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Theorem dsmmelbas 27205
Description: Membership in the finitely supported hull of a structure product in terms of the index set. (Contributed by Stefan O'Rear, 11-Jan-2015.)
Hypotheses
Ref Expression
dsmmelbas.p  |-  P  =  ( S X_s R )
dsmmelbas.c  |-  C  =  ( S  (+)m  R )
dsmmelbas.b  |-  B  =  ( Base `  P
)
dsmmelbas.h  |-  H  =  ( Base `  C
)
dsmmelbas.i  |-  ( ph  ->  I  e.  V )
dsmmelbas.r  |-  ( ph  ->  R  Fn  I )
Assertion
Ref Expression
dsmmelbas  |-  ( ph  ->  ( X  e.  H  <->  ( X  e.  B  /\  { a  e.  I  |  ( X `  a
)  =/=  ( 0g
`  ( R `  a ) ) }  e.  Fin ) ) )
Distinct variable groups:    S, a    R, a    X, a    I, a
Allowed substitution hints:    ph( a)    B( a)    C( a)    P( a)    H( a)    V( a)

Proof of Theorem dsmmelbas
Dummy variable  b is distinct from all other variables.
StepHypRef Expression
1 dsmmelbas.r . . . . . 6  |-  ( ph  ->  R  Fn  I )
2 dsmmelbas.i . . . . . 6  |-  ( ph  ->  I  e.  V )
3 fnex 5741 . . . . . 6  |-  ( ( R  Fn  I  /\  I  e.  V )  ->  R  e.  _V )
41, 2, 3syl2anc 642 . . . . 5  |-  ( ph  ->  R  e.  _V )
5 eqid 2283 . . . . . 6  |-  { b  e.  ( Base `  ( S X_s R ) )  |  { a  e.  dom  R  |  ( b `  a )  =/=  ( 0g `  ( R `  a ) ) }  e.  Fin }  =  { b  e.  (
Base `  ( S X_s R ) )  |  {
a  e.  dom  R  |  ( b `  a )  =/=  ( 0g `  ( R `  a ) ) }  e.  Fin }
65dsmmbase 27201 . . . . 5  |-  ( R  e.  _V  ->  { b  e.  ( Base `  ( S X_s R ) )  |  { a  e.  dom  R  |  ( b `  a )  =/=  ( 0g `  ( R `  a ) ) }  e.  Fin }  =  ( Base `  ( S  (+)m 
R ) ) )
74, 6syl 15 . . . 4  |-  ( ph  ->  { b  e.  (
Base `  ( S X_s R ) )  |  {
a  e.  dom  R  |  ( b `  a )  =/=  ( 0g `  ( R `  a ) ) }  e.  Fin }  =  ( Base `  ( S  (+)m 
R ) ) )
8 dsmmelbas.h . . . . 5  |-  H  =  ( Base `  C
)
9 dsmmelbas.c . . . . . 6  |-  C  =  ( S  (+)m  R )
109fveq2i 5528 . . . . 5  |-  ( Base `  C )  =  (
Base `  ( S  (+)m 
R ) )
118, 10eqtri 2303 . . . 4  |-  H  =  ( Base `  ( S  (+)m  R ) )
127, 11syl6reqr 2334 . . 3  |-  ( ph  ->  H  =  { b  e.  ( Base `  ( S X_s R ) )  |  { a  e.  dom  R  |  ( b `  a )  =/=  ( 0g `  ( R `  a ) ) }  e.  Fin } )
1312eleq2d 2350 . 2  |-  ( ph  ->  ( X  e.  H  <->  X  e.  { b  e.  ( Base `  ( S X_s R ) )  |  { a  e.  dom  R  |  ( b `  a )  =/=  ( 0g `  ( R `  a ) ) }  e.  Fin } ) )
14 fveq1 5524 . . . . . . 7  |-  ( b  =  X  ->  (
b `  a )  =  ( X `  a ) )
1514neeq1d 2459 . . . . . 6  |-  ( b  =  X  ->  (
( b `  a
)  =/=  ( 0g
`  ( R `  a ) )  <->  ( X `  a )  =/=  ( 0g `  ( R `  a ) ) ) )
1615rabbidv 2780 . . . . 5  |-  ( b  =  X  ->  { a  e.  dom  R  | 
( b `  a
)  =/=  ( 0g
`  ( R `  a ) ) }  =  { a  e. 
