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Theorem dsmmbas2 19231
Description: Base set of the direct sum module using the fndmin 6004 abbreviation. (Contributed by Stefan O'Rear, 1-Feb-2015.)
Hypotheses
Ref Expression
dsmmbas2.p  |-  P  =  ( S X_s R )
dsmmbas2.b  |-  B  =  { f  e.  (
Base `  P )  |  dom  ( f  \ 
( 0g  o.  R
) )  e.  Fin }
Assertion
Ref Expression
dsmmbas2  |-  ( ( R  Fn  I  /\  I  e.  V )  ->  B  =  ( Base `  ( S  (+)m  R ) ) )
Distinct variable groups:    S, f    R, f    P, f    f, I   
f, V
Allowed substitution hint:    B( f)

Proof of Theorem dsmmbas2
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 dsmmbas2.b . 2  |-  B  =  { f  e.  (
Base `  P )  |  dom  ( f  \ 
( 0g  o.  R
) )  e.  Fin }
2 dsmmbas2.p . . . . . 6  |-  P  =  ( S X_s R )
32fveq2i 5884 . . . . 5  |-  ( Base `  P )  =  (
Base `  ( S X_s R ) )
4 rabeq 3081 . . . . 5  |-  ( (
Base `  P )  =  ( Base `  ( S X_s R ) )  ->  { f  e.  (
Base `  P )  |  dom  ( f  \ 
( 0g  o.  R
) )  e.  Fin }  =  { f  e.  ( Base `  ( S X_s R ) )  |  dom  ( f  \ 
( 0g  o.  R
) )  e.  Fin } )
53, 4ax-mp 5 . . . 4  |-  { f  e.  ( Base `  P
)  |  dom  (
f  \  ( 0g  o.  R ) )  e. 
Fin }  =  {
f  e.  ( Base `  ( S X_s R ) )  |  dom  ( f  \ 
( 0g  o.  R
) )  e.  Fin }
6 simpll 758 . . . . . . . . . 10  |-  ( ( ( R  Fn  I  /\  I  e.  V
)  /\  f  e.  ( Base `  ( S X_s R ) ) )  ->  R  Fn  I )
7 fvco2 5956 . . . . . . . . . 10  |-  ( ( R  Fn  I  /\  x  e.  I )  ->  ( ( 0g  o.  R ) `  x
)  =  ( 0g
`  ( R `  x ) ) )
86, 7sylan 473 . . . . . . . . 9  |-  ( ( ( ( R  Fn  I  /\  I  e.  V
)  /\  f  e.  ( Base `  ( S X_s R ) ) )  /\  x  e.  I )  ->  ( ( 0g  o.  R ) `  x
)  =  ( 0g
`  ( R `  x ) ) )
98neeq2d 2709 . . . . . . . 8  |-  ( ( ( ( R  Fn  I  /\  I  e.  V
)  /\  f  e.  ( Base `  ( S X_s R ) ) )  /\  x  e.  I )  ->  ( ( f `  x )  =/=  (
( 0g  o.  R
) `  x )  <->  ( f `  x )  =/=  ( 0g `  ( R `  x ) ) ) )
109rabbidva 3078 . . . . . . 7  |-  ( ( ( R  Fn  I  /\  I  e.  V
)  /\  f  e.  ( Base `  ( S X_s R ) ) )  ->  { x  e.  I  |  ( f `  x )  =/=  (
( 0g  o.  R
) `  x ) }  =  { x  e.  I  |  (
f `  x )  =/=  ( 0g `  ( R `  x )
) } )
11 eqid 2429 . . . . . . . . 9  |-  ( S
X_s
R )  =  ( S X_s R )
12 eqid 2429 . . . . . . . . 9  |-  ( Base `  ( S X_s R ) )  =  ( Base `  ( S X_s R ) )
13 noel 3771 . . . . . . . . . . . 12  |-  -.  f  e.  (/)
14 reldmprds 15306 . . . . . . . . . . . . . . . 16  |-  Rel  dom  X_s
1514ovprc1 6336 . . . . . . . . . . . . . . 15  |-  ( -.  S  e.  _V  ->  ( S X_s R )  =  (/) )
1615fveq2d 5885 . . . . . . . . . . . . . 14  |-  ( -.  S  e.  _V  ->  (
Base `  ( S X_s R ) )  =  (
Base `  (/) ) )
17 base0 15125 . . . . . . . . . . . . . 14  |-  (/)  =  (
Base `  (/) )
1816, 17syl6eqr 2488 . . . . . . . . . . . . 13  |-  ( -.  S  e.  _V  ->  (
Base `  ( S X_s R ) )  =  (/) )
