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Theorem gsum2d 17682
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 17681 . 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 6946 . . . . . 6  |-  ( F supp 
.0.  )  C_  dom  F
12 fdm 5745 . . . . . . 7  |-  ( F : A --> B  ->  dom  F  =  A )
138, 12syl 17 . . . . . 6  |-  ( ph  ->  dom  F  =  A )
1411, 13syl5sseq 3466 . . . . 5  |-  ( ph  ->  ( F supp  .0.  )  C_  A )
15 relss 4927 . . . . . . 7  |-  ( ( F supp  .0.  )  C_  A  ->  ( Rel  A  ->  Rel  ( F supp  .0.  ) ) )
1614, 5, 15sylc 61 . . . . . 6  |-  ( ph  ->  Rel  ( F supp  .0.  ) )
17 relssdmrn 5363 . . . . . . 7  |-  ( Rel  ( F supp  .0.  )  ->  ( F supp  .0.  )  C_  ( dom  ( F supp 
.0.  )  X.  ran  ( F supp  .0.  ) ) )
18 ssv 3438 . . . . . . . 8  |-  ran  ( F supp  .0.  )  C_  _V
19 xpss2 4949 . . . . . . . 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 3430 . . . . . 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 3647 . . . 4  |-  ( ph  ->  ( F supp  .0.  )  C_  ( A  i^i  ( dom  ( F supp  .0.  )  X.  _V ) ) )
24 df-res 4851 . . . 4  |-  ( A  |`  dom  ( F supp  .0.  ) )  =  ( A  i^i  ( dom  ( F supp  .0.  )  X.  _V ) )
2523, 24syl6sseqr 3465 . . 3  |-  ( ph  ->  ( F supp  .0.  )  C_  ( A  |`  dom  ( F supp  .0.  ) ) )
261, 2, 3, 4, 8, 25, 9gsumres 17625 . 2  |-  ( ph  ->  ( G  gsumg  ( F  |`  ( A  |`  dom  ( F supp 
.0.  ) ) ) )  =  ( G 
gsumg  F ) )
27 dmss 5039 . . . . . . 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 3428 . . . . 5  |-  ( ph  ->  dom  ( F supp  .0.  )  C_  D )
3029resmptd 5162 . . . 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 6324 . . 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 17680 . . . . . 6  |-  ( ph  ->  ( G  gsumg  ( k  e.  ( A " { j } )  |->  ( j F k ) ) )  e.  B )
3332adantr 472 . . . . 5  |-  ( (
ph  /\  j  e.  D )  ->  ( G  gsumg  ( k  e.  ( A " { j } )  |->  ( j F k ) ) )  e.  B )
34 eqid 2471 . . . . 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 6061 . . . 4  |-  ( ph  ->  ( j  e.  D  |->  ( G  gsumg  ( k  e.  ( A " { j } )  |->  ( j F k ) ) ) ) : D --> B )
36 vex 3034 . . . . . . . . . . . . . 14  |-  j  e. 
_V
37 vex 3034 . . . . . . . . . . . . . 14  |-  k  e. 
_V
3836, 37elimasn 5199 . . . . . . . . . . . . 13  |-  ( k  e.  ( A " { j } )  <->  <. j ,  k >.  e.  A )
3938biimpi 199 . . . . . . . . . . . 12  |-  ( k  e.  ( A " { j } )  ->  <. j ,  k
>.  e.  A )
4039ad2antll 743 . . . . . . . . . . 11  |-  ( (
ph  /\  ( j  e.  ( D  \  dom  ( F supp  .0.  ) )  /\  k  e.  ( A " { j } ) ) )  ->  <. j ,  k
>.  e.  A )
41 eldifn 3545 . . . . . . . . . . . . 13  |-  ( j  e.  ( D  \  dom  ( F supp  .0.  )
)  ->  -.  j  e.  dom  ( F supp  .0.  ) )
4241ad2antrl 742 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( j  e.  ( D  \  dom  ( F supp  .0.  ) )  /\  k  e.  ( A " { j } ) ) )  ->  -.  j  e.  dom  ( F supp  .0.  )
)
4336, 37opeldm 5044 . . . . . . . . . . . 12  |-  ( <.
j ,  k >.  e.  ( F supp  .0.  )  ->  j  e.  dom  ( F supp  .0.  ) )
4442, 43nsyl 125 . . . . . . . . . . 11  |-  ( (
ph  /\  ( j  e.  ( D  \  dom  ( F supp  .0.  ) )  /\  k  e.  ( A " { j } ) ) )  ->  -.  <. j ,  k >.  e.  ( F supp  .0.  ) )
4540, 44eldifd 3401 . . . . . . . . . 10  |-  ( (
ph  /\  ( j  e.  ( D  \  dom  ( F supp  .0.  ) )  /\  k  e.  ( A " { j } ) ) )  ->  <. j ,  k
>.  e.  ( A  \ 
( F supp  .0.  )
) )
46 df-ov 6311 . . . . . . . . . . 11  |-  ( j F k )  =  ( F `  <. j ,  k >. )
47 ssid 3437 . . . . . . . . . . . . 13  |-  ( F supp 
.0.  )  C_  ( F supp  .0.  )
4847a1i 11 . . . . . . . . . . . 12  |-  ( ph  ->  ( F supp  .0.  )  C_  ( F supp  .0.  )
)
49 fvex 5889 . . . . . . . . . . . . . 14  |-  ( 0g
`  G )  e. 
