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Theorem gsumxp 15479
Description: Write a group sum over a cartesian product as a double sum. (Contributed by Mario Carneiro, 28-Dec-2014.)
Hypotheses
Ref Expression
gsumxp.b  |-  B  =  ( Base `  G
)
gsumxp.z  |-  .0.  =  ( 0g `  G )
gsumxp.g  |-  ( ph  ->  G  e. CMnd )
gsumxp.a  |-  ( ph  ->  A  e.  V )
gsumxp.r  |-  ( ph  ->  C  e.  W )
gsumxp.f  |-  ( ph  ->  F : ( A  X.  C ) --> B )
gsumxp.w  |-  ( ph  ->  ( `' F "
( _V  \  {  .0.  } ) )  e. 
Fin )
Assertion
Ref Expression
gsumxp  |-  ( ph  ->  ( G  gsumg  F )  =  ( G  gsumg  ( j  e.  A  |->  ( G  gsumg  ( k  e.  C  |->  ( j F k ) ) ) ) ) )
Distinct variable groups:    j, k,  .0.    j, G, k    ph, j,
k    A, j, k    B, j, k    C, j, k   
j, F, k    j, V
Allowed substitution hints:    V( k)    W( j, k)

Proof of Theorem gsumxp
StepHypRef Expression
1 gsumxp.b . . 3  |-  B  =  ( Base `  G
)
2 gsumxp.z . . 3  |-  .0.  =  ( 0g `  G )
3 gsumxp.g . . 3  |-  ( ph  ->  G  e. CMnd )
4 gsumxp.a . . . 4  |-  ( ph  ->  A  e.  V )
5 gsumxp.r . . . 4  |-  ( ph  ->  C  e.  W )
6 xpexg 4931 . . . 4  |-  ( ( A  e.  V  /\  C  e.  W )  ->  ( A  X.  C
)  e.  _V )
74, 5, 6syl2anc 643 . . 3  |-  ( ph  ->  ( A  X.  C
)  e.  _V )
8 relxp 4925 . . . 4  |-  Rel  ( A  X.  C )
98a1i 11 . . 3  |-  ( ph  ->  Rel  ( A  X.  C ) )
10 dmxpss 5242 . . . 4  |-  dom  ( A  X.  C )  C_  A
1110a1i 11 . . 3  |-  ( ph  ->  dom  ( A  X.  C )  C_  A
)
12 gsumxp.f . . 3  |-  ( ph  ->  F : ( A  X.  C ) --> B )
13 gsumxp.w . . 3  |-  ( ph  ->  ( `' F "
( _V  \  {  .0.  } ) )  e. 
Fin )
141, 2, 3, 7, 9, 4, 11, 12, 13gsum2d 15475 . 2  |-  ( ph  ->  ( G  gsumg  F )  =  ( G  gsumg  ( j  e.  A  |->  ( G  gsumg  ( k  e.  ( ( A  X.  C
) " { j } )  |->  ( j F k ) ) ) ) ) )
15 df-ima 4833 . . . . . . 7  |-  ( ( A  X.  C )
" { j } )  =  ran  (
( A  X.  C
)  |`  { j } )
16 df-res 4832 . . . . . . . . . . 11  |-  ( ( A  X.  C )  |`  { j } )  =  ( ( A  X.  C )  i^i  ( { j }  X.  _V ) )
17 inxp 4949 . . . . . . . . . . 11  |-  ( ( A  X.  C )  i^i  ( { j }  X.  _V )
)  =  ( ( A  i^i  { j } )  X.  ( C  i^i  _V ) )
1816, 17eqtri 2409 . . . . . . . . . 10  |-  ( ( A  X.  C )  |`  { j } )  =  ( ( A  i^i  { j } )  X.  ( C  i^i  _V ) )
19 simpr 448 . . . . . . . . . . . . 13  |-  ( (
ph  /\  j  e.  A )  ->  j  e.  A )
2019snssd 3888 . . . . . . . . . . . 12  |-  ( (
ph  /\  j  e.  A )  ->  { j }  C_  A )
21 sseqin2 3505 . . . . . . . . . . . 12  |-  ( { j }  C_  A  <->  ( A  i^i  { j } )  =  {
j } )
2220, 21sylib 189 . . . . . . . . . . 11  |-  ( (
ph  /\  j  e.  A )  ->  ( A  i^i  { j } )  =  { j } )
23 inv1 3599 . . . . . . . . . . . 12  |-  ( C  i^i  _V )  =  C
2423a1i 11 . . . . . . . . . . 11  |-  ( (
ph  /\  j  e.  A )  ->  ( C  i^i  _V )  =  C )
2522, 24xpeq12d 4845 . . . . . . . . . 10  |-  ( (
ph  /\  j  e.  A )  ->  (
( A  i^i  {
j } )  X.  ( C  i^i  _V ) )  =  ( { j }  X.  C ) )
2618, 25syl5eq 2433 . . . . . . . . 9  |-  ( (
ph  /\  j  e.  A )  ->  (
( A  X.  C
)  |`  { j } )  =  ( { j }  X.  C
) )
2726rneqd 5039 . . . . . . . 8  |-  ( (
ph  /\  j  e.  A )  ->  ran  ( ( A  X.  C )  |`  { j } )  =  ran  ( { j }  X.  C ) )
28 vex 2904 . . . . . . . . . 10  |-  j  e. 