dom  R  |  ( X `  a )  =/=  ( 0g `  ( R `  a )
) } )
1716eleq1d 2349 . . . 4  |-  ( b  =  X  ->  ( { a  e.  dom  R  |  ( b `  a )  =/=  ( 0g `  ( R `  a ) ) }  e.  Fin  <->  { a  e.  dom  R  |  ( X `  a )  =/=  ( 0g `  ( R `  a ) ) }  e.  Fin ) )
1817elrab 2923 . . 3  |-  ( X  e.  { b  e.  ( Base `  ( S X_s R ) )  |  { a  e.  dom  R  |  ( b `  a )  =/=  ( 0g `  ( R `  a ) ) }  e.  Fin }  <->  ( X  e.  ( Base `  ( S X_s R ) )  /\  { a  e.  dom  R  |  ( X `  a )  =/=  ( 0g `  ( R `  a ) ) }  e.  Fin ) )
19 dsmmelbas.b . . . . . . 7  |-  B  =  ( Base `  P
)
20 dsmmelbas.p . . . . . . . 8  |-  P  =  ( S X_s R )
2120fveq2i 5528 . . . . . . 7  |-  ( Base `  P )  =  (
Base `  ( S X_s R ) )
2219, 21eqtr2i 2304 . . . . . 6  |-  ( Base `  ( S X_s R ) )  =  B
2322eleq2i 2347 . . . . 5  |-  ( X  e.  ( Base `  ( S X_s R ) )  <->  X  e.  B )
2423a1i 10 . . . 4  |-  ( ph  ->  ( X  e.  (
Base `  ( S X_s R ) )  <->  X  e.  B ) )
25 fndm 5343 . . . . . 6  |-  ( R  Fn  I  ->  dom  R  =  I )
26 rabeq 2782 . . . . . 6  |-  ( dom 
R  =  I  ->  { a  e.  dom  R  |  ( X `  a )  =/=  ( 0g `  ( R `  a ) ) }  =  { a  e.  I  |  ( X `
 a )  =/=  ( 0g `  ( R `  a )
) } )
271, 25, 263syl 18 . . . . 5  |-  ( ph  ->  { a  e.  dom  R  |  ( X `  a )  =/=  ( 0g `  ( R `  a ) ) }  =  { a  e.  I  |  ( X `
 a )  =/=  ( 0g `  ( R `  a )
) } )
2827eleq1d 2349 . . . 4  |-  ( ph  ->  ( { a  e. 
dom  R  |  ( X `  a )  =/=  ( 0g `  ( R `  a )
) }  e.  Fin  <->  {
a  e.  I  |  ( X `  a
)  =/=  ( 0g
`  ( R `  a ) ) }  e.  Fin ) )
2924, 28anbi12d 691 . . 3  |-  ( ph  ->  ( ( X  e.  ( Base `  ( S X_s R ) )  /\  { a  e.  dom  R  |  ( X `  a )  =/=  ( 0g `  ( R `  a ) ) }  e.  Fin )  <->  ( X  e.  B  /\  { a  e.  I  |  ( X `  a )  =/=  ( 0g `  ( R `  a ) ) }  e.  Fin ) ) )
3018, 29syl5bb 248 . 2  |-  ( ph  ->  ( X  e.  {
b  e.  ( Base `  ( S X_s R ) )  |  { a  e.  dom  R  |  ( b `  a )  =/=  ( 0g `  ( R `  a ) ) }  e.  Fin }  <->  ( X  e.  B  /\  { a  e.  I  |  ( X `  a )  =/=  ( 0g `  ( R `  a ) ) }  e.  Fin ) ) )
3113, 30bitrd 244 1  |-  ( ph  ->  ( X  e.  H  <->  ( X  e.  B  /\  { a  e.  I  |  ( X `  a
)  =/=  ( 0g
`  ( R `  a ) ) }  e.  Fin ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   {crab 2547   _Vcvv 2788   dom cdm 4689    Fn wfn 5250   ` cfv 5255  (class class class)co 5858   Fincfn 6863   Basecbs 13148   X_scprds 13346   0gc0g 13400    (+)m cdsmm 27197
This theorem is referenced by:  dsmm0cl  27206  dsmmacl  27207  dsmmsubg  27209  dsmmlss  27210
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-fz 10783  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-hom 13232  df-cco 13233  df-prds 13348  df-dsmm 27198
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