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Theorem dsmmbas2 27180
Description: Base set of the direct sum module using the fndmin 5837 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 5731 . . . . 5  |-  ( Base `  P )  =  (
Base `  ( S X_s R ) )
4 rabeq 2950 . . . . 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 8 . . . 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 731 . . . . . . . . . 10  |-  ( ( ( R  Fn  I  /\  I  e.  V
)  /\  f  e.  ( Base `  ( S X_s R ) ) )  ->  R  Fn  I )
7 fvco2 5798 . . . . . . . . . 10  |-  ( ( R  Fn  I  /\  x  e.  I )  ->  ( ( 0g  o.  R ) `  x
)  =  ( 0g
`  ( R `  x ) ) )
86, 7sylan 458 . . . . . . . . 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 2615 . . . . . . . 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 2947 . . . . . . 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 2436 . . . . . . . . 9  |-  ( S
X_s
R )  =  ( S X_s R )
12 eqid 2436 . . . . . . . . 9  |-  ( Base `  ( S X_s R ) )  =  ( Base `  ( S X_s R ) )
13 noel 3632 . . . . . . . . . . . 12  |-  -.  f  e.  (/)
14 reldmprds 13672 . . . . . . . . . . . . . . . 16  |-  Rel  dom  X_s
1514ovprc1 6109 . . . . . . . . . . . . . . 15  |-  ( -.  S  e.  _V  ->  ( S X_s R )  =  (/) )
1615fveq2d 5732 . . . . . . . . . . . . . 14  |-  ( -.  S  e.  _V  ->  (
Base `  ( S X_s R ) )  =  (
Base `  (/) ) )
17 base0 13506 . . . . . . . . . . . . . 14  |-  (/)  =  (
Base `  (/) )
1816, 17syl6eqr 2486 . . . . . . . . . . . . 13  |-  ( -.  S  e.  _V  ->  (
Base `  ( S X_s R ) )  =  (/) )
1918eleq2d 2503 . . . . . . . . . . . 12  |-  ( -.  S  e.  _V  ->  ( f  e.  ( Base `  ( S X_s R ) )  <->  f  e.  (/) ) )
2013, 19mtbiri 295 . . . . . . . . . . 11  |-  ( -.  S  e.  _V  ->  -.  f  e.  ( Base `  ( S X_s R ) ) )
2120con4i 124 . . . . . . . . . 10  |-  ( f  e.  ( Base `  ( S X_s R ) )  ->  S  e.  _V )
2221adantl 453 . . . . . . . . 9  |-  ( ( ( R  Fn  I  /\  I  e.  V
)  /\  f  e.  ( Base `  ( S X_s R ) ) )  ->  S  e.  _V )
23 simplr 732 . . . . . . . . 9  |-  ( ( ( R  Fn  I  /\  I  e.  V
)  /\  f  e.  ( Base `  ( S X_s R ) ) )  ->  I  e.  V )
24 simpr 448 . . . . . . . . 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 13693 . . . . . . . 8  |-  ( ( ( R  Fn  I  /\  I  e.  V
)  /\  f  e.  ( Base `  ( S X_s R ) ) )  -> 
f  Fn  I )
26 fn0g 14708 . . . . . . . . . . . 12  |-  0g  Fn  _V
27 dffn2 5592 . . . . . . . . . . . 12  |-  ( 0g  Fn  _V  <->  0g : _V
--> _V )
2826, 27mpbi 200 . . . . . . . . . . 11  |-  0g : _V
--> _V
29 dffn2 5592 . . . . . . . . . . . 12  |-  ( R  Fn  I  <->  R :
I --> _V )
3029biimpi 187 . . . . . . . . . . 11  |-  ( R  Fn  I  ->  R : I --> _V )
31 fco 5600 . . . . . . . . . . 11  |-  ( ( 0g : _V --> _V  /\  R : I --> _V )  ->  ( 0g  o.  R
) : I --> _V )
3228, 30, 31sylancr 645 . . . . . . . . . 10  |-  ( R  Fn  I  ->  ( 0g  o.  R ) : I --> _V )
33 ffn 5591 . . . . . . . . . 10  |-  ( ( 0g  o.  R ) : I --> _V  ->  ( 0g  o.  R )  Fn  I )
3432, 33syl 16 . . . . . . . . 9  |-  ( R  Fn  I  ->  ( 0g  o.  R )  Fn  I )
3534ad2antrr 707 . . . . . . . 8  |-  ( ( ( R  Fn  I  /\  I  e.  V
)  /\  f  e.  ( Base `  ( S X_s R ) ) )  -> 
( 0g  o.  R
)  Fn  I )
36 fndmdif 5834 . . . . . . . 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 643 . . . . . . 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 5544 . . . . . . . . 9  |-  ( R  Fn  I  ->  dom  R  =  I )
39 rabeq 2950 . . . . . . . . 9  |-  ( dom 
R  =  I  ->  { x  e.  dom  R  |  ( f `  x )  =/=  ( 0g `  ( R `  x ) ) }  =  { x  e.  I  |  ( f `
 x )  =/=  ( 0g `  ( R `  x )
) } )
4038, 39syl 16 . . . . . . . 8  |-  ( R  Fn  I  ->  { x  e.  dom  R  |  ( f `  x )  =/=  ( 0g `  ( R `  x ) ) }  =  {
x  e.  I  |  ( f `  x
)  =/=  ( 0g
`  ( R `  x ) ) } )
4140ad2antrr 707 . . . . . . 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 2478 . . . . . 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 2502 . . . . 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 2947 . . . 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 2480 . . 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 5961 . . . 4  |-  ( ( R  Fn  I  /\  I  e.  V )  ->  R  e.  _V )
47 eqid 2436 . . . . 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 27178 . . . 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 16 . . 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 2468 . 2  |-  ( ( R  Fn  I  /\  I  e.  V )  ->  { f  e.  (
Base `  P )  |  dom  ( f  \ 
( 0g  o.  R
) )  e.  Fin }  =  ( Base `  ( S  (+)m  R ) ) )
511, 50syl5eq 2480 1  |-  ( ( R  Fn  I  /\  I  e.  V )  ->  B  =  ( Base `  ( S  (+)m  R ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2599   {crab 2709   _Vcvv 2956    \ cdif 3317   (/)c0 3628   dom cdm 4878    o. ccom 4882    Fn wfn 5449   -->wf 5450   ` cfv 5454  (class class class)co 6081   Fincfn 7109   Basecbs 13469   X_scprds 13669   0gc0g 13723    (+)m cdsmm 27174
This theorem is referenced by:  dsmmfi  27181  frlmbas  27200
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-0g 13727  df-dsmm 27175
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