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Theorem gsum2d 17592
Description: Write a sum over a two-dimensional region as a double sum. (Contributed by Mario Carneiro, 28-Dec-2014.) (Revised by AV, 8-Jun-2019.)
Hypotheses
Ref Expression
gsum2d.b  |-  B  =  ( Base `  G
)
gsum2d.z  |-  .0.  =  ( 0g `  G )
gsum2d.g  |-  ( ph  ->  G  e. CMnd )
gsum2d.a  |-  ( ph  ->  A  e.  V )
gsum2d.r  |-  ( ph  ->  Rel  A )
gsum2d.d  |-  ( ph  ->  D  e.  W )
gsum2d.s  |-  ( ph  ->  dom  A  C_  D
)
gsum2d.f  |-  ( ph  ->  F : A --> B )
gsum2d.w  |-  ( ph  ->  F finSupp  .0.  )
Assertion
Ref Expression
gsum2d  |-  ( ph  ->  ( G  gsumg  F )  =  ( G  gsumg  ( j  e.  D  |->  ( G  gsumg  ( k  e.  ( A " { j } )  |->  ( j F k ) ) ) ) ) )
Distinct variable groups:    j, k, A    j, F, k    j, G, k    ph, j, k    B, j, k    D, j, k    .0. , j, k
Allowed substitution hints:    V( j, k)    W( j, k)

Proof of Theorem gsum2d
StepHypRef Expression
1 gsum2d.b . . 3  |-  B  =  ( Base `  G
)
2 gsum2d.z . . 3  |-  .0.  =  ( 0g `  G )
3 gsum2d.g . . 3  |-  ( ph  ->  G  e. CMnd )
4 gsum2d.a . . 3  |-  ( ph  ->  A  e.  V )
5 gsum2d.r . . 3  |-  ( ph  ->  Rel  A )
6 gsum2d.d . . 3  |-  ( ph  ->  D  e.  W )
7 gsum2d.s . . 3  |-  ( ph  ->  dom  A  C_  D
)
8 gsum2d.f . . 3  |-  ( ph  ->  F : A --> B )
9 gsum2d.w . . 3  |-  ( ph  ->  F finSupp  .0.  )
101, 2, 3, 4, 5, 6, 7, 8, 9gsum2dlem2 17591 . 2  |-  ( ph  ->  ( G  gsumg  ( F  |`  ( A  |`  dom  ( F supp 
.0.  ) ) ) )  =  ( G 
gsumg  ( j  e.  dom  ( F supp  .0.  )  |->  ( G  gsumg  ( k  e.  ( A " { j } )  |->  ( j F k ) ) ) ) ) )
11 suppssdm 6935 . . . . . 6  |-  ( F supp 
.0.  )  C_  dom  F
12 fdm 5747 . . . . . . 7  |-  ( F : A --> B  ->  dom  F  =  A )
138, 12syl 17 . . . . . 6  |-  ( ph  ->  dom  F  =  A )
1411, 13syl5sseq 3512 . . . . 5  |-  ( ph  ->  ( F supp  .0.  )  C_  A )
15 relss 4938 . . . . . . 7  |-  ( ( F supp  .0.  )  C_  A  ->  ( Rel  A  ->  Rel  ( F supp  .0.  ) ) )
1614, 5, 15sylc 62 . . . . . 6  |-  ( ph  ->  Rel  ( F supp  .0.  ) )
17 relssdmrn 5372 . . . . . . 7  |-  ( Rel  ( F supp  .0.  )  ->  ( F supp  .0.  )  C_  ( dom  ( F supp 
.0.  )  X.  ran  ( F supp  .0.  ) ) )
18 ssv 3484 . . . . . . . 8  |-  ran  ( F supp  .0.  )  C_  _V
19 xpss2 4960 . . . . . . . 8  |-  ( ran  ( F supp  .0.  )  C_ 
_V  ->  ( dom  ( F supp  .0.  )  X.  ran  ( F supp  .0.  ) ) 
C_  ( dom  ( F supp  .0.  )  X.  _V ) )
2018, 19ax-mp 5 . . . . . . 7  |-  ( dom  ( F supp  .0.  )  X.  ran  ( F supp  .0.  ) )  C_  ( dom  ( F supp  .0.  )  X.  _V )
2117, 20syl6ss 3476 . . . . . 6  |-  ( Rel  ( F supp  .0.  )  ->  ( F supp  .0.  )  C_  ( dom  ( F supp 
.0.  )  X.  _V ) )
2216, 21syl 17 . . . . 5  |-  ( ph  ->  ( F supp  .0.  )  C_  ( dom  ( F supp 
