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Theorem dsmmbas2 18162
Description: Base set of the direct sum module using the fndmin 5810 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 5694 . . . . 5  |-  ( Base `  P )  =  (
Base `  ( S X_s R ) )
4 rabeq 2966 . . . . 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 753 . . . . . . . . . 10  |-  ( ( ( R  Fn  I  /\  I  e.  V
)  /\  f  e.  ( Base `  ( S X_s R ) ) )  ->  R  Fn  I )
7 fvco2 5766 . . . . . . . . . 10  |-  ( ( R  Fn  I  /\  x  e.  I )  ->  ( ( 0g  o.  R ) `  x
)  =  ( 0g
`  ( R `  x ) ) )
86, 7sylan 471 . . . . . . . . 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 2622 . . . . . . . 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 2963 . . . . . . 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 2443 . . . . . . . . 9  |-  ( S
X_s
R )  =  ( S X_s R )
12 eqid 2443 . . . . . . . . 9  |-  ( Base `  ( S X_s R ) )  =  ( Base `  ( S X_s R ) )
13 noel 3641 . . . . . . . . . . . 12  |-  -.  f  e.  (/)
14 reldmprds 14387 . . . . . . . . . . . . . . . 16  |-  Rel  dom  X_s
1514ovprc1 6119 . . . . . . . . . . . . . . 15  |-  ( -.  S  e.  _V  ->  ( S X_s R )  =  (/) )
1615fveq2d 5695 . . . . . . . . . . . . . 14  |-  ( -.  S  e.  _V  ->  (
Base `  ( S X_s R ) )  =  (
Base `  (/) ) )
17 base0 14213 . . . . . . . . . . . . . 14  |-  (/)  =  (
Base `  (/) )
1816, 17syl6eqr 2493 . . . . . . . . . . . . 13  |-  ( -.  S  e.  _V  ->  (
Base `  ( S X_s R ) )  =  (/) )
1918eleq2d 2510 . . . . . . . . . . . 12  |-  ( -.  S  e.  _V  ->  ( f  e.  ( Base `  ( S X_s R ) )  <->  f  e.  (/) ) )
2013, 19mtbiri 303 . . . . . . . . . . 11  |-  ( -.  S  e.  _V  ->  -.  f  e.  ( Base `  ( S X_s R ) ) )
2120con4i 130 . . . . . . . . . 10  |-  ( f  e.  ( Base `  ( S X_s R ) )  ->  S  e.  _V )
2221adantl 466 . . . . . . . . 9  |-  ( ( ( R  Fn  I  /\  I  e.  V
)  /\  f  e.  ( Base `  ( S X_s R ) ) )  ->  S  e.  _V )
23 simplr 754 . . . . . . . . 9  |-  ( ( ( R  Fn  I  /\  I  e.  V
)  /\  f  e.  ( Base `  ( S X_s R ) ) )  ->  I  e.  V )
24 simpr 461 . . . . . . . . 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 14409 . . . . . . . 8  |-  ( ( ( R  Fn  I  /\  I  e.  V
)  /\  f  e.  ( Base `  ( S X_s R ) ) )  -> 
f  Fn  I )
26 fn0g 15433 . . . . . . . . . . . 12  |-  0g  Fn  _V
27 dffn2 5560 . . . . . . . . . . . 12  |-  ( 0g  Fn  _V  <->  0g : _V
--> _V )
2826, 27mpbi 208 . . . . . . . . . . 11  |-  0g : _V
--> _V
29 dffn2 5560 . . . . . . . . . . . 12  |-  ( R  Fn  I  <->  R :
I --> _V )
3029biimpi 194 . . . . . . . . . . 11  |-  ( R  Fn  I  ->  R : I --> _V )
31 fco 5568 . . . . . . . . . . 11  |-  ( ( 0g : _V --> _V  /\  R : I --> _V )  ->  ( 0g  o.  R
) : I --> _V )
3228, 30, 31sylancr 663 . . . . . . . . . 10  |-  ( R  Fn  I  ->  ( 0g  o.  R ) : I --> _V )
33 ffn 5559 . . . . . . . . . 10  |-  ( ( 0g  o.  R ) : I --> _V  ->  ( 0g  o.  R )  Fn  I )
3432, 33syl 16 . . . . . . . . 9  |-  ( R  Fn  I  ->  ( 0g  o.  R )  Fn  I )
3534ad2antrr 725 . . . . . . . 8  |-  ( ( ( R  Fn  I  /\  I  e.  V
)  /\  f  e.  ( Base `  ( S X_s R ) ) )  -> 
( 0g  o.  R
)  Fn  I )
36 fndmdif 5807 . . . . . . . 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 661 . . . . . . 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 5510 . . . . . . . . 9  |-  ( R  Fn  I  ->  dom  R  =  I )
39 rabeq 2966 . . . . . . . . 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 725 . . . . . . 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 2485 . . . . . 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 2509 . . . . 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 2963 . . . 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 2487 . . 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 5944 . . . 4  |-  ( ( R  Fn  I  /\  I  e.  V )  ->  R  e.  _V )
47 eqid 2443 . . . . 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 18160 . . . 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 2475 . 2  |-  ( ( R  Fn  I  /\  I  e.  V )  ->  { f  e.  (
Base `  P )  |  dom  ( f  \ 
( 0g  o.  R
) )  e.  Fin }  =  ( Base `  ( S  (+)m  R ) ) )
511, 50syl5eq 2487 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 369    = wceq 1369    e. wcel 1756    =/= wne 2606   {crab 2719   _Vcvv 2972    \ cdif 3325   (/)c0 3637   dom cdm 4840    o. ccom 4844    Fn wfn 5413   -->wf 5414   ` cfv 5418  (class class class)co 6091   Fincfn 7310   Basecbs 14174   0gc0g 14378   X_scprds 14384    (+)m cdsmm 18156
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-recs 6832  df-rdg 6866  df-1o 6920  df-oadd 6924  df-er 7101  df-map 7216  df-ixp 7264  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-sup 7691  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-nn 10323  df-2 10380  df-3 10381  df-4 10382  df-5 10383  df-6 10384  df-7 10385  df-8 10386  df-9 10387  df-10 10388  df-n0 10580  df-z 10647  df-dec 10756  df-uz 10862  df-fz 11438  df-struct 14176  df-ndx 14177  df-slot 14178  df-base 14179  df-sets 14180  df-ress 14181  df-plusg 14251  df-mulr 14252  df-sca 14254  df-vsca 14255  df-ip 14256  df-tset 14257  df-ple 14258  df-ds 14260  df-hom 14262  df-cco 14263  df-0g 14380  df-prds 14386  df-dsmm 18157
This theorem is referenced by:  dsmmfi  18163  frlmbas  18180  frlmbasOLD  18181
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