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Theorem gsumxp 16458
Description: Write a group sum over a cartesian product as a double sum. (Contributed by Mario Carneiro, 28-Dec-2014.) (Revised by AV, 9-Jun-2019.)
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 finSupp  .0.  )
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 6506 . . . 4  |-  ( ( A  e.  V  /\  C  e.  W )  ->  ( A  X.  C
)  e.  _V )
74, 5, 6syl2anc 656 . . 3  |-  ( ph  ->  ( A  X.  C
)  e.  _V )
8 relxp 4943 . . . 4  |-  Rel  ( A  X.  C )
98a1i 11 . . 3  |-  ( ph  ->  Rel  ( A  X.  C ) )
10 dmxpss 5266 . . . 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 finSupp  .0.  )
141, 2, 3, 7, 9, 4, 11, 12, 13gsum2d 16453 . 2  |-  ( ph  ->  ( G  gsumg  F )  =  ( G  gsumg  ( j  e.  A  |->  ( G  gsumg  ( k  e.  ( ( A  X.  C
) " { j } )  |->  ( j F k ) ) ) ) ) )
15 df-ima 4849 . . . . . . 7  |-  ( ( A  X.  C )
" { j } )  =  ran  (
( A  X.  C
)  |`  { j } )
16 df-res 4848 . . . . . . . . . . 11  |-  ( ( A  X.  C )  |`  { j } )  =  ( ( A  X.  C )  i^i  ( { j }  X.  _V ) )
17 inxp 4968 . . . . . . . . . . 11  |-  ( ( A  X.  C )  i^i  ( { j }  X.  _V )
)  =  ( ( A  i^i  { j } )  X.  ( C  i^i  _V ) )
1816, 17eqtri 2461 . . . . . . . . . 10  |-  ( ( A  X.  C )  |`  { j } )  =  ( ( A  i^i  { j } )  X.  ( C  i^i  _V ) )
19 simpr 458 . . . . . . . . . . . . 13  |-  ( (
ph  /\  j  e.  A )  ->  j  e.  A )
2019snssd 4015 . . . . . . . . . . . 12  |-  ( (
ph  /\  j  e.  A )  ->  { j }  C_  A )
21 sseqin2 3566 . . . . . . . . . . . 12  |-  ( { j }  C_  A  <->  ( A  i^i  { j } )  =  {
j } )
2220, 21sylib 196 . . . . . . . . . . 11  |-  ( (
ph  /\  j  e.  A )  ->  ( A  i^i  { j } )  =  { j } )
23 inv1 3661 . . . . . . . . . . . 12  |-  ( C  i^i  _V )  =  C
2423a1i 11 . . . . . . . . . . 11  |-  ( (
ph  /\  j  e.  A )  ->  ( C  i^i  _V )  =  C )
2522, 24xpeq12d 4861 . . . . . . . . . 10  |-  ( (
ph  /\  j  e.  A )  ->  (
( A  i^i  {
j } )  X.  ( C  i^i  _V ) )  =  ( { j }  X.  C ) )
2618, 25syl5eq 2485 . . . . . . . . 9  |-  ( (
ph  /\  j  e.  A )  ->  (
( A  X.  C
)  |`  { j } )  =  ( { j }  X.  C
) )
2726rneqd 5063 . . . . . . . 8  |-  ( (
ph  /\  j  e.  A )  ->  ran  ( ( A  X.  C )  |`  { j } )  =  ran  ( { j }  X.  C ) )
28 vex 2973 . . . . . . . . . 10  |-  j  e. 
_V
2928snnz 3990 . . . . . . . . 9  |-  { j }  =/=  (/)
30 rnxp 5265 . . . . . . . . 9  |-  ( { j }  =/=  (/)  ->  ran  ( { j }  X.  C )  =  C )
3129, 30ax-mp 5 . . . . . . . 8  |-  ran  ( { j }  X.  C )  =  C
3227, 31syl6eq 2489 . . . . . . 7  |-  ( (
ph  /\  j  e.  A )  ->  ran  ( ( A  X.  C )  |`  { j } )  =  C )
3315, 32syl5eq 2485 . . . . . 6  |-  ( (
ph  /\  j  e.  A )  ->  (
( A  X.  C
) " { j } )  =  C )
3433mpteq1d 4370 . . . . 5  |-  ( (
ph  /\  j  e.  A )  ->  (
k  e.  ( ( A  X.  C )
" { j } )  |->  ( j F k ) )  =  ( k  e.  C  |->  ( j F k ) ) )
3534oveq2d 6106 . . . 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 4375 . . 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 6106 . 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 2473 1  |-  ( ph  ->  ( G  gsumg  F )  =  ( G  gsumg  ( j  e.  A  |->  ( G  gsumg  ( k  e.  C  |->  ( j F k ) ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1364    e. wcel 1761    =/= wne 2604   _Vcvv 2970    i^i cin 3324    C_ wss 3325   (/)c0 3634   {csn 3874   class class class wbr 4289    e. cmpt 4347    X. cxp 4834   dom cdm 4836   ran crn 4837    |` cres 4838   "cima 4839   Rel wrel 4841   -->wf 5411   ` cfv 5415  (class class class)co 6090   finSupp cfsupp 7616   Basecbs 14170   0gc0g 14374    gsumg cgsu 14375  CMndccmn 16270
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-inf2 7843  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-iin 4171  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-se 4676  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-isom 5424  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-of 6319  df-om 6476  df-1st 6576  df-2nd 6577  df-supp 6690  df-recs 6828  df-rdg 6862  df-1o 6916  df-oadd 6920  df-er 7097  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-fsupp 7617  df-oi 7720  df-card 8105  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-nn 10319  df-2 10376  df-n0 10576  df-z 10643  df-uz 10858  df-fz 11434  df-fzo 11545  df-seq 11803  df-hash 12100  df-ndx 14173  df-slot 14174  df-base 14175  df-sets 14176  df-ress 14177  df-plusg 14247  df-0g 14376  df-gsum 14377  df-mre 14520  df-mrc 14521  df-acs 14523  df-mnd 15411  df-submnd 15461  df-mulg 15541  df-cntz 15828  df-cmn 16272
This theorem is referenced by:  tsmsxplem1  19627  tsmsxplem2  19628
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