.0.  )  X.  _V ) )
2314, 22ssind 3686 . . . 4  |-  ( ph  ->  ( F supp  .0.  )  C_  ( A  i^i  ( dom  ( F supp  .0.  )  X.  _V ) ) )
24 df-res 4862 . . . 4  |-  ( A  |`  dom  ( F supp  .0.  ) )  =  ( A  i^i  ( dom  ( F supp  .0.  )  X.  _V ) )
2523, 24syl6sseqr 3511 . . 3  |-  ( ph  ->  ( F supp  .0.  )  C_  ( A  |`  dom  ( F supp  .0.  ) ) )
261, 2, 3, 4, 8, 25, 9gsumres 17535 . 2  |-  ( ph  ->  ( G  gsumg  ( F  |`  ( A  |`  dom  ( F supp 
.0.  ) ) ) )  =  ( G 
gsumg  F ) )
27 dmss 5050 . . . . . . 7  |-  ( ( F supp  .0.  )  C_  A  ->  dom  ( F supp  .0.  )  C_  dom  A )
2814, 27syl 17 . . . . . 6  |-  ( ph  ->  dom  ( F supp  .0.  )  C_  dom  A )
2928, 7sstrd 3474 . . . . 5  |-  ( ph  ->  dom  ( F supp  .0.  )  C_  D )
3029resmptd 5172 . . . 4  |-  ( ph  ->  ( ( j  e.  D  |->  ( G  gsumg  ( k  e.  ( A " { j } ) 
|->  ( j F k ) ) ) )  |`  dom  ( F supp  .0.  ) )  =  ( j  e.  dom  ( F supp  .0.  )  |->  ( G 
gsumg  ( k  e.  ( A " { j } )  |->  ( j F k ) ) ) ) )
3130oveq2d 6318 . . 3  |-  ( ph  ->  ( G  gsumg  ( ( j  e.  D  |->  ( G  gsumg  ( k  e.  ( A " { j } ) 
|->  ( j F k ) ) ) )  |`  dom  ( F supp  .0.  ) ) )  =  ( G  gsumg  ( j  e.  dom  ( F supp  .0.  )  |->  ( G  gsumg  ( k  e.  ( A " { j } )  |->  ( j F k ) ) ) ) ) )
321, 2, 3, 4, 5, 6, 7, 8, 9gsum2dlem1 17590 . . . . . 6  |-  ( ph  ->  ( G  gsumg  ( k  e.  ( A " { j } )  |->  ( j F k ) ) )  e.  B )
3332adantr 466 . . . . 5  |-  ( (
ph  /\  j  e.  D )  ->  ( G  gsumg  ( k  e.  ( A " { j } )  |->  ( j F k ) ) )  e.  B )
34 eqid 2422 . . . . 5  |-  ( j  e.  D  |->  ( G 
gsumg  ( k  e.  ( A " { j } )  |->  ( j F k ) ) ) )  =  ( j  e.  D  |->  ( G  gsumg  ( k  e.  ( A " { j } )  |->  ( j F k ) ) ) )
3533, 34fmptd 6058 . . . 4  |-  ( ph  ->  ( j  e.  D  |->  ( G  gsumg  ( k  e.  ( A " { j } )  |->  ( j F k ) ) ) ) : D --> B )
36 vex 3084 . . . . . . . . . . . . . 14  |-  j  e. 
_V
37 vex 3084 . . . . . . . . . . . . . 14  |-  k  e. 
_V
3836, 37elimasn 5209 . . . . . . . . . . . . 13  |-  ( k  e.  ( A " { j } )  <->  <. j ,  k >.  e.  A )
3938biimpi 197 . . . . . . . . . . . 12  |-  ( k  e.  ( A " { j } )  ->  <. j ,  k
>.  e.  A )
4039ad2antll 733 . . . . . . . . . . 11  |-  ( (
ph  /\  ( j  e.  ( D  \  dom  ( F supp  .0.  ) )  /\  k  e.  ( A " { j } ) ) )  ->  <. j ,  k
>.  e.  A )
41 eldifn 3588 . . . . . . . . . . . . 13  |-  ( j  e.  ( D  \  dom  ( F supp  .0.  )
)  ->  -.  j  e.  dom  ( F supp  .0.  ) )
4241ad2antrl 732 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( j  e.  ( D  \  dom  ( F supp  .0.  ) )  /\  k  e.  ( A " { j } ) ) )  ->  -.  j  e.  dom  ( F supp  .0.  )
)
4336, 37opeldm 5054 . . . . . . . . . . . 12  |-  ( <.
