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Theorem gsumxp 15227
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 4800 . . . 4  |-  ( ( A  e.  V  /\  C  e.  W )  ->  ( A  X.  C
)  e.  _V )
74, 5, 6syl2anc 642 . . 3  |-  ( ph  ->  ( A  X.  C
)  e.  _V )
8 relxp 4794 . . . 4  |-  Rel  ( A  X.  C )
98a1i 10 . . 3  |-  ( ph  ->  Rel  ( A  X.  C ) )
10 dmxpss 5107 . . . 4  |-  dom  ( A  X.  C )  C_  A
1110a1i 10 . . 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 15223 . 2  |-  ( ph  ->  ( G  gsumg  F )  =  ( G  gsumg  ( j  e.  A  |->  ( G  gsumg  ( k  e.  ( ( A  X.  C
) " { j } )  |->  ( j F k ) ) ) ) ) )
15 df-ima 4702 . . . . . . 7  |-  ( ( A  X.  C )
" { j } )  =  ran  (
( A  X.  C
)  |`  { j } )
16 df-res 4701 . . . . . . . . . . 11  |-  ( ( A  X.  C )  |`  { j } )  =  ( ( A  X.  C )  i^i  ( { j }  X.  _V ) )
17 inxp 4818 . . . . . . . . . . 11  |-  ( ( A  X.  C )  i^i  ( { j }  X.  _V )
)  =  ( ( A  i^i  { j } )  X.  ( C  i^i  _V ) )
1816, 17eqtri 2303 . . . . . . . . . 10  |-  ( ( A  X.  C )  |`  { j } )  =  ( ( A  i^i  { j } )  X.  ( C  i^i  _V ) )
19 simpr 447 . . . . . . . . . . . . 13  |-  ( (
ph  /\  j  e.  A )  ->  j  e.  A )
2019snssd 3760 . . . . . . . . . . . 12  |-  ( (
ph  /\  j  e.  A )  ->  { j }  C_  A )
21 sseqin2 3388 . . . . . . . . . . . 12  |-  ( { j }  C_  A  <->  ( A  i^i  { j } )  =  {
j } )
2220, 21sylib 188 . . . . . . . . . . 11  |-  ( (
ph  /\  j  e.  A )  ->  ( A  i^i  { j } )  =  { j } )
23 inv1 3481 . . . . . . . . . . . 12  |-  ( C  i^i  _V )  =  C
2423a1i 10 . . . . . . . . . . 11  |-  ( (
ph  /\  j  e.  A )  ->  ( C  i^i  _V )  =  C )
2522, 24xpeq12d 4714 . . . . . . . . . 10  |-  ( (
ph  /\  j  e.  A )  ->  (
( A  i^i  {
j } )  X.  ( C  i^i  _V ) )  =  ( { j }  X.  C ) )
2618, 25syl5eq 2327 . . . . . . . . 9  |-  ( (
ph  /\  j  e.  A )  ->  (
( A  X.  C
)  |`  { j } )  =  ( { j }  X.  C
) )
2726rneqd 4906 . . . . . . . 8  |-  ( (
ph  /\  j  e.  A )  ->  ran  ( ( A  X.  C )  |`  { j } )  =  ran  ( { j }  X.  C ) )
28 vex 2791 . . . . . . . . . 10  |-  j  e. 
_V
2928snnz 3744 . . . . . . . . 9  |-  { j }  =/=  (/)
30 rnxp 5106 . . . . . . . . 9  |-  ( { j }  =/=  (/)  ->  ran  ( { j }  X.  C )  =  C )
3129, 30ax-mp 8 . . . . . . . 8  |-  ran  ( { j }  X.  C )  =  C
3227, 31syl6eq 2331 . . . . . . 7  |-  ( (
ph  /\  j  e.  A )  ->  ran  ( ( A  X.  C )  |`  { j } )  =  C )
3315, 32syl5eq 2327 . . . . . 6  |-  ( (
ph  /\  j  e.  A )  ->  (
( A  X.  C
) " { j } )  =  C )
34 mpteq1 4100 . . . . . 6  |-  ( ( ( A  X.  C
) " { j } )  =  C  ->  ( k  e.  ( ( A  X.  C ) " {
j } )  |->  ( j F k ) )  =  ( k  e.  C  |->  ( j F k ) ) )
3533, 34syl 15 . . . . 5  |-  ( (
ph  /\  j  e.  A )  ->  (
k  e.  ( ( A  X.  C )
" { j } )  |->  ( j F k ) )  =  ( k  e.  C  |->  ( j F k ) ) )
3635oveq2d 5874 . . . 4  |-  ( (
ph  /\  j  e.  A )  ->  ( G  gsumg  ( k  e.  ( ( A  X.  C
) " { j } )  |->  ( j F k ) ) )  =  ( G 
gsumg  ( k  e.  C  |->  ( j F k ) ) ) )
3736mpteq2dva 4106 . . 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 ) ) ) ) )
3837oveq2d 5874 . 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 ) ) ) ) ) )
3914, 38eqtrd 2315 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 358    = wceq 1623    e. wcel 1684    =/= wne 2446   _Vcvv 2788    \ cdif 3149    i^i cin 3151    C_ wss 3152   (/)c0 3455   {csn 3640    e. cmpt 4077    X. cxp 4687   `'ccnv 4688   dom cdm 4689   ran crn 4690    |` cres 4691   "cima 4692   Rel wrel 4694   -->wf 5251   ` cfv 5255  (class class class)co 5858   Fincfn 6863   Basecbs 13148   0gc0g 13400    gsumg cgsu 13401  CMndccmn 15089
This theorem is referenced by:  tsmsxplem1  17835  tsmsxplem2  17836
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-oi 7225  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-fzo 10871  df-seq 11047  df-hash 11338  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-0g 13404  df-gsum 13405  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-submnd 14416  df-mulg 14492  df-cntz 14793  df-cmn 15091
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