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Theorem gsum2d 17198
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 17197 . 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 6904 . . . . . 6  |-  ( F supp 
.0.  )  C_  dom  F
12 fdm 5717 . . . . . . 7  |-  ( F : A --> B  ->  dom  F  =  A )
138, 12syl 16 . . . . . 6  |-  ( ph  ->  dom  F  =  A )
1411, 13syl5sseq 3537 . . . . 5  |-  ( ph  ->  ( F supp  .0.  )  C_  A )
15 relss 5078 . . . . . . 7  |-  ( ( F supp  .0.  )  C_  A  ->  ( Rel  A  ->  Rel  ( F supp  .0.  ) ) )
1614, 5, 15sylc 60 . . . . . 6  |-  ( ph  ->  Rel  ( F supp  .0.  ) )
17 relssdmrn 5511 . . . . . . 7  |-  ( Rel  ( F supp  .0.  )  ->  ( F supp  .0.  )  C_  ( dom  ( F supp 
.0.  )  X.  ran  ( F supp  .0.  ) ) )
18 ssv 3509 . . . . . . . 8  |-  ran  ( F supp  .0.  )  C_  _V
19 xpss2 5100 . . . . . . . 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 3501 . . . . . 6  |-  ( Rel  ( F supp  .0.  )  ->  ( F supp  .0.  )  C_  ( dom  ( F supp 
.0.  )  X.  _V ) )
2216, 21syl 16 . . . . 5  |-  ( ph  ->  ( F supp  .0.  )  C_  ( dom  ( F supp 
.0.  )  X.  _V ) )
2314, 22ssind 3708 . . . 4  |-  ( ph  ->  ( F supp  .0.  )  C_  ( A  i^i  ( dom  ( F supp  .0.  )  X.  _V ) ) )
24 df-res 5000 . . . 4  |-  ( A  |`  dom  ( F supp  .0.  ) )  =  ( A  i^i  ( dom  ( F supp  .0.  )  X.  _V ) )
2523, 24syl6sseqr 3536 . . 3  |-  ( ph  ->  ( F supp  .0.  )  C_  ( A  |`  dom  ( F supp  .0.  ) ) )
261, 2, 3, 4, 8, 25, 9gsumres 17123 . 2  |-  ( ph  ->  ( G  gsumg  ( F  |`  ( A  |`  dom  ( F supp 
.0.  ) ) ) )  =  ( G 
gsumg  F ) )
27 dmss 5191 . . . . . . 7  |-  ( ( F supp  .0.  )  C_  A  ->  dom  ( F supp  .0.  )  C_  dom  A )
2814, 27syl 16 . . . . . 6  |-  ( ph  ->  dom  ( F supp  .0.  )  C_  dom  A )
2928, 7sstrd 3499 . . . . 5  |-  ( ph  ->  dom  ( F supp  .0.  )  C_  D )
3029resmptd 5313 . . . 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 6286 . . 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 17196 . . . . . 6  |-  ( ph  ->  ( G  gsumg  ( k  e.  ( A " { j } )  |->  ( j F k ) ) )  e.  B )
3332adantr 463 . . . . 5  |-  ( (
ph  /\  j  e.  D )  ->  ( G  gsumg  ( k  e.  ( A " { j } )  |->  ( j F k ) ) )  e.  B )
34 eqid 2454 . . . . 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 6031 . . . 4  |-  ( ph  ->  ( j  e.  D  |->  ( G  gsumg  ( k  e.  ( A " { j } )  |->  ( j F k ) ) ) ) : D --> B )
36 vex 3109 . . . . . . . . . . . . . 14  |-  j  e. 
_V
37 vex 3109 . . . . . . . . . . . . . 14  |-  k  e. 
