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Theorem dsmmval 18634
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 3127 . 2  |-  ( R  e.  V  ->  R  e.  _V )
2 oveq12 6304 . . . . 5  |-  ( ( s  =  S  /\  r  =  R )  ->  ( s X_s r )  =  ( S X_s R ) )
3 eqid 2467 . . . . . . . . 9  |-  ( s
X_s r )  =  ( s X_s r )
4 vex 3121 . . . . . . . . . 10  |-  s  e. 
_V
54a1i 11 . . . . . . . . 9  |-  ( ( s  =  S  /\  r  =  R )  ->  s  e.  _V )
6 vex 3121 . . . . . . . . . 10  |-  r  e. 
_V
76a1i 11 . . . . . . . . 9  |-  ( ( s  =  S  /\  r  =  R )  ->  r  e.  _V )
8 eqid 2467 . . . . . . . . 9  |-  ( Base `  ( s X_s r ) )  =  ( Base `  (
s X_s r ) )
9 eqidd 2468 . . . . . . . . 9  |-  ( ( s  =  S  /\  r  =  R )  ->  dom  r  =  dom  r )
103, 5, 7, 8, 9prdsbas 14729 . . . . . . . 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 2510 . . . . . . 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 5211 . . . . . . . . 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 2745 . . . . . . . . 9  |-  ( ( s  =  S  /\  r  =  R )  ->  ( ( f `  x )  =/=  ( 0g `  ( r `  x ) )  <->  ( f `  x )  =/=  ( 0g `  ( R `  x ) ) ) )
1814, 17rabeqbidv 3113 . . . . . . . 8  |-  ( ( s  =  S  /\  r  =  R )  ->  { x  e.  dom  r  |  ( f `  x )  =/=  ( 0g `  ( r `  x ) ) }  =  { x  e. 
dom  R  |  (
f `  x )  =/=  ( 0g `  ( R `  x )
) } )
1918eleq1d 2536 . . . . . . 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 3113 . . . . . 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 2526 . . . . 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 6313 . . . 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 18632 . . . 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 6320 . . . 4  |-  ( ( S X_s R )s  B )  e.  _V
2623, 24, 25ovmpt2a 6428 . . 3  |-  ( ( S  e.  _V  /\  R  e.  _V )  ->  ( S  (+)m  R )  =  ( ( S
X_s
R )s  B ) )
27 reldmdsmm 18633 . . . . . . 7  |-  Rel  dom  (+)m
2827ovprc1 6323 . . . . . 6  |-  ( -.  S  e.  _V  ->  ( S  (+)m  R )  =  (/) )
29 ress0 14566 . . . . . 6  |-  ( (/)s  B )  =  (/)
3028, 29syl6eqr 2526 . . . . 5  |-  ( -.  S  e.  _V  ->  ( S  (+)m  R )  =  (
(/)s  B ) )
31 reldmprds 14721 . . . . . . 7  |-  Rel  dom  X_s
3231ovprc1 6323 . . . . . 6  |-  ( -.  S  e.  _V  ->  ( S X_s R )  =  (/) )
3332oveq1d 6310 . . . . 5  |-  ( -.  S  e.  _V  ->  ( ( S X_s R )s  B )  =  (
(/)s  B ) )
3430, 33eqtr4d 2511 . . . 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 788 . 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 1379    e. wcel 1767    =/= wne 2662   {crab 2821   _Vcvv 3118   (/)c0 3790   dom cdm 5005   ` cfv 5594  (class class class)co 6295   X_cixp 7481   Fincfn 7528   Basecbs 14507   ↾s cress 14508   0gc0g 14712   X_scprds 14718    (+)m cdsmm 18631
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  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 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-er 7323  df-map 7434  df-ixp 7482  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-sup 7913  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-2 10606  df-3 10607  df-4 10608  df-5 10609  df-6 10610  df-7 10611  df-8 10612  df-9 10613  df-10 10614  df-n0 10808  df-z 10877  df-dec 10989  df-uz 11095  df-fz 11685  df-struct 14509  df-ndx 14510  df-slot 14511  df-base 14512  df-ress 14514  df-plusg 14585  df-mulr 14586  df-sca 14588  df-vsca 14589  df-ip 14590  df-tset 14591  df-ple 14592  df-ds 14594  df-hom 14596  df-cco 14597  df-prds 14720  df-dsmm 18632
This theorem is referenced by:  dsmmbase  18635  dsmmval2  18636
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