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Theorem gsumxp 16873
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 6583 . . . 4  |-  ( ( A  e.  V  /\  C  e.  W )  ->  ( A  X.  C
)  e.  _V )
74, 5, 6syl2anc 661 . . 3  |-  ( ph  ->  ( A  X.  C
)  e.  _V )
8 relxp 5096 . . . 4  |-  Rel  ( A  X.  C )
98a1i 11 . . 3  |-  ( ph  ->  Rel  ( A  X.  C ) )
10 dmxpss 5424 . . . 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 16868 . 2  |-  ( ph  ->  ( G  gsumg  F )  =  ( G  gsumg  ( j  e.  A  |->  ( G  gsumg  ( k  e.  ( ( A  X.  C
) " { j } )  |->  ( j F k ) ) ) ) ) )
15 df-ima 4998 . . . . . . 7  |-  ( ( A  X.  C )
" { j } )  =  ran  (
( A  X.  C
)  |`  { j } )
16 df-res 4997 . . . . . . . . . . 11  |-  ( ( A  X.  C )  |`  { j } )  =  ( ( A  X.  C )  i^i  ( { j }  X.  _V ) )
17 inxp 5121 . . . . . . . . . . 11  |-  ( ( A  X.  C )  i^i  ( { j }  X.  _V )
)  =  ( ( A  i^i  { j } )  X.  ( C  i^i  _V ) )
1816, 17eqtri 2470 . . . . . . . . . 10  |-  ( ( A  X.  C )  |`  { j } )  =  ( ( A  i^i  { j } )  X.  ( C  i^i  _V ) )
19 simpr 461 . . . . . . . . . . . . 13  |-  ( (
ph  /\  j  e.  A )  ->  j  e.  A )
2019snssd 4156 . . . . . . . . . . . 12  |-  ( (
ph  /\  j  e.  A )  ->  { j }  C_  A )
21 sseqin2 3699 . . . . . . . . . . . 12  |-  ( { j }  C_  A  <->  ( A  i^i  { j } )  =  {
j } )
2220, 21sylib 196 . . . . . . . . . . 11  |-  ( (
ph  /\  j  e.  A )  ->  ( A  i^i  { j } )  =  { j } )
23 inv1 3794 . . . . . . . . . . . 12  |-  ( C  i^i  _V )  =  C
2423a1i 11 . . . . . . . . . . 11  |-  ( (
ph  /\  j  e.  A )  ->  ( C  i^i  _V )  =  C )
2522, 24xpeq12d 5010 . . . . . . . . . 10  |-  ( (
ph  /\  j  e.  A )  ->  (
( A  i^i  {
j } )  X.  ( C  i^i  _V ) )  =  ( { j }  X.  C ) )
2618, 25syl5eq 2494 . . . . . . . . 9  |-  ( (
ph  /\  j  e.  A )  ->  (
( A  X.  C
)  |`  { j } )  =  ( { j }  X.  C
) )
2726rneqd 5216 . . . . . . . 8  |-  ( (
ph  /\  j  e.  A )  ->  ran  ( ( A  X.  C )  |`  { j } )  =  ran  ( { j }  X.  C ) )
28 vex 3096 . . . . . . . . . 10  |-  j  e. 
_V
2928snnz 4129 . . . . . . . . 9  |-  { j }  =/=  (/)
30 rnxp 5423 . . . . . . . . 9  |-  ( { j }  =/=  (/)  ->  ran  ( { j }  X.  C )  =  C )
3129, 30ax-mp 5 . . . . . . . 8  |-  ran  ( { j }  X.  C )  =  C
3227, 31syl6eq 2498 . . . . . . 7  |-  ( (
ph  /\  j  e.  A )  ->  ran  ( ( A  X.  C )  |`  { j } )  =  C )
3315, 32syl5eq 2494 . . . . . 6  |-  ( (
ph  /\  j  e.  A )  ->  (
( A  X.  C
) " { j } )  =  C )
3433mpteq1d 4514 . . . . 5  |-  ( (
ph  /\  j  e.  A )  ->  (
k  e.  ( ( A  X.  C )
" { j } )  |->  ( j F k ) )  =  ( k  e.  C  |->  ( j F k ) ) )
3534oveq2d 6293 . . . 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 4519 . . 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 6293 . 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 2482 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 1381    e. wcel 1802    =/= wne 2636   _Vcvv 3093    i^i cin 3457    C_ wss 3458   (/)c0 3767   {csn 4010   class class class wbr 4433    |-> cmpt 4491    X. cxp 4983   dom cdm 4985   ran crn 4986    |` cres 4987   "cima 4988   Rel wrel 4990   -->wf 5570   ` cfv 5574  (class class class)co 6277   finSupp cfsupp 7827   Basecbs 14504   0gc0g 14709    gsumg cgsu 14710  CMndccmn 16667
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573  ax-inf2 8056  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-int 4268  df-iun 4313  df-iin 4314  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-se 4825  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-isom 5583  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-of 6521  df-om 6682  df-1st 6781  df-2nd 6782  df-supp 6900  df-recs 7040  df-rdg 7074  df-1o 7128  df-oadd 7132  df-er 7309  df-en 7515  df-dom 7516  df-sdom 7517  df-fin 7518  df-fsupp 7828  df-oi 7933  df-card 8318  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9807  df-neg 9808  df-nn 10538  df-2 10595  df-n0 10797  df-z 10866  df-uz 11086  df-fz 11677  df-fzo 11799  df-seq 12082  df-hash 12380  df-ndx 14507  df-slot 14508  df-base 14509  df-sets 14510  df-ress 14511  df-plusg 14582  df-0g 14711  df-gsum 14712  df-mre 14855  df-mrc 14856  df-acs 14858  df-mgm 15741  df-sgrp 15780  df-mnd 15790  df-submnd 15836  df-mulg 15929  df-cntz 16224  df-cmn 16669
This theorem is referenced by:  tsmsxplem1  20521  tsmsxplem2  20522
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