Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dsmmbas2 Unicode version

Theorem dsmmbas2 27306
Description: Base set of the direct sum module using the fndmin 5648 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 5544 . . . . 5  |-  ( Base `  P )  =  (
Base `  ( S X_s R ) )
4 rabeq 2795 . . . . 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 730 . . . . . . . . . 10  |-  ( ( ( R  Fn  I  /\  I  e.  V
)  /\  f  e.  ( Base `  ( S X_s R ) ) )  ->  R  Fn  I )
7 fvco2 5610 . . . . . . . . . 10  |-  ( ( R  Fn  I  /\  x  e.  I )  ->  ( ( 0g  o.  R ) `  x
)  =  ( 0g
`  ( R `  x ) ) )
86, 7sylan 457 . . . . . . . . 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 2473 . . . . . . . 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 2792 . . . . . . 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 2296 . . . . . . . . 9  |-  ( S
X_s
R )  =  ( S X_s R )
12 eqid 2296 . . . . . . . . 9  |-  ( Base `  ( S X_s R ) )  =  ( Base `  ( S X_s R ) )
13 noel 3472 . . . . . . . . . . . 12  |-  -.  f  e.  (/)
14 reldmprds 13365 . . . . . . . . . . . . . . . 16  |-  Rel  dom  X_s
1514ovprc1 5902 . . . . . . . . . . . . . . 15  |-  ( -.  S  e.  _V  ->  ( S X_s R )  =  (/) )
1615fveq2d 5545 . . . . . . . . . . . . . 14  |-  ( -.  S  e.  _V  ->  (
Base `  ( S X_s R ) )  =  (
Base `  (/) ) )
17 base0 13201 . . . . . . . . . . . . . 14  |-  (/)  =  (
Base `  (/) )
1816, 17syl6eqr 2346 . . . . . . . . . . . . 13  |-  ( -.  S  e.  _V  ->  (
Base `  ( S X_s R ) )  =  (/) )
1918eleq2d 2363 . . . . . . . . . . . 12  |-  ( -.  S  e.  _V  ->  ( f  e.  ( Base `  ( S X_s R ) )  <->  f  e.  (/) ) )
2013, 19mtbiri 294 . . . . . . . . . . 11  |-  ( -.  S  e.  _V  ->  -.  f  e.  ( Base `  ( S X_s R ) ) )
2120con4i 122 . . . . . . . . . 10  |-  ( f  e.  ( Base `  ( S X_s R ) )  ->  S  e.  _V )
2221adantl 452 . . . . . . . . 9  |-  ( ( ( R  Fn  I  /\  I  e.  V
)  /\  f  e.  ( Base `  ( S X_s R ) ) )  ->  S  e.  _V )
23 simplr 731 . . . . . . . . 9  |-  ( ( ( R  Fn  I  /\  I  e.  V
)  /\  f  e.  ( Base `  ( S X_s R ) ) )  ->  I  e.  V )
24 simpr 447 . . . . . . . . 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 13386 . . . . . . . 8  |-  ( ( ( R  Fn  I  /\  I  e.  V
)  /\  f  e.  ( Base `  ( S X_s R ) ) )  -> 
f  Fn  I )
26 fn0g 14401 . . . . . . . . . . . 12  |-  0g  Fn  _V
27 dffn2 5406 . . . . . . . . . . . 12  |-  ( 0g  Fn  _V  <->  0g : _V
--> _V )
2826, 27mpbi 199 . . . . . . . . . . 11  |-  0g : _V
--> _V
29 dffn2 5406 . . . . . . . . . . . 12  |-  ( R  Fn  I  <->  R :
I --> _V )
3029biimpi 186 . . . . . . . . . . 11  |-  ( R  Fn  I  ->  R : I --> _V )
31 fco 5414 . . . . . . . . . . 11  |-  ( ( 0g : _V --> _V  /\  R : I --> _V )  ->  ( 0g  o.  R
) : I --> _V )
3228, 30, 31sylancr 644 . . . . . . . . . 10  |-  ( R  Fn  I  ->  ( 0g  o.  R ) : I --> _V )
33 ffn 5405 . . . . . . . . . 10  |-  ( ( 0g  o.  R ) : I --> _V  ->  ( 0g  o.  R )  Fn  I )
3432, 33syl 15 . . . . . . . . 9  |-  ( R  Fn  I  ->  ( 0g  o.  R )  Fn  I )
3534ad2antrr 706 . . . . . . . 8  |-  ( ( ( R  Fn  I  /\  I  e.  V
)  /\  f  e.  ( Base `  ( S X_s R ) ) )  -> 
( 0g  o.  R
)  Fn  I )
36 fndmdif 5645 . . . . . . . 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 642 . . . . . . 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 5359 . . . . . . . . 9  |-  ( R  Fn  I  ->  dom  R  =  I )
39 rabeq 2795 . . . . . . . . 9  |-  ( dom 
R  =  I  ->  { x  e.  dom  R  |  ( f `  x )  =/=  ( 0g `  ( R `  x ) ) }  =  { x  e.  I  |  ( f `
 x )  =/=  ( 0g `  ( R `  x )
) } )
4038, 39syl 15 . . . . . . . 8  |-  ( R  Fn  I  ->  { x  e.  dom  R  |  ( f `  x )  =/=  ( 0g `  ( R `  x ) ) }  =  {
x  e.  I  |  ( f `  x
)  =/=  ( 0g
`  ( R `  x ) ) } )
4140ad2antrr 706 . . . . . . 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 2338 . . . . . 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 2362 . . . . 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 2792 . . . 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 2340 . . 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 5757 . . . 4  |-  ( ( R  Fn  I  /\  I  e.  V )  ->  R  e.  _V )
47 eqid 2296 . . . . 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 27304 . . . 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 15 . . 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 2328 . 2  |-  ( ( R  Fn  I  /\  I  e.  V )  ->  { f  e.  (
Base `  P )  |  dom  ( f  \ 
( 0g  o.  R
) )  e.  Fin }  =  ( Base `  ( S  (+)m  R ) ) )
511, 50syl5eq 2340 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 358    = wceq 1632    e. wcel 1696    =/= wne 2459   {crab 2560   _Vcvv 2801    \ cdif 3162   (/)c0 3468   dom cdm 4705    o. ccom 4709    Fn wfn 5266   -->wf 5267   ` cfv 5271  (class class class)co 5874   Fincfn 6879   Basecbs 13164   X_scprds 13362   0gc0g 13416    (+)m cdsmm 27300
This theorem is referenced by:  dsmmfi  27307  frlmbas  27326
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-map 6790  df-ixp 6834  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-fz 10799  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-sca 13240  df-vsca 13241  df-tset 13243  df-ple 13244  df-ds 13246  df-hom 13248  df-cco 13249  df-prds 13364  df-0g 13420  df-dsmm 27301
  Copyright terms: Public domain W3C validator