_V
3836, 37elimasn 5350 . . . . . . . . . . . . 13  |-  ( k  e.  ( A " { j } )  <->  <. j ,  k >.  e.  A )
3938biimpi 194 . . . . . . . . . . . 12  |-  ( k  e.  ( A " { j } )  ->  <. j ,  k
>.  e.  A )
4039ad2antll 726 . . . . . . . . . . 11  |-  ( (
ph  /\  ( j  e.  ( D  \  dom  ( F supp  .0.  ) )  /\  k  e.  ( A " { j } ) ) )  ->  <. j ,  k
>.  e.  A )
41 eldifn 3613 . . . . . . . . . . . . 13  |-  ( j  e.  ( D  \  dom  ( F supp  .0.  )
)  ->  -.  j  e.  dom  ( F supp  .0.  ) )
4241ad2antrl 725 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( j  e.  ( D  \  dom  ( F supp  .0.  ) )  /\  k  e.  ( A " { j } ) ) )  ->  -.  j  e.  dom  ( F supp  .0.  )
)
4336, 37opeldm 5195 . . . . . . . . . . . 12  |-  ( <.
j ,  k >.  e.  ( F supp  .0.  )  ->  j  e.  dom  ( F supp  .0.  ) )
4442, 43nsyl 121 . . . . . . . . . . 11  |-  ( (
ph  /\  ( j  e.  ( D  \  dom  ( F supp  .0.  ) )  /\  k  e.  ( A " { j } ) ) )  ->  -.  <. j ,  k >.  e.  ( F supp  .0.  ) )
4540, 44eldifd 3472 . . . . . . . . . 10  |-  ( (
ph  /\  ( j  e.  ( D  \  dom  ( F supp  .0.  ) )  /\  k  e.  ( A " { j } ) ) )  ->  <. j ,  k
>.  e.  ( A  \ 
( F supp  .0.  )
) )
46 df-ov 6273 . . . . . . . . . . 11  |-  ( j F k )  =  ( F `  <. j ,  k >. )
47 ssid 3508 . . . . . . . . . . . . 13  |-  ( F supp 
.0.  )  C_  ( F supp  .0.  )
4847a1i 11 . . . . . . . . . . . 12  |-  ( ph  ->  ( F supp  .0.  )  C_  ( F supp  .0.  )
)
49 fvex 5858 . . . . . . . . . . . . . 14  |-  ( 0g
`  G )  e. 
_V
502, 49eqeltri 2538 . . . . . . . . . . . . 13  |-  .0.  e.  _V
5150a1i 11 . . . . . . . . . . . 12  |-  ( ph  ->  .0.  e.  _V )
528, 48, 4, 51suppssr 6923 . . . . . . . . . . 11  |-  ( (
ph  /\  <. j ,  k >.  e.  ( A  \  ( F supp  .0.  ) ) )  -> 
( F `  <. j ,  k >. )  =  .0.  )
5346, 52syl5eq 2507 . . . . . . . . . 10  |-  ( (
ph  /\  <. j ,  k >.  e.  ( A  \  ( F supp  .0.  ) ) )  -> 
( j F k )  =  .0.  )
5445, 53syldan 468 . . . . . . . . 9  |-  ( (
ph  /\  ( j  e.  ( D  \  dom  ( F supp  .0.  ) )  /\  k  e.  ( A " { j } ) ) )  ->  ( j F k )  =  .0.  )
5554anassrs 646 . . . . . . . 8  |-  ( ( ( ph  /\  j  e.  ( D  \  dom  ( F supp  .0.  ) ) )  /\  k  e.  ( A " {
j } ) )  ->  ( j F k )  =  .0.  )
5655mpteq2dva 4525 . . . . . . 7  |-  ( (
ph  /\  j  e.  ( D  \  dom  ( F supp  .0.  ) ) )  ->  ( k  e.  ( A " {
j } )  |->  ( j F k ) )  =  ( k  e.  ( A " { j } ) 
|->  .0.  ) )
5756oveq2d 6286 . . . . . 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 17015 . . . . . . . . 9  |-  ( G  e. CMnd  ->  G  e.  Mnd )
593, 58syl 16 . . . . . . . 8  |-  ( ph  ->  G  e.  Mnd )
60 imaexg 6710 . . . . . . . . 9  |-  ( A  e.  V  ->  ( A " { j } )  e.  _V )
614, 60syl 16 . . . . . . . 8  |-  ( ph  ->  ( A " {
j } )  e. 
