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

Theorem dsmmval 26869
Description: Value of the module direct sum. (Contributed by Stefan O'Rear, 7-Jan-2015.)
Hypothesis
Ref Expression
dsmmval.b  |-  B  =  { f  e.  (
Base `  ( S X_s R ) )  |  {
x  e.  dom  R  |  ( f `  x )  =/=  ( 0g `  ( R `  x ) ) }  e.  Fin }
Assertion
Ref Expression
dsmmval  |-  ( R  e.  V  ->  ( S  (+)m  R )  =  ( ( S X_s R )s  B ) )
Distinct variable groups:    S, f, x    R, f, x
Allowed substitution hints:    B( x, f)    V( x, f)

Proof of Theorem dsmmval
Dummy variables  s 
r are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 2907 . 2  |-  ( R  e.  V  ->  R  e.  _V )
2 oveq12 6029 . . . . 5  |-  ( ( s  =  S  /\  r  =  R )  ->  ( s X_s r )  =  ( S X_s R ) )
3 eqid 2387 . . . . . . . . 9  |-  ( s
X_s r )  =  ( s X_s r )
4 vex 2902 . . . . . . . . . 10  |-  s  e. 
_V
54a1i 11 . . . . . . . . 9  |-  ( ( s  =  S  /\  r  =  R )  ->  s  e.  _V )
6 vex 2902 . . . . . . . . . 10  |-  r  e. 
_V
76a1i 11 . . . . . . . . 9  |-  ( ( s  =  S  /\  r  =  R )  ->  r  e.  _V )
8 eqid 2387 . . . . . . . . 9  |-  ( Base `  ( s X_s r ) )  =  ( Base `  (
s X_s r ) )
9 eqidd 2388 . . . . . . . . 9  |-  ( ( s  =  S  /\  r  =  R )  ->  dom  r  =  dom  r )
103, 5, 7, 8, 9prdsbas 13607 . . . . . . . 8  |-  ( ( s  =  S  /\  r  =  R )  ->  ( Base `  (
s X_s r ) )  = 
X_ x  e.  dom  r ( Base `  (
r `  x )
) )
112fveq2d 5672 . . . . . . . 8  |-  ( ( s  =  S  /\  r  =  R )  ->  ( Base `  (
s X_s r ) )  =  ( Base `  ( S X_s R ) ) )
1210, 11eqtr3d 2421 . . . . . . 7  |-  ( ( s  =  S  /\  r  =  R )  -> 
X_ x  e.  dom  r ( Base `  (
r `  x )
)  =  ( Base `  ( S X_s R ) ) )
13 simpr 448 . . . . . . . . . 10  |-  ( ( s  =  S  /\  r  =  R )  ->  r  =  R )
1413dmeqd 5012 . . . . . . . . 9  |-  ( ( s  =  S  /\  r  =  R )  ->  dom  r  =  dom  R )
1513fveq1d 5670 . . . . . . . . . . 11  |-  ( ( s  =  S  /\  r  =  R )  ->  ( r `  x
)  =  ( R `
 x ) )
1615fveq2d 5672 . . . . . . . . . 10  |-  ( ( s  =  S  /\  r  =  R )  ->  ( 0g `  (
r `  x )
)  =  ( 0g
`  ( R `  x ) ) )
1716neeq2d 2564 . . . . . . . . 9  |-  ( ( s  =  S  /\  r  =  R )  ->  ( ( f `  x )  =/=  ( 0g `  ( r `  x ) )  <->  ( f `  x )  =/=  ( 0g `  ( R `  x ) ) ) )
1814, 17rabeqbidv 2894 . . . . . . . 8  |-  ( ( s  =  S  /\  r  =  R )  ->  { x  e.  dom  r  |  ( f `  x )  =/=  ( 0g `  ( r `  x ) ) }  =  { x  e. 
dom  R  |  (
f `  x )  =/=  ( 0g `  ( R `  x )
) } )
1918eleq1d 2453 . . . . . . 7  |-  ( ( s  =  S  /\  r  =  R )  ->  ( { x  e. 