j ,  k >.  e.  ( F supp  .0.  )  ->  j  e.  dom  ( F supp  .0.  ) )
4442, 43nsyl 124 . . . . . . . . . . 11  |-  ( (
ph  /\  ( j  e.  ( D  \  dom  ( F supp  .0.  ) )  /\  k  e.  ( A " { j } ) ) )  ->  -.  <. j ,  k >.  e.  ( F supp  .0.  ) )
4540, 44eldifd 3447 . . . . . . . . . 10  |-  ( (
ph  /\  ( j  e.  ( D  \  dom  ( F supp  .0.  ) )  /\  k  e.  ( A " { j } ) ) )  ->  <. j ,  k
>.  e.  ( A  \ 
( F supp  .0.  )
) )
46 df-ov 6305 . . . . . . . . . . 11  |-  ( j F k )  =  ( F `  <. j ,  k >. )
47 ssid 3483 . . . . . . . . . . . . 13  |-  ( F supp 
.0.  )  C_  ( F supp  .0.  )
4847a1i 11 . . . . . . . . . . . 12  |-  ( ph  ->  ( F supp  .0.  )  C_  ( F supp  .0.  )
)
49 fvex 5888 . . . . . . . . . . . . . 14  |-  ( 0g
`  G )  e. 
_V
502, 49eqeltri 2506 . . . . . . . . . . . . 13  |-  .0.  e.  _V
5150a1i 11 . . . . . . . . . . . 12  |-  ( ph  ->  .0.  e.  _V )
528, 48, 4, 51suppssr 6954 . . . . . . . . . . 11  |-  ( (
ph  /\  <. j ,  k >.  e.  ( A  \  ( F supp  .0.  ) ) )  -> 
( F `  <. j ,  k >. )  =  .0.  )
5346, 52syl5eq 2475 . . . . . . . . . 10  |-  ( (
ph  /\  <. j ,  k >.  e.  ( A  \  ( F supp  .0.  ) ) )  -> 
( j F k )  =  .0.  )
5445, 53syldan 472 . . . . . . . . 9  |-  ( (
ph  /\  ( j  e.  ( D  \  dom  ( F supp  .0.  ) )  /\  k  e.  ( A " { j } ) ) )  ->  ( j F k )  =  .0.  )
5554anassrs 652 . . . . . . . 8  |-  ( ( ( ph  /\  j  e.  ( D  \  dom  ( F supp  .0.  ) ) )  /\  k  e.  ( A " {
j } ) )  ->  ( j F k )  =  .0.  )
5655mpteq2dva 4507 . . . . . . 7  |-  ( (
ph  /\  j  e.  ( D  \  dom  ( F supp  .0.  ) ) )  ->  ( k  e.  ( A " {
j } )  |->  ( j F k ) )  =  ( k  e.  ( A " { j } ) 
|->  .0.  ) )
5756oveq2d 6318 . . . . . 6  |-  ( (
ph  /\  j  e.  ( D  \  dom  ( F supp  .0.  ) ) )  ->  ( G  gsumg  ( k  e.  ( A " { j } ) 
|->  ( j F k ) ) )  =  ( G  gsumg  ( k  e.  ( A " { j } )  |->  .0.  )
) )
58 cmnmnd 17433 . . . . . . . . 9  |-  ( G  e. CMnd  ->  G  e.  Mnd )
593, 58syl 17 . . . . . . . 8  |-  ( ph  ->  G  e.  Mnd )
60 imaexg 6741 . . . . . . . . 9  |-  ( A  e.  V  ->  ( A " { j } )  e.  _V )
614, 60syl 17 . . . . . . . 8  |-  ( ph  ->  ( A " {
j } )  e. 