_V )
622gsumz 16207 . . . . . . . 8  |-  ( ( G  e.  Mnd  /\  ( A " { j } )  e.  _V )  ->  ( G  gsumg  ( k  e.  ( A " { j } ) 
|->  .0.  ) )  =  .0.  )
6359, 61, 62syl2anc 659 . . . . . . 7  |-  ( ph  ->  ( G  gsumg  ( k  e.  ( A " { j } )  |->  .0.  )
)  =  .0.  )
6463adantr 463 . . . . . 6  |-  ( (
ph  /\  j  e.  ( D  \  dom  ( F supp  .0.  ) ) )  ->  ( G  gsumg  ( k  e.  ( A " { j } ) 
|->  .0.  ) )  =  .0.  )
6557, 64eqtrd 2495 . . . . 5  |-  ( (
ph  /\  j  e.  ( D  \  dom  ( F supp  .0.  ) ) )  ->  ( G  gsumg  ( k  e.  ( A " { j } ) 
|->  ( j F k ) ) )  =  .0.  )
6665, 6suppss2 6926 . . . 4  |-  ( ph  ->  ( ( j  e.  D  |->  ( G  gsumg  ( k  e.  ( A " { j } ) 
|->  ( j F k ) ) ) ) supp 
.0.  )  C_  dom  ( F supp  .0.  ) )
67 funmpt 5606 . . . . . 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 7828 . . . . . . 7  |-  ( ph  ->  ( F supp  .0.  )  e.  Fin )
70 dmfi 7795 . . . . . . 7  |-  ( ( F supp  .0.  )  e.  Fin  ->  dom  ( F supp  .0.  )  e.  Fin )
7169, 70syl 16 . . . . . 6  |-  ( ph  ->  dom  ( F supp  .0.  )  e.  Fin )
72 ssfi 7733 . . . . . 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 659 . . . . 5  |-  ( ph  ->  ( ( j  e.  D  |->  ( G  gsumg  ( k  e.  ( A " { j } ) 
|->  ( j F k ) ) ) ) supp 
.0.  )  e.  Fin )
74 mptexg 6117 . . . . . . 7  |-  ( D  e.  W  ->  (
j  e.  D  |->  ( G  gsumg  ( k  e.  ( A " { j } )  |->  ( j F k ) ) ) )  e.  _V )
756, 74syl 16 . . . . . 6  |-  ( ph  ->  ( j  e.  D  |->  ( G  gsumg  ( k  e.  ( A " { j } )  |->  ( j F k ) ) ) )  e.  _V )
76 isfsupp 7825 . . . . . 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 659 . . . . 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 920 . . . 4  |-  ( ph  ->  ( j  e.  D  |->  ( G  gsumg  ( k  e.  ( A " { j } )  |->  ( j F k ) ) ) ) finSupp  .0.  )
791, 2, 3, 6, 35, 66, 78gsumres 17123 . . 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 2497 . 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 2503 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 184    /\ wa 367    = wceq 1398    e. wcel 1823   _Vcvv 3106    \ cdif 3458    i^i cin 3460    C_ wss 3461   {csn 4016   <.cop 4022   class class class wbr 4439    |-> cmpt 4497    X. cxp 4986   dom cdm 4988   ran crn 4989    |` cres 4990   "cima 4991   Rel wrel 4993   Fun wfun 5564   -->wf 5566   ` cfv 5570  (class class class)co 6270   supp csupp 6891   Fincfn 7509   finSupp cfsupp 7821   Basecbs 14719   0gc0g 14932    gsumg cgsu 14933   Mndcmnd 16121  CMndccmn 17000
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-inf2 8049  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-iin 4318  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-of 6513  df-om 6674  df-1st 6773  df-2nd 6774  df-supp 6892  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-fsupp 7822  df-oi 7927  df-card 8311  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-n0 10792  df-z 10861  df-uz 11083  df-fz 11676  df-fzo 11800  df-seq 12093  df-hash 12391  df-ndx 14722  df-slot 14723  df-base 14724  df-sets 14725  df-ress 14726  df-plusg 14800  df-0g 14934  df-gsum 14935  df-mre 15078  df-mrc 15079  df-acs 15081  df-mgm 16074  df-sgrp 16113  df-mnd 16123  df-submnd 16169  df-mulg 16262  df-cntz 16557  df-cmn 17002
This theorem is referenced by:  gsum2d2  17201  gsumxp  17203
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