dom  r  |  ( f `  x )  =/=  ( 0g `  ( r `  x
) ) }  e.  Fin 
<->  { x  e.  dom  R  |  ( f `  x )  =/=  ( 0g `  ( R `  x ) ) }  e.  Fin ) )
2012, 19rabeqbidv 2894 . . . . . 6  |-  ( ( s  =  S  /\  r  =  R )  ->  { f  e.  X_ x  e.  dom  r (
Base `  ( r `  x ) )  |  { 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 } )
21 dsmmval.b . . . . . 6  |-  B  =  { f  e.  (
Base `  ( S X_s R ) )  |  {
x  e.  dom  R  |  ( f `  x )  =/=  ( 0g `  ( R `  x ) ) }  e.  Fin }
2220, 21syl6eqr 2437 . . . . 5  |-  ( ( s  =  S  /\  r  =  R )  ->  { f  e.  X_ x  e.  dom  r (
Base `  ( r `  x ) )  |  { x  e.  dom  r  |  ( f `  x )  =/=  ( 0g `  ( r `  x ) ) }  e.  Fin }  =  B )
232, 22oveq12d 6038 . . . 4  |-  ( ( s  =  S  /\  r  =  R )  ->  ( ( s X_s r
)s 
{ f  e.  X_ x  e.  dom  r (
Base `  ( r `  x ) )  |  { x  e.  dom  r  |  ( f `  x )  =/=  ( 0g `  ( r `  x ) ) }  e.  Fin } )  =  ( ( S
X_s
R )s  B ) )
24 df-dsmm 26867 . . . 4  |-  (+)m  =  ( s  e.  _V , 
r  e.  _V  |->  ( ( s X_s r )s  { f  e.  X_ x  e.  dom  r (
Base `  ( r `  x ) )  |  { x  e.  dom  r  |  ( f `  x )  =/=  ( 0g `  ( r `  x ) ) }  e.  Fin } ) )
25 ovex 6045 . . . 4  |-  ( ( S X_s R )s  B )  e.  _V
2623, 24, 25ovmpt2a 6143 . . 3  |-  ( ( S  e.  _V  /\  R  e.  _V )  ->  ( S  (+)m  R )  =  ( ( S
X_s
R )s  B ) )
27 reldmdsmm 26868 . . . . . . 7  |-  Rel  dom  (+)m
2827ovprc1 6048 . . . . . 6  |-  ( -.  S  e.  _V  ->  ( S  (+)m  R )  =  (/) )
29 ress0 13450 . . . . . 6  |-  ( (/)s  B )  =  (/)
3028, 29syl6eqr 2437 . . . . 5  |-  ( -.  S  e.  _V  ->  ( S  (+)m  R )  =  (
(/)s  B ) )
31 reldmprds 13599 . . . . . . 7  |-  Rel  dom  X_s
3231ovprc1 6048 . . . . . 6  |-  ( -.  S  e.  _V  ->  ( S X_s R )  =  (/) )
3332oveq1d 6035 . . . . 5  |-  ( -.  S  e.  _V  ->  ( ( S X_s R )s  B )  =  (
(/)s  B ) )
3430, 33eqtr4d 2422 . . . 4  |-  ( -.  S  e.  _V  ->  ( S  (+)m  R )  =  ( ( S X_s R )s  B ) )
3534adantr 452 . . 3  |-  ( ( -.  S  e.  _V  /\  R  e.  _V )  ->  ( S  (+)m  R )  =  ( ( S
X_s
R )s  B ) )
3626, 35pm2.61ian 766 . 2  |-  ( R  e.  _V  ->  ( S  (+)m  R )  =  ( ( S X_s R )s  B ) )
371, 36syl 16 1  |-  ( R  e.  V  ->  ( S  (+)m  R )  =  ( ( S X_s R )s  B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717    =/= wne 2550   {crab 2653   _Vcvv 2899   (/)c0 3571   dom cdm 4818   ` cfv 5394  (class class class)co 6020   X_cixp 6999   Fincfn 7045   Basecbs 13396   ↾s cress 13397   X_scprds 13596   0gc0g 13650    (+)m cdsmm 26866
This theorem is referenced by:  dsmmbase  26870  dsmmval2  26871
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-int 3993  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-riota 6485  df-recs 6569  df-rdg 6604  df-1o 6660  df-oadd 6664  df-er 6841  df-map 6956  df-ixp 7000  df-en 7046  df-dom 7047  df-sdom 7048  df-fin 7049  df-sup 7381  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-nn 9933  df-2 9990  df-3 9991  df-4 9992  df-5 9993  df-6 9994  df-7 9995  df-8 9996  df-9 9997  df-10 9998  df-n0 10154  df-z 10215  df-dec 10315  df-uz 10421  df-fz 10976  df-struct 13398  df-ndx 13399  df-slot 13400  df-base 13401  df-ress 13403  df-plusg 13469  df-mulr 13470  df-sca 13472  df-vsca 13473  df-tset 13475  df-ple 13476  df-ds 13478  df-hom 13480  df-cco 13481  df-prds 13598  df-dsmm 26867
  Copyright terms: Public domain W3C validator