1918eleq2d 2499 . . . . . . . . . . . 12  |-  ( -.  S  e.  _V  ->  ( f  e.  ( Base `  ( S X_s R ) )  <->  f  e.  (/) ) )
2013, 19mtbiri 304 . . . . . . . . . . 11  |-  ( -.  S  e.  _V  ->  -.  f  e.  ( Base `  ( S X_s R ) ) )
2120con4i 133 . . . . . . . . . 10  |-  ( f  e.  ( Base `  ( S X_s R ) )  ->  S  e.  _V )
2221adantl 467 . . . . . . . . 9  |-  ( ( ( R  Fn  I  /\  I  e.  V
)  /\  f  e.  ( Base `  ( S X_s R ) ) )  ->  S  e.  _V )
23 simplr 760 . . . . . . . . 9  |-  ( ( ( R  Fn  I  /\  I  e.  V
)  /\  f  e.  ( Base `  ( S X_s R ) ) )  ->  I  e.  V )
24 simpr 462 . . . . . . . . 9  |-  ( ( ( R  Fn  I  /\  I  e.  V
)  /\  f  e.  ( Base `  ( S X_s R ) ) )  -> 
f  e.  ( Base `  ( S X_s R ) ) )
2511, 12, 22, 23, 6, 24prdsbasfn 15328 . . . . . . . 8  |-  ( ( ( R  Fn  I  /\  I  e.  V
)  /\  f  e.  ( Base `  ( S X_s R ) ) )  -> 
f  Fn  I )
26 fn0g 16456 . . . . . . . . . . . 12  |-  0g  Fn  _V
27 dffn2 5747 . . . . . . . . . . . 12  |-  ( 0g  Fn  _V  <->  0g : _V
--> _V )
2826, 27mpbi 211 . . . . . . . . . . 11  |-  0g : _V
--> _V
29 dffn2 5747 . . . . . . . . . . . 12  |-  ( R  Fn  I  <->  R :
I --> _V )
3029biimpi 197 . . . . . . . . . . 11  |-  ( R  Fn  I  ->  R : I --> _V )
31 fco 5756 . . . . . . . . . . 11  |-  ( ( 0g : _V --> _V  /\  R : I --> _V )  ->  ( 0g  o.  R
) : I --> _V )
3228, 30, 31sylancr 667 . . . . . . . . . 10  |-  ( R  Fn  I  ->  ( 0g  o.  R ) : I --> _V )
33 ffn 5746 . . . . . . . . . 10  |-  ( ( 0g  o.  R ) : I --> _V  ->  ( 0g  o.  R )  Fn  I )
3432, 33syl 17 . . . . . . . . 9  |-  ( R  Fn  I  ->  ( 0g  o.  R )  Fn  I )
3534ad2antrr 730 . . . . . . . 8  |-  ( ( ( R  Fn  I  /\  I  e.  V
)  /\  f  e.  ( Base `  ( S X_s R ) ) )  -> 
( 0g  o.  R
)  Fn  I )
36 fndmdif 6001 . . . . . . . 8  |-  ( ( f  Fn  I  /\  ( 0g  o.  R
)  Fn  I )  ->  dom  ( f  \  ( 0g  o.  R ) )  =  { x  e.  I  |  ( f `  x )  =/=  (
( 0g  o.  R
) `  x ) } )
3725, 35, 36syl2anc 665 . . . . . . 7  |-  ( ( ( R  Fn  I  /\  I  e.  V
)  /\  f  e.  ( Base `  ( S X_s R ) ) )  ->  dom  ( f  \  ( 0g  o.  R ) )  =  { x  e.  I  |  ( f `
 x )  =/=  ( ( 0g  o.  R ) `  x
) } )
38 fndm 5693 . . . . . . . . 9  |-  ( R  Fn  I  ->  dom  R  =  I )
39 rabeq 3081 . . . . . . . . 9  |-  ( dom 
R  =  I  ->  { x  e.  dom  R  |  ( f `  x )  =/=  ( 0g `  ( R `  x ) ) }  =  { x  e.  I  |  ( f `
 x )  =/=  ( 0g `  ( R `  x )
) } )
4038, 39syl 17 . . . . . . . 8  |-  ( R  Fn  I  ->  { x  e.  dom  R  |  ( f `  x )  =/=  ( 0g `  ( R `  x ) ) }  =  {
x  e.  I  |  ( f `  x
)  =/=  ( 0g
`  ( R `  x ) ) } )
4140ad2antrr 730 . . . . . . 7  |-  ( ( ( R  Fn  I  /\  I  e.  V
)  /\  f  e.  ( Base `  ( S X_s R ) ) )  ->  { x  e.  dom  R  |  ( f `  x )  =/=  ( 0g `  ( R `  x ) ) }  =  { x  e.  I  |  ( f `
 x )  =/=  ( 0g `  ( R `  x )
) } )
4210, 37, 413eqtr4d 2480 . . . . . 6  |-  ( ( ( R  Fn  I  /\  I  e.  V
)  /\  f  e.  ( Base `  ( S X_s R ) ) )  ->  dom  ( f  \  ( 0g  o.  R ) )  =  { x  e. 