_V )
622gsumz 16609 . . . . . . . 8  |-  ( ( G  e.  Mnd  /\  ( A " { j } )  e.  _V )  ->  ( G  gsumg  ( k  e.  ( A " { j } ) 
|->  .0.  ) )  =  .0.  )
6359, 61, 62syl2anc 665 . . . . . . 7  |-  ( ph  ->  ( G  gsumg  ( k  e.  ( A " { j } )  |->  .0.  )
)  =  .0.  )
6463adantr 466 . . . . . 6  |-  ( (
ph  /\  j  e.  ( D  \  dom  ( F supp  .0.  ) ) )  ->  ( G  gsumg  ( k  e.  ( A " { j } ) 
|->  .0.  ) )  =  .0.  )
6557, 64eqtrd 2463 . . . . 5  |-  ( (
ph  /\  j  e.  ( D  \  dom  ( F supp  .0.  ) ) )  ->  ( G  gsumg  ( k  e.  ( A " { j } ) 
|->  ( j F k ) ) )  =  .0.  )
6665, 6suppss2 6957 . . . 4  |-  ( ph  ->  ( ( j  e.  D  |->  ( G  gsumg  ( k  e.  ( A " { j } ) 
|->  ( j F k ) ) ) ) supp 
.0.  )  C_  dom  ( F supp  .0.  ) )
67 funmpt 5634 . . . . . 6  |-  Fun  (
j  e.  D  |->  ( G  gsumg  ( k  e.  ( A " { j } )  |->  ( j F k ) ) ) )
6867a1i 11 . . . . 5  |-  ( ph  ->  Fun  ( j  e.  D  |->  ( G  gsumg  ( k  e.  ( A " { j } ) 
|->  ( j F k ) ) ) ) )
699fsuppimpd 7893 . . . . . . 7  |-  ( ph  ->  ( F supp  .0.  )  e.  Fin )
70 dmfi 7857 . . . . . . 7  |-  ( ( F supp  .0.  )  e.  Fin  ->  dom  ( F supp  .0.  )  e.  Fin )
7169, 70syl 17 . . . . . 6  |-  ( ph  ->  dom  ( F supp  .0.  )  e.  Fin )
72 ssfi 7795 . . . . . 6  |-  ( ( dom  ( F supp  .0.  )  e.  Fin  /\  (
( j  e.  D  |->  ( G  gsumg  ( k  e.  ( A " { j } )  |->  ( j F k ) ) ) ) supp  .0.  )  C_ 
dom  ( F supp  .0.  ) )  ->  (
( j  e.  D  |->  ( G  gsumg  ( k  e.  ( A " { j } )  |->  ( j F k ) ) ) ) supp  .0.  )  e.  Fin )
7371, 66, 72syl2anc 665 . . . . 5  |-  ( ph  ->  ( ( j  e.  D  |->  ( G  gsumg  ( k  e.  ( A " { j } ) 
|->  ( j F k ) ) ) ) supp 
.0.  )  e.  Fin )
74 mptexg 6147 . . . . . . 7  |-  ( D  e.  W  ->  (
j  e.  D  |->  ( G  gsumg  ( k  e.  ( A " { j } )  |->  ( j F k ) ) ) )  e.  _V )
756, 74syl 17 . . . . . 6  |-  ( ph  ->  ( j  e.  D  |->  ( G  gsumg  ( k  e.  ( A " { j } )  |->  ( j F k ) ) ) )  e.  _V )
76 isfsupp 7890 . . . . . 6  |-  ( ( ( j  e.  D  |->  ( G  gsumg  ( k  e.  ( A " { j } )  |->  ( j F k ) ) ) )  e.  _V  /\  .0.  e.  _V )  ->  ( ( j  e.  D  |->  ( G  gsumg  ( k  e.  ( A " { j } ) 
|->  ( j F k ) ) ) ) finSupp  .0. 
<->  ( Fun  ( j  e.  D  |->  ( G 
gsumg  ( k  e.  ( A " { j } )  |->  ( j F k ) ) ) )  /\  (
( j  e.  D  |->  ( G  gsumg  ( k  e.  ( A " { j } )  |->  ( j F k ) ) ) ) supp  .0.  )  e.  Fin ) ) )
7775, 51, 76syl2anc 665 . . . . 5  |-  ( ph  ->  ( ( j  e.  D  |->  ( G  gsumg  ( k  e.  ( A " { j } ) 
|->  ( j F k ) ) ) ) finSupp  .0. 
