ( .r `  M ) ( 0g `  M
) ) )
2 simpll1 1030 . . . . . . . 8  |-  ( ( ( ( M  e. 
Ring  /\  V  e.  X  /\  C  e.  R
)  /\  A  e.  ( R  ^m  V ) )  /\  w  e.  V )  ->  M  e.  Ring )
3 simpll3 1032 . . . . . . . 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 2460 . . . . . . . . 9  |-  ( .r
`  M )  =  ( .r `  M
)
6 eqid 2460 . . . . . . . . 9  |-  ( 0g
`  M )  =  ( 0g `  M
)
74, 5, 6rngrz 17016 . . . . . . . 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 2523 . . . . . 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 2684 . . . 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 3574 . . 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 7430 . . . . . 6  |-  ( A  e.  ( R  ^m  V )  ->  A : V --> R )
14 fdm 5726 . . . . . 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 3100 . . . 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 3534 . 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 5857 . . . . 5  |-  ( v  =  w  ->  ( A `  v )  =  ( A `  w ) )
2120oveq2d 6291 . . . 4  |-  ( v  =  w  ->  ( C ( .r `  M ) ( A `
 v ) )  =  ( C ( .r `  M ) ( A `  w
) ) )
2221cbvmptv 4531 . . 3  |-  ( v  e.  V  |->  ( C ( .r `  M
) ( A `  v ) ) )  =  ( w  e.  V  |->  ( C ( .r `  M ) ( A `  w
) ) )
23 simpl2 995 . . 3  |-  ( ( ( M  e.  Ring  /\  V  e.  X  /\  C  e.  R )  /\  A  e.  ( R  ^m  V ) )  ->  V  e.  X
)
24 fvex 5867 . . . 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 6300 . . . 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 6913 . 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 ffun 5724 . . . . 5  |-  ( A : V --> R  ->  Fun  A )
3013, 29syl 16 . . . 4  |-  ( A  e.  ( R  ^m  V )  ->  Fun  A )
3130adantl 466 . . 3  |-  ( ( ( M  e.  Ring  /\  V  e.  X  /\  C  e.  R )  /\  A  e.  ( R  ^m  V ) )  ->  Fun  A )
32 simpr 461 . . 3  |-  ( ( ( M  e.  Ring  /\  V  e.  X  /\  C  e.  R )  /\  A  e.  ( R  ^m  V ) )  ->  A  e.  ( R  ^m  V ) )
33 suppval1 6897 . . 3  |-  ( ( Fun  A  /\  A  e.  ( R  ^m  V
)  /\  ( 0g `  M )  e.  _V )  ->  ( A supp  ( 0g `  M ) )  =  { w  e. 
dom  A  |  ( A `  w )  =/=  ( 0g `  M
) } )
3431, 32, 25, 33syl3anc 1223 . 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
) } )
3519, 28, 343sstr4d 3540 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 968    = wceq 1374    e. wcel 1762    =/= wne 2655   {crab 2811   _Vcvv 3106    C_ wss 3469    |-> cmpt 4498   dom cdm 4992   Fun wfun 5573   -->wf 5575   ` cfv 5579  (class class class)co 6275   supp csupp 6891    ^m cmap 7410   Basecbs 14479   .rcmulr 14545   0gc0g 14684   Ringcrg 16979
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 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
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 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-supp 6892  df-recs 7032  df-rdg 7066  df-er 7301  df-map 7412  df-en 7507  df-dom 7508  df-sdom 7509  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-nn 10526  df-2 10583  df-ndx 14482  df-slot 14483  df-base 14484  df-sets 14485  df-plusg 14557  df-0g 14686  df-mnd 15721  df-grp 15851  df-mgp 16925  df-rng 16981
This theorem is referenced by:  rmsuppfi  31914
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