_V
2928snnz 3867 . . . . . . . . 9  |-  { j }  =/=  (/)
30 rnxp 5241 . . . . . . . . 9  |-  ( { j }  =/=  (/)  ->  ran  ( { j }  X.  C )  =  C )
3129, 30ax-mp 8 . . . . . . . 8  |-  ran  ( { j }  X.  C )  =  C
3227, 31syl6eq 2437 . . . . . . 7  |-  ( (
ph  /\  j  e.  A )  ->  ran  ( ( A  X.  C )  |`  { j } )  =  C )
3315, 32syl5eq 2433 . . . . . 6  |-  ( (
ph  /\  j  e.  A )  ->  (
( A  X.  C
) " { j } )  =  C )
3433mpteq1d 4233 . . . . 5  |-  ( (
ph  /\  j  e.  A )  ->  (
k  e.  ( ( A  X.  C )
" { j } )  |->  ( j F k ) )  =  ( k  e.  C  |->  ( j F k ) ) )
3534oveq2d 6038 . . . 4  |-  ( (
ph  /\  j  e.  A )  ->  ( G  gsumg  ( k  e.  ( ( A  X.  C
) " { j } )  |->  ( j F k ) ) )  =  ( G 
gsumg  ( k  e.  C  |->  ( j F k ) ) ) )
3635mpteq2dva 4238 . . 3  |-  ( ph  ->  ( j  e.  A  |->  ( G  gsumg  ( k  e.  ( ( A  X.  C
) " { j } )  |->  ( j F k ) ) ) )  =  ( j  e.  A  |->  ( G  gsumg  ( k  e.  C  |->  ( j F k ) ) ) ) )
3736oveq2d 6038 . 2  |-  ( ph  ->  ( G  gsumg  ( j  e.  A  |->  ( G  gsumg  ( k  e.  ( ( A  X.  C
) " { j } )  |->  ( j F k ) ) ) ) )  =  ( G  gsumg  ( j  e.  A  |->  ( G  gsumg  ( k  e.  C  |->  ( j F k ) ) ) ) ) )
3814, 37eqtrd 2421 1  |-  ( ph  ->  ( G  gsumg  F )  =  ( G  gsumg  ( j  e.  A  |->  ( G  gsumg  ( k  e.  C  |->  ( j F k ) ) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717    =/= wne 2552   _Vcvv 2901    \ cdif 3262    i^i cin 3264    C_ wss 3265   (/)c0 3573   {csn 3759    e. cmpt 4209    X. cxp 4818   `'ccnv 4819   dom cdm 4820   ran crn 4821    |` cres 4822   "cima 4823   Rel wrel 4825   -->wf 5392   ` cfv 5396  (class class class)co 6022   Fincfn 7047   Basecbs 13398   0gc0g 13652    gsumg cgsu 13653  CMndccmn 15341
This theorem is referenced by:  tsmsxplem1  18105  tsmsxplem2  18106
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643  ax-inf2 7531  ax-cnex 8981  ax-resscn 8982  ax-1cn 8983  ax-icn 8984  ax-addcl 8985  ax-addrcl 8986  ax-mulcl 8987  ax-mulrcl 8988  ax-mulcom 8989  ax-addass 8990  ax-mulass 8991  ax-distr 8992  ax-i2m1 8993  ax-1ne0 8994  ax-1rid 8995  ax-rnegex 8996  ax-rrecex 8997  ax-cnre 8998  ax-pre-lttri 8999  ax-pre-lttrn 9000  ax-pre-ltadd 9001  ax-pre-mulgt0 9002
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rmo 2659  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-pss 3281  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-tp 3767  df-op 3768  df-uni 3960  df-int 3995  df-iun 4039  df-iin 4040  df-br 4156  df-opab 4210  df-mpt 4211  df-tr 4246  df-eprel 4437  df-id 4441  df-po 4446  df-so 4447  df-fr 4484  df-se 4485  df-we 4486  df-ord 4527  df-on 4528  df-lim 4529  df-suc 4530  df-om 4788  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-isom 5405  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-of 6246  df-1st 6290  df-2nd 6291  df-riota 6487  df-recs 6571  df-rdg 6606  df-1o 6662  df-oadd 6666  df-er 6843  df-en 7048  df-dom 7049  df-sdom 7050  df-fin 7051  df-oi 7414  df-card 7761  df-pnf 9057  df-mnf 9058  df-xr 9059  df-ltxr 9060  df-le 9061  df-sub 9227  df-neg 9228  df-nn 9935  df-2 9992  df-n0 10156  df-z 10217  df-uz 10423  df-fz 10978  df-fzo 11068  df-seq 11253  df-hash 11548  df-ndx 13401  df-slot 13402  df-base 13403  df-sets 13404  df-ress 13405  df-plusg 13471  df-0g 13656  df-gsum 13657  df-mre 13740  df-mrc 13741  df-acs 13743  df-mnd 14619  df-submnd 14668  df-mulg 14744  df-cntz 15045  df-cmn 15343
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