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Theorem dsmmval 27303
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 2809 . 2  |-  ( R  e.  V  ->  R  e.  _V )
2 oveq12 5883 . . . . 5  |-  ( ( s  =  S  /\  r  =  R )  ->  ( s X_s r )  =  ( S X_s R ) )
3 eqid 2296 . . . . . . . . 9  |-  ( s
X_s r )  =  ( s X_s r )
4 vex 2804 . . . . . . . . . 10  |-  s  e. 
_V
54a1i 10 . . . . . . . . 9  |-  ( ( s  =  S  /\  r  =  R )  ->  s  e.  _V )
6 vex 2804 . . . . . . . . . 10  |-  r  e. 
_V
76a1i 10 . . . . . . . . 9  |-  ( ( s  =  S  /\  r  =  R )  ->  r  e.  _V )
8 eqid 2296 . . . . . . . . 9  |-  ( Base `  ( s X_s r ) )  =  ( Base `  (
s X_s r ) )
9 eqidd 2297 . . . . . . . . 9  |-  ( ( s  =  S  /\  r  =  R )  ->  dom  r  =  dom  r )
103, 5, 7, 8, 9prdsbas 13373 . . . . . . . 8  |-  ( ( s  =  S  /\  r  =  R )  ->  ( Base `  (
s X_s r ) )  = 
X_ x  e.  dom  r ( Base `  (
r `  x )
) )
112fveq2d 5545 . . . . . . . 8  |-  ( ( s  =  S  /\  r  =  R )  ->  ( Base `  (
s X_s r ) )  =  ( Base `  ( S X_s R ) ) )
1210, 11eqtr3d 2330 . . . . . . 7  |-  ( ( s  =  S  /\  r  =  R )  -> 
X_ x  e.  dom  r ( Base `  (
r `  x )
)  =  ( Base `  ( S X_s R ) ) )
13 simpr 447 . . . . . . . . . 10  |-  ( ( s  =  S  /\  r  =  R )  ->  r  =  R )
1413dmeqd 4897 . . . . . . . . 9  |-  ( ( s  =  S  /\  r  =  R )  ->  dom  r  =  dom  R )
1513fveq1d 5543 . . . . . . . . . . 11  |-  ( ( s  =  S  /\  r  =  R )  ->  ( r `  x
)  =  ( R `
 x ) )
1615fveq2d 5545 . . . . . . . . . 10  |-  ( ( s  =  S  /\  r  =  R )  ->  ( 0g `  (
r `  x )
)  =  ( 0g
`  ( R `  x ) ) )
1716neeq2d 2473 . . . . . . . . 9  |-  ( ( s  =  S  /\  r  =  R )  ->  ( ( f `  x )  =/=  ( 0g `  ( r `  x ) )  <->  ( f `  x )  =/=  ( 0g `  ( R `  x ) ) ) )
1814, 17rabeqbidv 2796 . . . . . . . 8  |-  ( ( s  =  S  /\  r  =  R )  ->  { x  e.  dom  r  |  ( f `  x )  =/=  ( 0g `  ( r `  x ) ) }  =  { x  e. 
dom  R  |  (
f `  x )  =/=  ( 0g `  ( R `  x )
) } )
1918eleq1d 2362 . . . . . . 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 2796 . . . . . 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 2346 . . . . 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 5892 . . . 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 27301 . . . 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 5899 . . . 4  |-  ( ( S X_s R )s  B )  e.  _V
2623, 24, 25ovmpt2a 5994 . . 3  |-  ( ( S  e.  _V  /\  R  e.  _V )  ->  ( S  (+)m  R )  =  ( ( S
X_s
R )s  B ) )
27 reldmdsmm 27302 . . . . . . 7  |-  Rel  dom  (+)m
2827ovprc1 5902 . . . . . 6  |-  ( -.  S  e.  _V  ->  ( S  (+)m  R )  =  (/) )
29 ress0 13218 . . . . . 6  |-  ( (/)s  B )  =  (/)
3028, 29syl6eqr 2346 . . . . 5  |-  ( -.  S  e.  _V  ->  ( S  (+)m  R )  =  (
(/)s  B ) )
31 reldmprds 13365 . . . . . . 7  |-  Rel  dom  X_s
3231ovprc1 5902 . . . . . 6  |-  ( -.  S  e.  _V  ->  ( S X_s R )  =  (/) )
3332oveq1d 5889 . . . . 5  |-  ( -.  S  e.  _V  ->  ( ( S X_s R )s  B )  =  (
(/)s  B ) )
3430, 33eqtr4d 2331 . . . 4  |-  ( -.  S  e.  _V  ->  ( S  (+)m  R )  =  ( ( S X_s R )s  B ) )
3534adantr 451 . . 3  |-  ( ( -.  S  e.  _V  /\  R  e.  _V )  ->  ( S  (+)m  R )  =  ( ( S
X_s
R )s  B ) )
3626, 35pm2.61ian 765 . 2  |-  ( R  e.  _V  ->  ( S  (+)m  R )  =  ( ( S X_s R )s  B ) )
371, 36syl 15 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 358    = wceq 1632    e. wcel 1696    =/= wne 2459   {crab 2560   _Vcvv 2801   (/)c0 3468   dom cdm 4705   ` cfv 5271  (class class class)co 5874   X_cixp 6833   Fincfn 6879   Basecbs 13164   ↾s cress 13165   X_scprds 13362   0gc0g 13416    (+)m cdsmm 27300
This theorem is referenced by:  dsmmbase  27304  dsmmval2  27305
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-map 6790  df-ixp 6834  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-fz 10799  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-ress 13171  df-plusg 13237  df-mulr 13238  df-sca 13240  df-vsca 13241  df-tset 13243  df-ple 13244  df-ds 13246  df-hom 13248  df-cco 13249  df-prds 13364  df-dsmm 27301
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