<->  ( Fun  ( j  e.  D  |->  ( G 
gsumg  ( k  e.  ( A " { j } )  |->  ( j F k ) ) ) )  /\  (
( j  e.  D  |->  ( G  gsumg  ( k  e.  ( A " { j } )  |->  ( j F k ) ) ) ) supp  .0.  )  e.  Fin ) ) )
7868, 73, 77mpbir2and 930 . . . 4  |-  ( ph  ->  ( j  e.  D  |->  ( G  gsumg  ( k  e.  ( A " { j } )  |->  ( j F k ) ) ) ) finSupp  .0.  )
791, 2, 3, 6, 35, 66, 78gsumres 17535 . . 3  |-  ( ph  ->  ( G  gsumg  ( ( j  e.  D  |->  ( G  gsumg  ( k  e.  ( A " { j } ) 
|->  ( j F k ) ) ) )  |`  dom  ( F supp  .0.  ) ) )  =  ( G  gsumg  ( j  e.  D  |->  ( G  gsumg  ( k  e.  ( A " { j } )  |->  ( j F k ) ) ) ) ) )
8031, 79eqtr3d 2465 . 2  |-  ( ph  ->  ( G  gsumg  ( j  e.  dom  ( F supp  .0.  )  |->  ( G  gsumg  ( k  e.  ( A " { j } )  |->  ( j F k ) ) ) ) )  =  ( G  gsumg  ( j  e.  D  |->  ( G  gsumg  ( k  e.  ( A " { j } )  |->  ( j F k ) ) ) ) ) )
8110, 26, 803eqtr3d 2471 1  |-  ( ph  ->  ( G  gsumg  F )  =  ( G  gsumg  ( j  e.  D  |->  ( G  gsumg  ( k  e.  ( A " { j } )  |->  ( j F k ) ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1868   _Vcvv 3081    \ cdif 3433    i^i cin 3435    C_ wss 3436   {csn 3996   <.cop 4002   class class class wbr 4420    |-> cmpt 4479    X. cxp 4848   dom cdm 4850   ran crn 4851    |` cres 4852   "cima 4853   Rel wrel 4855   Fun wfun 5592   -->wf 5594   ` cfv 5598  (class class class)co 6302   supp csupp 6922   Fincfn 7574   finSupp cfsupp 7886   Basecbs 15109   0gc0g 15326    gsumg cgsu 15327   Mndcmnd 16523  CMndccmn 17418
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-rep 4533  ax-sep 4543  ax-nul 4552  ax-pow 4599  ax-pr 4657  ax-un 6594  ax-inf2 8149  ax-cnex 9596  ax-resscn 9597  ax-1cn 9598  ax-icn 9599  ax-addcl 9600  ax-addrcl 9601  ax-mulcl 9602  ax-mulrcl 9603  ax-mulcom 9604  ax-addass 9605  ax-mulass 9606  ax-distr 9607  ax-i2m1 9608  ax-1ne0 9609  ax-1rid 9610  ax-rnegex 9611  ax-rrecex 9612  ax-cnre 9613  ax-pre-lttri 9614  ax-pre-lttrn 9615  ax-pre-ltadd 9616  ax-pre-mulgt0 9617
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-nel 2621  df-ral 2780  df-rex 2781  df-reu 2782  df-rmo 2783  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-tp 4001  df-op 4003  df-uni 4217  df-int 4253  df-iun 4298  df-iin 4299  df-br 4421  df-opab 4480  df-mpt 4481  df-tr 4516  df-eprel 4761  df-id 4765  df-po 4771  df-so 4772  df-fr 4809  df-se 4810  df-we 4811  df-xp 4856  df-rel 4857  df-cnv 4858  df-co 4859  df-dm 4860  df-rn 4861  df-res 4862  df-ima 4863  df-pred 5396  df-ord 5442  df-on 5443  df-lim 5444  df-suc 5445  df-iota 5562  df-fun 5600  df-fn 5601  df-f 5602  df-f1 5603  df-fo 5604  df-f1o 5605  df-fv 5606  df-isom 5607  df-riota 6264  df-ov 6305  df-oprab 6306  df-mpt2 6307  df-of 6542  df-om 6704  df-1st 6804  df-2nd 6805  df-supp 6923  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-1o 7187  df-oadd 7191  df-er 7368  df-en 7575  df-dom 7576  df-sdom 7577  df-fin 7578  df-fsupp 7887  df-oi 8028  df-card 8375  df-pnf 9678  df-mnf 9679  df-xr 9680  df-ltxr 9681  df-le 9682  df-sub 9863  df-neg 9864  df-nn 10611  df-2 10669  df-n0 10871  df-z 10939  df-uz 11161  df-fz 11786  df-fzo 11917  df-seq 12214  df-hash 12516  df-ndx 15112  df-slot 15113  df-base 15114  df-sets 15115  df-ress 15116  df-plusg 15191  df-0g 15328  df-gsum 15329  df-mre 15480  df-mrc 15481  df-acs 15483  df-mgm 16476  df-sgrp 16515  df-mnd 16525  df-submnd 16571  df-mulg 16664  df-cntz 16959  df-cmn 17420
This theorem is referenced by:  gsum2d2  17594  gsumxp  17596
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