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Theorem dsmmval 18892
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 3118 . 2  |-  ( R  e.  V  ->  R  e.  _V )
2 oveq12 6305 . . . . 5  |-  ( ( s  =  S  /\  r  =  R )  ->  ( s X_s r )  =  ( S X_s R ) )
3 eqid 2457 . . . . . . . . 9  |-  ( s
X_s r )  =  ( s X_s r )
4 vex 3112 . . . . . . . . . 10  |-  s  e. 
_V
54a1i 11 . . . . . . . . 9  |-  ( ( s  =  S  /\  r  =  R )  ->  s  e.  _V )
6 vex 3112 . . . . . . . . . 10  |-  r  e. 
_V
76a1i 11 . . . . . . . . 9  |-  ( ( s  =  S  /\  r  =  R )  ->  r  e.  _V )
8 eqid 2457 . . . . . . . . 9  |-  ( Base `  ( s X_s r ) )  =  ( Base `  (
s X_s r ) )
9 eqidd 2458 . . . . . . . . 9  |-  ( ( s  =  S  /\  r  =  R )  ->  dom  r  =  dom  r )
103, 5, 7, 8, 9prdsbas 14874 . . . . . . . 8  |-  ( ( s  =  S  /\  r  =  R )  ->  ( Base `  (
s X_s r ) )  = 
X_ x  e.  dom  r ( Base `  (
r `  x )
) )
112fveq2d 5876 . . . . . . . 8  |-  ( ( s  =  S  /\  r  =  R )  ->  ( Base `  (
s X_s r ) )  =  ( Base `  ( S X_s R ) ) )
1210, 11eqtr3d 2500 . . . . . . 7  |-  ( ( s  =  S  /\  r  =  R )  -> 
X_ x  e.  dom  r ( Base `  (
r `  x )
)  =  ( Base `  ( S X_s R ) ) )
13 simpr 461 . . . . . . . . . 10  |-  ( ( s  =  S  /\  r  =  R )  ->  r  =  R )
1413dmeqd 5215 . . . . . . . . 9  |-  ( ( s  =  S  /\  r  =  R )  ->  dom  r  =  dom  R )
1513fveq1d 5874 . . . . . . . . . . 11  |-  ( ( s  =  S  /\  r  =  R )  ->  ( r `  x
)  =  ( R `
 x ) )
1615fveq2d 5876 . . . . . . . . . 10  |-  ( ( s  =  S  /\  r  =  R )  ->  ( 0g `  (
r `  x )
)  =  ( 0g
`  ( R `  x ) ) )
1716neeq2d 2735 . . . . . . . . 9  |-  ( ( s  =  S  /\  r  =  R )  ->  ( ( f `  x )  =/=  ( 0g `  ( r `  x ) )  <->  ( f `  x )  =/=  ( 0g `  ( R `  x ) ) ) )
1814, 17rabeqbidv 3104 . . . . . . . 8  |-  ( ( s  =  S  /\  r  =  R )  ->  { x  e.  dom  r  |  ( f `  x )  =/=  ( 0g `  ( r `  x ) ) }  =  { x  e. 
dom  R  |  (
f `  x )  =/=  ( 0g `  ( R `  x )
) } )
1918eleq1d 2526 . . . . . . 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 3104 . . . . . 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 2516 . . . . 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 6314 . . . 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 18890 . . . 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 6324 . . . 4  |-  ( ( S X_s R )s  B )  e.  _V
2623, 24, 25ovmpt2a 6432 . . 3  |-  ( ( S  e.  _V  /\  R  e.  _V )  ->  ( S  (+)m  R )  =  ( ( S
X_s
R )s  B ) )
27 reldmdsmm 18891 . . . . . . 7  |-  Rel  dom  (+)m
2827ovprc1 6327 . . . . . 6  |-  ( -.  S  e.  _V  ->  ( S  (+)m  R )  =  (/) )
29 ress0 14705 . . . . . 6  |-  ( (/)s  B )  =  (/)
3028, 29syl6eqr 2516 . . . . 5  |-  ( -.  S  e.  _V  ->  ( S  (+)m  R )  =  (
(/)s  B ) )
31 reldmprds 14866 . . . . . . 7  |-  Rel  dom  X_s
3231ovprc1 6327 . . . . . 6  |-  ( -.  S  e.  _V  ->  ( S X_s R )  =  (/) )
3332oveq1d 6311 . . . . 5  |-  ( -.  S  e.  _V  ->  ( ( S X_s R )s  B )  =  (
(/)s  B ) )
3430, 33eqtr4d 2501 . . . 4  |-  ( -.  S  e.  _V  ->  ( S  (+)m  R )  =  ( ( S X_s R )s  B ) )
3534adantr 465 . . 3  |-  ( ( -.  S  e.  _V  /\  R  e.  _V )  ->  ( S  (+)m  R )  =  ( ( S
X_s
R )s  B ) )
3626, 35pm2.61ian 790 . 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1395    e. wcel 1819    =/= wne 2652   {crab 2811   _Vcvv 3109   (/)c0 3793   dom cdm 5008   ` cfv 5594  (class class class)co 6296   X_cixp 7488   Fincfn 7535   Basecbs 14644   ↾s cress 14645   0gc0g 14857   X_scprds 14863    (+)m cdsmm 18889
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-map 7440  df-ixp 7489  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-sup 7919  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-3 10616  df-4 10617  df-5 10618  df-6 10619  df-7 10620  df-8 10621  df-9 10622  df-10 10623  df-n0 10817  df-z 10886  df-dec 11001  df-uz 11107  df-fz 11698  df-struct 14646  df-ndx 14647  df-slot 14648  df-base 14649  df-ress 14651  df-plusg 14725  df-mulr 14726  df-sca 14728  df-vsca 14729  df-ip 14730  df-tset 14731  df-ple 14732  df-ds 14734  df-hom 14736  df-cco 14737  df-prds 14865  df-dsmm 18890
This theorem is referenced by:  dsmmbase  18893  dsmmval2  18894
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