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Theorem gsumxp 15540
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 4981 . . . 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 4975 . . . 4  |-  Rel  ( A  X.  C )
98a1i 11 . . 3  |-  ( ph  ->  Rel  ( A  X.  C ) )
10 dmxpss 5292 . . . 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 15536 . 2  |-  ( ph  ->  ( G  gsumg  F )  =  ( G  gsumg  ( j  e.  A  |->  ( G  gsumg  ( k  e.  ( ( A  X.  C
) " { j } )  |->  ( j F k ) ) ) ) ) )
15 df-ima 4883 . . . . . . 7  |-  ( ( A  X.  C )
" { j } )  =  ran  (
( A  X.  C
)  |`  { j } )
16 df-res 4882 . . . . . . . . . . 11  |-  ( ( A  X.  C )  |`  { j } )  =  ( ( A  X.  C )  i^i  ( { j }  X.  _V ) )
17 inxp 4999 . . . . . . . . . . 11  |-  ( ( A  X.  C )  i^i  ( { j }  X.  _V )
)  =  ( ( A  i^i  { j } )  X.  ( C  i^i  _V ) )
1816, 17eqtri 2455 . . . . . . . . . 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 3935 . . . . . . . . . . . 12  |-  ( (
ph  /\  j  e.  A )  ->  { j }  C_  A )
21 sseqin2 3552 . . . . . . . . . . . 12  |-  ( { j }  C_  A  <->  ( A  i^i  { j } )  =  {
j } )
2220, 21sylib 189 . . . . . . . . . . 11  |-  ( (
ph  /\  j  e.  A )  ->  ( A  i^i  { j } )  =  { j } )
23 inv1 3646 . . . . . . . . . . . 12  |-  ( C  i^i  _V )  =  C
2423a1i 11 . . . . . . . . . . 11  |-  ( (
ph  /\  j  e.  A )  ->  ( C  i^i  _V )  =  C )
2522, 24xpeq12d 4895 . . . . . . . . . 10  |-  ( (
ph  /\  j  e.  A )  ->  (
( A  i^i  {
j } )  X.  ( C  i^i  _V ) )  =  ( { j }  X.  C ) )
2618, 25syl5eq 2479 . . . . . . . . 9  |-  ( (
ph  /\  j  e.  A )  ->  (
( A  X.  C
)  |`  { j } )  =  ( { j }  X.  C
) )
2726rneqd 5089 . . . . . . . 8  |-  ( (
ph  /\  j  e.  A )  ->  ran  ( ( A  X.  C )  |`  { j } )  =  ran  ( { j }  X.  C ) )
28 vex 2951 . . . . . . . . . 10  |-  j  e. 
_V
2928snnz 3914 . . . . . . . . 9  |-  { j }  =/=  (/)
30 rnxp 5291 . . . . . . . . 9  |-  ( { j }  =/=  (/)  ->  ran  ( { j }  X.  C )  =  C )
3129, 30ax-mp 8 . . . . . . . 8  |-  ran  ( { j }  X.  C )  =  C
3227, 31syl6eq 2483 . . . . . . 7  |-  ( (
ph  /\  j  e.  A )  ->  ran  ( ( A  X.  C )  |`  { j } )  =  C )
3315, 32syl5eq 2479 . . . . . 6  |-  ( (
ph  /\  j  e.  A )  ->  (
( A  X.  C
) " { j } )  =  C )
3433mpteq1d 4282 . . . . 5  |-  ( (
ph  /\  j  e.  A )  ->  (
k  e.  ( ( A  X.  C )
" { j } )  |->  ( j F k ) )  =  ( k  e.  C  |->  ( j F k ) ) )
3534oveq2d 6089 . . . 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 4287 . . 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 6089 . 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 2467 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 1652    e. wcel 1725    =/= wne 2598   _Vcvv 2948    \ cdif 3309    i^i cin 3311    C_ wss 3312   (/)c0 3620   {csn 3806    e. cmpt 4258    X. cxp 4868   `'ccnv 4869   dom cdm 4870   ran crn 4871    |` cres 4872   "cima 4873   Rel wrel 4875   -->wf 5442   ` cfv 5446  (class class class)co 6073   Fincfn 7101   Basecbs 13459   0gc0g 13713    gsumg cgsu 13714  CMndccmn 15402
This theorem is referenced by:  tsmsxplem1  18172  tsmsxplem2  18173
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7586  ax-cnex 9036  ax-resscn 9037  ax-1cn 9038  ax-icn 9039  ax-addcl 9040  ax-addrcl 9041  ax-mulcl 9042  ax-mulrcl 9043  ax-mulcom 9044  ax-addass 9045  ax-mulass 9046  ax-distr 9047  ax-i2m1 9048  ax-1ne0 9049  ax-1rid 9050  ax-rnegex 9051  ax-rrecex 9052  ax-cnre 9053  ax-pre-lttri 9054  ax-pre-lttrn 9055  ax-pre-ltadd 9056  ax-pre-mulgt0 9057
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-of 6297  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-oi 7469  df-card 7816  df-pnf 9112  df-mnf 9113  df-xr 9114  df-ltxr 9115  df-le 9116  df-sub 9283  df-neg 9284  df-nn 9991  df-2 10048  df-n0 10212  df-z 10273  df-uz 10479  df-fz 11034  df-fzo 11126  df-seq 11314  df-hash 11609  df-ndx 13462  df-slot 13463  df-base 13464  df-sets 13465  df-ress 13466  df-plusg 13532  df-0g 13717  df-gsum 13718  df-mre 13801  df-mrc 13802  df-acs 13804  df-mnd 14680  df-submnd 14729  df-mulg 14805  df-cntz 15106  df-cmn 15404
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