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Theorem dsmmbas2 27203
Description: Base set of the direct sum module using the fndmin 5632 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 5528 . . . . 5  |-  ( Base `  P )  =  (
Base `  ( S X_s R ) )
4 rabeq 2782 . . . . 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 5594 . . . . . . . . . 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 2460 . . . . . . . 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 2779 . . . . . . 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 2283 . . . . . . . . 9  |-  ( S
X_s
R )  =  ( S X_s R )
12 eqid 2283 . . . . . . . . 9  |-  ( Base `  ( S X_s R ) )  =  ( Base `  ( S X_s R ) )
13 noel 3459 . . . . . . . . . . . 12  |-  -.  f  e.  (/)
14 reldmprds 13349 . . . . . . . . . . . . . . . 16  |-  Rel  dom  X_s
1514ovprc1 5886 . . . . . . . . . . . . . . 15  |-  ( -.  S  e.  _V  ->  ( S X_s R )  =  (/) )
1615fveq2d 5529 . . . . . . . . . . . . . 14  |-  ( -.  S  e.  _V  ->  (
Base `  ( S X_s R ) )  =  (
Base `  (/) ) )
17 base0 13185 . . . . . . . . . . . . . 14  |-  (/)  =  (
Base `  (/) )
1816, 17syl6eqr 2333 . . . . . . . . . . . . 13  |-  ( -.  S  e.  _V  ->  (
Base `  ( S X_s R ) )  =  (/) )
1918eleq2d 2350 . . . . . . . . . . . 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 13370 . . . . . . . 8  |-  ( ( ( R  Fn  I  /\  I  e.  V
)  /\  f  e.  ( Base `  ( S X_s R ) ) )  -> 
f  Fn  I )
26 fn0g 14385 . . . . . . . . . . . 12  |-  0g  Fn  _V
27 dffn2 5390 . . . . . . . . . . . 12  |-  ( 0g  Fn  _V  <->  0g : _V
--> _V )
2826, 27mpbi 199 . . . . . . . . . . 11  |-  0g : _V
--> _V
29 dffn2 5390 . . . . . . . . . . . 12  |-  ( R  Fn  I  <->  R :
I --> _V )
3029biimpi 186 . . . . . . . . . . 11  |-  ( R  Fn  I  ->  R : I --> _V )
31 fco 5398 . . . . . . . . . . 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 5389 . . . . . . . . . 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 5629 . . . . . . . 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 5343 . . . . . . . . 9  |-  ( R  Fn  I  ->  dom  R  =  I )
39 rabeq 2782 . . . . . . . . 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 2325 . . . . . 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 2349 . . . . 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 2779 . . . 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 2327 . . 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 5741 . . . 4  |-  ( ( R  Fn  I  /\  I  e.  V )  ->  R  e.  _V )
47 eqid 2283 . . . . 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 27201 . . . 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 2315 . 2  |-  ( ( R  Fn  I  /\  I  e.  V )  ->  { f  e.  (
Base `  P )  |  dom  ( f  \ 
( 0g  o.  R
) )  e.  Fin }  =  ( Base `  ( S  (+)m  R ) ) )
511, 50syl5eq 2327 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 1623    e. wcel 1684    =/= wne 2446   {crab 2547   _Vcvv 2788    \ cdif 3149   (/)c0 3455   dom cdm 4689    o. ccom 4693    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858   Fincfn 6863   Basecbs 13148   X_scprds 13346   0gc0g 13400    (+)m cdsmm 27197
This theorem is referenced by:  dsmmfi  27204  frlmbas  27223
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-fz 10783  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-hom 13232  df-cco 13233  df-prds 13348  df-0g 13404  df-dsmm 27198
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