_V
502, 49eqeltri 2545 . . . . . . . . . . . . 13  |-  .0.  e.  _V
5150a1i 11 . . . . . . . . . . . 12  |-  ( ph  ->  .0.  e.  _V )
528, 48, 4, 51suppssr 6965 . . . . . . . . . . 11  |-  ( (
ph  /\  <. j ,  k >.  e.  ( A  \  ( F supp  .0.  ) ) )  -> 
( F `  <. j ,  k >. )  =  .0.  )
5346, 52syl5eq 2517 . . . . . . . . . 10  |-  ( (
ph  /\  <. j ,  k >.  e.  ( A  \  ( F supp  .0.  ) ) )  -> 
( j F k )  =  .0.  )
5445, 53syldan 478 . . . . . . . . 9  |-  ( (
ph  /\  ( j  e.  ( D  \  dom  ( F supp  .0.  ) )  /\  k  e.  ( A " { j } ) ) )  ->  ( j F k )  =  .0.  )
5554anassrs 660 . . . . . . . 8  |-  ( ( ( ph  /\  j  e.  ( D  \  dom  ( F supp  .0.  ) ) )  /\  k  e.  ( A " {
j } ) )  ->  ( j F k )  =  .0.  )
5655mpteq2dva 4482 . . . . . . 7  |-  ( (
ph  /\  j  e.  ( D  \  dom  ( F supp  .0.  ) ) )  ->  ( k  e.  ( A " {
j } )  |->  ( j F k ) )  =  ( k  e.  ( A " { j } ) 
|->  .0.  ) )
5756oveq2d 6324 . . . . . 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 17523 . . . . . . . . 9  |-  ( G  e. CMnd  ->  G  e.  Mnd )
593, 58syl 17 . . . . . . . 8  |-  ( ph  ->  G  e.  Mnd )
60 imaexg 6749 . . . . . . . . 9  |-  ( A  e.  V  ->  ( A " { j } )  e.  _V )
614, 60syl 17 . . . . . . . 8  |-  ( ph  ->  ( A " {
j } )  e. 
_V )
622gsumz 16699 . . . . . . . 8  |-  ( ( G  e.  Mnd  /\  ( A " { j } )  e.  _V )  ->  ( G  gsumg  ( k  e.  ( A " { j } ) 
|->  .0.  ) )  =  .0.  )
6359, 61, 62syl2anc 673 . . . . . . 7  |-  ( ph  ->  ( G  gsumg  ( k  e.  ( A " { j } )  |->  .0.  )
)  =  .0.  )
6463adantr 472 . . . . . 6  |-  ( (
ph  /\  j  e.  ( D  \  dom  ( F supp  .0.  ) ) )  ->  ( G  gsumg  ( k  e.  ( A " { j } ) 
|->  .0.  ) )  =  .0.  )
6557, 64eqtrd 2505 . . . . 5  |-  ( (
ph  /\  j  e.  ( D  \  dom  ( F supp  .0.  ) ) )  ->  ( G  gsumg  ( k  e.  ( A " { j } ) 
|->  ( j F k ) ) )  =  .0.  )
6665, 6suppss2 6968 . . . 4  |-  ( ph  ->  ( ( j  e.  D  |->  ( G  gsumg  ( k  e.  ( A " { j } ) 
|->  ( j F k ) ) ) ) supp 
.0.  )  C_  dom  ( F supp  .0.  ) )
67 funmpt 5625 . . . . . 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 7908 . . . . . . 7  |-  ( ph  ->  ( F supp  .0.  )  e.  Fin )
70 dmfi 7872 . . . . . . 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 7810 . . . . . 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 673 . . . . 5  |-  ( ph  ->  ( ( j  e.  D  |->  ( G  gsumg  ( k  e.  ( A " { j } ) 
|->  ( j F k ) ) ) ) supp 
.0.  )  e.  Fin )
74 mptexg 6151 . . . . . . 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 7905 . . . . . 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 673 . . . . 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 936 . . . 4  |-  ( ph  ->  ( j  e.  D  |->  ( G  gsumg  ( k  e.  ( A " { j } )  |->  ( j F k ) ) ) ) finSupp  .0.  )
791, 2, 3, 6, 35, 66, 78gsumres 17625 . . 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 2507 . 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 2513 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 189    /\ wa 376    = wceq 1452    e. wcel 1904   _Vcvv 3031    \ cdif 3387    i^i cin 3389    C_ wss 3390   {csn 3959   <.cop 3965   class class class wbr 4395    |-> cmpt 4454    X. cxp 4837   dom cdm 4839   ran crn 4840    |` cres 4841   "cima 4842   Rel wrel 4844   Fun wfun 5583   -->wf 5585   ` cfv 5589  (class class class)co 6308   supp csupp 6933   Fincfn 7587   finSupp cfsupp 7901   Basecbs 15199   0gc0g 15416    gsumg cgsu 15417   Mndcmnd 16613  CMndccmn 17508
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-iin 4272  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-of 6550  df-om 6712  df-1st 6812  df-2nd 6813  df-supp 6934  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-fsupp 7902  df-oi 8043  df-card 8391  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-2 10690  df-n0 10894  df-z 10962  df-uz 11183  df-fz 11811  df-fzo 11943  df-seq 12252  df-hash 12554  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-ress 15206  df-plusg 15281  df-0g 15418  df-gsum 15419  df-mre 15570  df-mrc 15571  df-acs 15573  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-submnd 16661  df-mulg 16754  df-cntz 17049  df-cmn 17510
This theorem is referenced by:  gsum2d2  17684  gsumxp  17686
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