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Theorem dsmmelbas 18958
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 6074 . . . . . 6  |-  ( ( R  Fn  I  /\  I  e.  V )  ->  R  e.  _V )
41, 2, 3syl2anc 659 . . . . 5  |-  ( ph  ->  R  e.  _V )
5 eqid 2400 . . . . . 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 18954 . . . . 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 17 . . . 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 5806 . . . . 5  |-  ( Base `  C )  =  (
Base `  ( S  (+)m 
R ) )
118, 10eqtri 2429 . . . 4  |-  H  =  ( Base `  ( S  (+)m  R ) )
127, 11syl6reqr 2460 . . 3  |-  ( ph  ->  H  =  { b  e.  ( Base `  ( S X_s R ) )  |  { a  e.  dom  R  |  ( b `  a )  =/=  ( 0g `  ( R `  a ) ) }  e.  Fin } )
1312eleq2d 2470 . 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 5802 . . . . . . 7  |-  ( b  =  X  ->  (
b `  a )  =  ( X `  a ) )
1514neeq1d 2678 . . . . . 6  |-  ( b  =  X  ->  (
( b `  a
)  =/=  ( 0g
`  ( R `  a ) )  <->  ( X `  a )  =/=  ( 0g `  ( R `  a ) ) ) )
1615rabbidv 3048 . . . . 5  |-  ( b  =  X  ->  { a  e.  dom  R  | 
( b `  a
)  =/=  ( 0g
`  ( R `  a ) ) }  =  { a  e. 
dom  R  |  ( X `  a )  =/=  ( 0g `  ( R `  a )
) } )
1716eleq1d 2469 . . . 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 3204 . . 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 5806 . . . . . . 7  |-  ( Base `  P )  =  (
Base `  ( S X_s R ) )
2219, 21eqtr2i 2430 . . . . . 6  |-  ( Base `  ( S X_s R ) )  =  B
2322eleq2i 2478 . . . . 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 5615 . . . . . 6  |-  ( R  Fn  I  ->  dom  R  =  I )
26 rabeq 3050 . . . . . 6  |-  ( dom 
R  =  I  ->  { a  e.  dom  R  |  ( X `  a )  =/=  ( 0g `  ( R `  a ) ) }  =  { a  e.  I  |  ( X `
 a )  =/=  ( 0g `  ( R `  a )
) } )
271, 25, 263syl 20 . . . . 5  |-  ( ph  ->  { a  e.  dom  R  |  ( X `  a )  =/=  ( 0g `  ( R `  a ) ) }  =  { a  e.  I  |  ( X `
 a )  =/=  ( 0g `  ( R `  a )
) } )
2827eleq1d 2469 . . . 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 709 . . 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 257 . 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 253 1  |-  ( ph  ->  ( X  e.  H  <->  ( X  e.  B  /\  { a  e.  I  |  ( X `  a
)  =/=  ( 0g
`  ( R `  a ) ) }  e.  Fin ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1403    e. wcel 1840    =/= wne 2596   {crab 2755   _Vcvv 3056   dom cdm 4940    Fn wfn 5518   ` cfv 5523  (class class class)co 6232   Fincfn 7472   Basecbs 14731   0gc0g 14944   X_scprds 14950    (+)m cdsmm 18950
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627  ax-un 6528  ax-cnex 9496  ax-resscn 9497  ax-1cn 9498  ax-icn 9499  ax-addcl 9500  ax-addrcl 9501  ax-mulcl 9502  ax-mulrcl 9503  ax-mulcom 9504  ax-addass 9505  ax-mulass 9506  ax-distr 9507  ax-i2m1 9508  ax-1ne0 9509  ax-1rid 9510  ax-rnegex 9511  ax-rrecex 9512  ax-cnre 9513  ax-pre-lttri 9514  ax-pre-lttrn 9515  ax-pre-ltadd 9516  ax-pre-mulgt0 9517
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 973  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-nel 2599  df-ral 2756  df-rex 2757  df-reu 2758  df-rab 2760  df-v 3058  df-sbc 3275  df-csb 3371  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-pss 3427  df-nul 3736  df-if 3883  df-pw 3954  df-sn 3970  df-pr 3972  df-tp 3974  df-op 3976  df-uni 4189  df-int 4225  df-iun 4270  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4487  df-eprel 4731  df-id 4735  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5487  df-fun 5525  df-fn 5526  df-f 5527  df-f1 5528  df-fo 5529  df-f1o 5530  df-fv 5531  df-riota 6194  df-ov 6235  df-oprab 6236  df-mpt2 6237  df-om 6637  df-1st 6736  df-2nd 6737  df-recs 6997  df-rdg 7031  df-1o 7085  df-oadd 7089  df-er 7266  df-map 7377  df-ixp 7426  df-en 7473  df-dom 7474  df-sdom 7475  df-fin 7476  df-sup 7853  df-pnf 9578  df-mnf 9579  df-xr 9580  df-ltxr 9581  df-le 9582  df-sub 9761  df-neg 9762  df-nn 10495  df-2 10553  df-3 10554  df-4 10555  df-5 10556  df-6 10557  df-7 10558  df-8 10559  df-9 10560  df-10 10561  df-n0 10755  df-z 10824  df-dec 10938  df-uz 11044  df-fz 11642  df-struct 14733  df-ndx 14734  df-slot 14735  df-base 14736  df-sets 14737  df-ress 14738  df-plusg 14812  df-mulr 14813  df-sca 14815  df-vsca 14816  df-ip 14817  df-tset 14818  df-ple 14819  df-ds 14821  df-hom 14823  df-cco 14824  df-prds 14952  df-dsmm 18951
This theorem is referenced by:  dsmm0cl  18959  dsmmacl  18960  dsmmsubg  18962  dsmmlss  18963
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