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Theorem dsmmfi 19378
Description: For finite products, the direct sum is just the module product. See also the observation in [Lang] p. 129. (Contributed by Stefan O'Rear, 1-Feb-2015.)
Assertion
Ref Expression
dsmmfi  |-  ( ( R  Fn  I  /\  I  e.  Fin )  ->  ( S  (+)m  R )  =  ( S X_s R
) )

Proof of Theorem dsmmfi
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 eqid 2471 . . 3  |-  ( Base `  ( S  (+)m  R ) )  =  ( Base `  ( S  (+)m  R ) )
21dsmmval2 19376 . 2  |-  ( S 
(+)m  R )  =  ( ( S X_s R )s  ( Base `  ( S  (+)m  R ) ) )
3 eqid 2471 . . . . . . . . . . 11  |-  ( S
X_s
R )  =  ( S X_s R )
4 eqid 2471 . . . . . . . . . . 11  |-  ( Base `  ( S X_s R ) )  =  ( Base `  ( S X_s R ) )
5 noel 3726 . . . . . . . . . . . . . 14  |-  -.  f  e.  (/)
6 reldmprds 15425 . . . . . . . . . . . . . . . . . 18  |-  Rel  dom  X_s
76ovprc1 6339 . . . . . . . . . . . . . . . . 17  |-  ( -.  S  e.  _V  ->  ( S X_s R )  =  (/) )
87fveq2d 5883 . . . . . . . . . . . . . . . 16  |-  ( -.  S  e.  _V  ->  (
Base `  ( S X_s R ) )  =  (
Base `  (/) ) )
9 base0 15240 . . . . . . . . . . . . . . . 16  |-  (/)  =  (
Base `  (/) )
108, 9syl6eqr 2523 . . . . . . . . . . . . . . 15  |-  ( -.  S  e.  _V  ->  (
Base `  ( S X_s R ) )  =  (/) )
1110eleq2d 2534 . . . . . . . . . . . . . 14  |-  ( -.  S  e.  _V  ->  ( f  e.  ( Base `  ( S X_s R ) )  <->  f  e.  (/) ) )
125, 11mtbiri 310 . . . . . . . . . . . . 13  |-  ( -.  S  e.  _V  ->  -.  f  e.  ( Base `  ( S X_s R ) ) )
1312con4i 135 . . . . . . . . . . . 12  |-  ( f  e.  ( Base `  ( S X_s R ) )  ->  S  e.  _V )
1413adantl 473 . . . . . . . . . . 11  |-  ( ( ( R  Fn  I  /\  I  e.  Fin )  /\  f  e.  (
Base `  ( S X_s R ) ) )  ->  S  e.  _V )
15 simplr 770 . . . . . . . . . . 11  |-  ( ( ( R  Fn  I  /\  I  e.  Fin )  /\  f  e.  (
Base `  ( S X_s R ) ) )  ->  I  e.  Fin )
16 simpll 768 . . . . . . . . . . 11  |-  ( ( ( R  Fn  I  /\  I  e.  Fin )  /\  f  e.  (
Base `  ( S X_s R ) ) )  ->  R  Fn  I )
17 simpr 468 . . . . . . . . . . 11  |-  ( ( ( R  Fn  I  /\  I  e.  Fin )  /\  f  e.  (
Base `  ( S X_s R ) ) )  -> 
f  e.  ( Base `  ( S X_s R ) ) )
183, 4, 14, 15, 16, 17prdsbasfn 15447 . . . . . . . . . 10  |-  ( ( ( R  Fn  I  /\  I  e.  Fin )  /\  f  e.  (
Base `  ( S X_s R ) ) )  -> 
f  Fn  I )
19 fndm 5685 . . . . . . . . . 10  |-  ( f  Fn  I  ->  dom  f  =  I )
2018, 19syl 17 . . . . . . . . 9  |-  ( ( ( R  Fn  I  /\  I  e.  Fin )  /\  f  e.  (
Base `  ( S X_s R ) ) )  ->  dom  f  =  I
)
2120, 15eqeltrd 2549 . . . . . . . 8  |-  ( ( ( R  Fn  I  /\  I  e.  Fin )  /\  f  e.  (
Base `  ( S X_s R ) ) )  ->  dom  f  e.  Fin )
22 difss 3549 . . . . . . . . 9  |-  ( f 
\  ( 0g  o.  R ) )  C_  f
23 dmss 5039 . . . . . . . . 9  |-  ( ( f  \  ( 0g  o.  R ) ) 
C_  f  ->  dom  ( f  \  ( 0g  o.  R ) ) 
C_  dom  f )
2422, 23ax-mp 5 . . . . . . . 8  |-  dom  (
f  \  ( 0g  o.  R ) )  C_  dom  f
25 ssfi 7810 . . . . . . . 8  |-  ( ( dom  f  e.  Fin  /\ 
dom  ( f  \ 
( 0g  o.  R
) )  C_  dom  f )  ->  dom  ( f  \  ( 0g  o.  R ) )  e.  Fin )
2621, 24, 25sylancl 675 . . . . . . 7  |-  ( ( ( R  Fn  I  /\  I  e.  Fin )  /\  f  e.  (
Base `  ( S X_s R ) ) )  ->  dom  ( f  \  ( 0g  o.  R ) )  e.  Fin )
2726ralrimiva 2809 . . . . . 6  |-  ( ( R  Fn  I  /\  I  e.  Fin )  ->  A. f  e.  (
Base `  ( S X_s R ) ) dom  (
f  \  ( 0g  o.  R ) )  e. 
