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Theorem dsmmelbas 27182
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 5961 . . . . . 6  |-  ( ( R  Fn  I  /\  I  e.  V )  ->  R  e.  _V )
41, 2, 3syl2anc 643 . . . . 5  |-  ( ph  ->  R  e.  _V )
5 eqid 2436 . . . . . 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 27178 . . . . 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 16 . . . 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 5731 . . . . 5  |-  ( Base `  C )  =  (
Base `  ( S  (+)m 
R ) )
118, 10eqtri 2456 . . . 4  |-  H  =  ( Base `  ( S  (+)m  R ) )
127, 11syl6reqr 2487 . . 3  |-  ( ph  ->  H  =  { b  e.  ( Base `  ( S X_s R ) )  |  { a  e.  dom  R  |  ( b `  a )  =/=  ( 0g `  ( R `  a ) ) }  e.  Fin } )
1312eleq2d 2503 . 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 5727 . . . . . . 7  |-  ( b  =  X  ->  (
b `  a )  =  ( X `  a ) )
1514neeq1d 2614 . . . . . 6  |-  ( b  =  X  ->  (
( b `  a
)  =/=  ( 0g
`  ( R `  a ) )  <->  ( X `  a )  =/=  ( 0g `  ( R `  a ) ) ) )
1615rabbidv 2948 . . . . 5  |-  ( b  =  X  ->  { a  e.  dom  R  | 
( b `  a
)  =/=  ( 0g
`  ( R `  a ) ) }  =  { a  e. 
dom  R  |  ( X `  a )  =/=  ( 0g `  ( R `  a )
) } )
1716eleq1d 2502 . . . 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 3092 . . 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 5731 . . . . . . 7  |-  ( Base `  P )  =  (
Base `  ( S X_s R ) )
2219, 21eqtr2i 2457 . . . . . 6  |-  ( Base `  ( S X_s R ) )  =  B
2322eleq2i 2500 . . . . 5  |-  ( X  e.  ( Base `  ( S X_s R ) )  <->  X  e.  B )
2423a1i 11 . . . 4  |-  ( ph  ->  ( X  e.  (
Base `  ( S X_s R ) )  <->  X  e.  B ) )
25 fndm 5544 . . . . . 6  |-  ( R  Fn  I  ->  dom  R  =  I )
26 rabeq 2950 . . . . . 6  |-  ( dom 
R  =  I  ->  { a  e.  dom  R  |  ( X `  a )  =/=  ( 0g `  ( R `  a ) ) }  =  { a  e.  I  |  ( X `
 a )  =/=  ( 0g `  ( R `  a )
) } )
271, 25, 263syl 19 . . . . 5  |-  ( ph  ->  { a  e.  dom  R  |  ( X `  a )  =/=  ( 0g `  ( R `  a ) ) }  =  { a  e.  I  |  ( X `
 a )  =/=  ( 0g `  ( R `  a )
) } )
2827eleq1d 2502 . . . 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 692 . . 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 249 . 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 245 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 177    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2599   {crab 2709   _Vcvv 2956   dom cdm 4878    Fn wfn 5449   ` cfv 5454  (class class class)co 6081   Fincfn 7109   Basecbs 13469   X_scprds 13669   0gc0g 13723    (+)m cdsmm 27174
This theorem is referenced by:  dsmm0cl  27183  dsmmacl  27184  dsmmsubg  27186  dsmmlss  27187
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-er 6905  df-map 7020  df-ixp 7064  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-sup 7446  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-2 10058  df-3 10059  df-4 10060  df-5 10061  df-6 10062  df-7 10063  df-8 10064  df-9 10065  df-10 10066  df-n0 10222  df-z 10283  df-dec 10383  df-uz 10489  df-fz 11044  df-struct 13471  df-ndx 13472  df-slot 13473  df-base 13474  df-sets 13475  df-ress 13476  df-plusg 13542  df-mulr 13543  df-sca 13545  df-vsca 13546  df-tset 13548  df-ple 13549  df-ds 13551  df-hom 13553  df-cco 13554  df-prds 13671  df-dsmm 27175
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