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Theorem dsmmval 19374
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 3040 . 2  |-  ( R  e.  V  ->  R  e.  _V )
2 oveq12 6317 . . . . 5  |-  ( ( s  =  S  /\  r  =  R )  ->  ( s X_s r )  =  ( S X_s R ) )
3 eqid 2471 . . . . . . . . 9  |-  ( s
X_s r )  =  ( s X_s r )
4 vex 3034 . . . . . . . . . 10  |-  s  e. 
_V
54a1i 11 . . . . . . . . 9  |-  ( ( s  =  S  /\  r  =  R )  ->  s  e.  _V )
6 vex 3034 . . . . . . . . . 10  |-  r  e. 
_V
76a1i 11 . . . . . . . . 9  |-  ( ( s  =  S  /\  r  =  R )  ->  r  e.  _V )
8 eqid 2471 . . . . . . . . 9  |-  ( Base `  ( s X_s r ) )  =  ( Base `  (
s X_s r ) )
9 eqidd 2472 . . . . . . . . 9  |-  ( ( s  =  S  /\  r  =  R )  ->  dom  r  =  dom  r )
103, 5, 7, 8, 9prdsbas 15433 . . . . . . . 8  |-  ( ( s  =  S  /\  r  =  R )  ->  ( Base `  (
s X_s r ) )  = 
X_ x  e.  dom  r ( Base `  (
r `  x )
) )
112fveq2d 5883 . . . . . . . 8  |-  ( ( s  =  S  /\  r  =  R )  ->  ( Base `  (
s X_s r ) )  =  ( Base `  ( S X_s R ) ) )
1210, 11eqtr3d 2507 . . . . . . 7  |-  ( ( s  =  S  /\  r  =  R )  -> 
X_ x  e.  dom  r ( Base `  (
r `  x )
)  =  ( Base `  ( S X_s R ) ) )
13 simpr 468 . . . . . . . . . 10  |-  ( ( s  =  S  /\  r  =  R )  ->  r  =  R )
1413dmeqd 5042 . . . . . . . . 9  |-  ( ( s  =  S  /\  r  =  R )  ->  dom  r  =  dom  R )
1513fveq1d 5881 . . . . . . . . . . 11  |-  ( ( s  =  S  /\  r  =  R )  ->  ( r `  x
)  =  ( R `
 x ) )
1615fveq2d 5883 . . . . . . . . . 10  |-  ( ( s  =  S  /\  r  =  R )  ->  ( 0g `  (
r `  x )
)  =  ( 0g
`  ( R `  x ) ) )
1716neeq2d 2703 . . . . . . . . 9  |-  ( ( s  =  S  /\  r  =  R )  ->  ( ( f `  x )  =/=  ( 0g `  ( r `  x ) )  <->  ( f `  x )  =/=  ( 0g `  ( R `  x ) ) ) )
1814, 17rabeqbidv 3026 . . . . . . . 8  |-  ( ( s  =  S  /\  r  =  R )  ->  { x  e.  dom  r  |  ( f `  x )  =/=  ( 0g `  ( r `  x ) ) }  =  { x  e. 
dom  R  |  (
f `  x )  =/=  ( 0g `  ( R `  x )
) } )
1918eleq1d 2533 . . . . . . 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 3026 . . . . . 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 2523 . . . . 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 6326 . . . 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 19372 . . . 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 6336 . . . 4  |-  ( ( S X_s R )s  B )  e.  _V
2623, 24, 25ovmpt2a 6446 . . 3  |-  ( ( S  e.  _V  /\  R  e.  _V )  ->  ( S  (+)m  R )  =  ( ( S
X_s
R )s  B ) )
27 reldmdsmm 19373 . . . . . . 7  |-  Rel  dom  (+)m
2827ovprc1 6339 . . . . . 6  |-  ( -.  S  e.  _V  ->  ( S  (+)m  R )  =  (/) )
29 ress0 15261 . . . . . 6  |-  ( (/)s  B )  =  (/)
3028, 29syl6eqr 2523 . . . . 5  |-  ( -.  S  e.  _V  ->  ( S  (+)m  R )  =  (
(/)s  B ) )
31 reldmprds 15425 . . . . . . 7  |-  Rel  dom  X_s
3231ovprc1 6339 . . . . . 6  |-  ( -.  S  e.  _V  ->  ( S X_s R )  =  (/) )
3332oveq1d 6323 . . . . 5  |-  ( -.  S  e.  _V  ->  ( ( S X_s R )s  B )  =  (
(/)s  B ) )
3430, 33eqtr4d 2508 . . . 4  |-  ( -.  S  e.  _V  ->  ( S  (+)m  R )  =  ( ( S X_s R )s  B ) )
3534adantr 472 . . 3  |-  ( ( -.  S  e.  _V  /\  R  e.  _V )  ->  ( S  (+)m  R )  =  ( ( S
X_s
R )s  B ) )
3626, 35pm2.61ian 807 . 2  |-  ( R  e.  _V  ->  ( S  (+)m  R )  =  ( ( S X_s R )s  B ) )
371, 36syl 17 1  |-  ( R  e.  V  ->  ( S  (+)m  R )  =  ( ( S X_s R )s  B ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 376    = wceq 1452    e. wcel 1904    =/= wne 2641   {crab 2760   _Vcvv 3031   (/)c0 3722   dom cdm 4839   ` cfv 5589  (class class class)co 6308   X_cixp 7540   Fincfn 7587   Basecbs 15199   ↾s cress 15200   0gc0g 15416   X_scprds 15422    (+)m cdsmm 19371
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-map 7492  df-ixp 7541  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-sup 7974  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-2 10690  df-3 10691  df-4 10692  df-5 10693  df-6 10694  df-7 10695  df-8 10696  df-9 10697  df-10 10698  df-n0 10894  df-z 10962  df-dec 11075  df-uz 11183  df-fz 11811  df-struct 15201  df-ndx 15202  df-slot 15203  df-base 15204  df-ress 15206  df-plusg 15281  df-mulr 15282  df-sca 15284  df-vsca 15285  df-ip 15286  df-tset 15287  df-ple 15288  df-ds 15290  df-hom 15292  df-cco 15293  df-prds 15424  df-dsmm 19372
This theorem is referenced by:  dsmmbase  19375  dsmmval2  19376
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