Fin )
28 rabid2 2954 . . . . . 6  |-  ( (
Base `  ( S X_s R ) )  =  {
f  e.  ( Base `  ( S X_s R ) )  |  dom  ( f  \ 
( 0g  o.  R
) )  e.  Fin }  <->  A. f  e.  ( Base `  ( S X_s R
) ) dom  (
f  \  ( 0g  o.  R ) )  e. 
Fin )
2927, 28sylibr 217 . . . . 5  |-  ( ( R  Fn  I  /\  I  e.  Fin )  ->  ( Base `  ( S X_s R ) )  =  { f  e.  (
Base `  ( S X_s R ) )  |  dom  ( f  \  ( 0g  o.  R ) )  e.  Fin } )
30 eqid 2471 . . . . . 6  |-  { f  e.  ( Base `  ( S X_s R ) )  |  dom  ( f  \ 
( 0g  o.  R
) )  e.  Fin }  =  { f  e.  ( Base `  ( S X_s R ) )  |  dom  ( f  \ 
( 0g  o.  R
) )  e.  Fin }
313, 30dsmmbas2 19377 . . . . 5  |-  ( ( R  Fn  I  /\  I  e.  Fin )  ->  { f  e.  (
Base `  ( S X_s R ) )  |  dom  ( f  \  ( 0g  o.  R ) )  e.  Fin }  =  ( Base `  ( S  (+)m 
R ) ) )
3229, 31eqtr2d 2506 . . . 4  |-  ( ( R  Fn  I  /\  I  e.  Fin )  ->  ( Base `  ( S  (+)m  R ) )  =  ( Base `  ( S X_s R ) ) )
3332oveq2d 6324 . . 3  |-  ( ( R  Fn  I  /\  I  e.  Fin )  ->  ( ( S X_s R
)s  ( Base `  ( S  (+)m  R ) ) )  =  ( ( S
X_s
R )s  ( Base `  ( S X_s R ) ) ) )
34 ovex 6336 . . . 4  |-  ( S
X_s
R )  e.  _V
354ressid 15262 . . . 4  |-  ( ( S X_s R )  e.  _V  ->  ( ( S X_s R
)s  ( Base `  ( S X_s R ) ) )  =  ( S X_s R
) )
3634, 35ax-mp 5 . . 3  |-  ( ( S X_s R )s  ( Base `  ( S X_s R ) ) )  =  ( S X_s R
)
3733, 36syl6eq 2521 . 2  |-  ( ( R  Fn  I  /\  I  e.  Fin )  ->  ( ( S X_s R
)s  ( Base `  ( S  (+)m  R ) ) )  =  ( S X_s R
) )
382, 37syl5eq 2517 1  |-  ( ( R  Fn  I  /\  I  e.  Fin )  ->  ( S  (+)m  R )  =  ( S X_s R
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 376    = wceq 1452    e. wcel 1904   A.wral 2756   {crab 2760   _Vcvv 3031    \ cdif 3387    C_ wss 3390   (/)c0 3722   dom cdm 4839    o. ccom 4843    Fn wfn 5584   ` cfv 5589  (class class class)co 6308   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-rep 4508  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-sets 15205  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-0g 15418  df-prds 15424  df-dsmm 19372
This theorem is referenced by:  frlmpwsfi  19392
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