dom  R  |  (
f `  x )  =/=  ( 0g `  ( R `  x )
) } )
4342eleq1d 2498 . . . . 5  |-  ( ( ( R  Fn  I  /\  I  e.  V
)  /\  f  e.  ( Base `  ( S X_s R ) ) )  -> 
( dom  ( f  \  ( 0g  o.  R ) )  e. 
Fin 
<->  { x  e.  dom  R  |  ( f `  x )  =/=  ( 0g `  ( R `  x ) ) }  e.  Fin ) )
4443rabbidva 3078 . . . 4  |-  ( ( R  Fn  I  /\  I  e.  V )  ->  { f  e.  (
Base `  ( S X_s R ) )  |  dom  ( f  \  ( 0g  o.  R ) )  e.  Fin }  =  { f  e.  (
Base `  ( S X_s R ) )  |  {
x  e.  dom  R  |  ( f `  x )  =/=  ( 0g `  ( R `  x ) ) }  e.  Fin } )
455, 44syl5eq 2482 . . 3  |-  ( ( R  Fn  I  /\  I  e.  V )  ->  { f  e.  (
Base `  P )  |  dom  ( f  \ 
( 0g  o.  R
) )  e.  Fin }  =  { f  e.  ( Base `  ( S X_s R ) )  |  { x  e.  dom  R  |  ( f `  x )  =/=  ( 0g `  ( R `  x ) ) }  e.  Fin } )
46 fnex 6147 . . . 4  |-  ( ( R  Fn  I  /\  I  e.  V )  ->  R  e.  _V )
47 eqid 2429 . . . . 5  |-  { f  e.  ( Base `  ( S X_s R ) )  |  { x  e.  dom  R  |  ( f `  x )  =/=  ( 0g `  ( R `  x ) ) }  e.  Fin }  =  { f  e.  (
Base `  ( S X_s R ) )  |  {
x  e.  dom  R  |  ( f `  x )  =/=  ( 0g `  ( R `  x ) ) }  e.  Fin }
4847dsmmbase 19229 . . . 4  |-  ( R  e.  _V  ->  { f  e.  ( Base `  ( S X_s R ) )  |  { x  e.  dom  R  |  ( f `  x )  =/=  ( 0g `  ( R `  x ) ) }  e.  Fin }  =  ( Base `  ( S  (+)m 
R ) ) )
4946, 48syl 17 . . 3  |-  ( ( R  Fn  I  /\  I  e.  V )  ->  { f  e.  (
Base `  ( S X_s R ) )  |  {
x  e.  dom  R  |  ( f `  x )  =/=  ( 0g `  ( R `  x ) ) }  e.  Fin }  =  ( Base `  ( S  (+)m 
R ) ) )
5045, 49eqtrd 2470 . 2  |-  ( ( R  Fn  I  /\  I  e.  V )  ->  { f  e.  (
Base `  P )  |  dom  ( f  \ 
( 0g  o.  R
) )  e.  Fin }  =  ( Base `  ( S  (+)m  R ) ) )
511, 50syl5eq 2482 1  |-  ( ( R  Fn  I  /\  I  e.  V )  ->  B  =  ( Base `  ( S  (+)m  R ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1870    =/= wne 2625   {crab 2786   _Vcvv 3087    \ cdif 3439   (/)c0 3767   dom cdm 4854    o. ccom 4858    Fn wfn 5596   -->wf 5597   ` cfv 5601  (class class class)co 6305   Fincfn 7577   Basecbs 15084   0gc0g 15297   X_scprds 15303    (+)m cdsmm 19225
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615
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 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-oadd 7194  df-er 7371  df-map 7482  df-ixp 7531  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-sup 7962  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-fz 11783  df-struct 15086  df-ndx 15087  df-slot 15088  df-base 15089  df-sets 15090  df-ress 15091  df-plusg 15165  df-mulr 15166  df-sca 15168  df-vsca 15169  df-ip 15170  df-tset 15171  df-ple 15172  df-ds 15174  df-hom 15176  df-cco 15177  df-0g 15299  df-prds 15305  df-dsmm 19226
This theorem is referenced by:  dsmmfi  19232  frlmbas  19249
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