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Theorem gsum2d 16463
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 16462 . 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 6703 . . . . . 6  |-  ( F supp 
.0.  )  C_  dom  F
12 fdm 5563 . . . . . . 7  |-  ( F : A --> B  ->  dom  F  =  A )
138, 12syl 16 . . . . . 6  |-  ( ph  ->  dom  F  =  A )
1411, 13syl5sseq 3404 . . . . 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 60 . . . . . 6  |-  ( ph  ->  Rel  ( F supp  .0.  ) )
17 relssdmrn 5358 . . . . . . 7  |-  ( Rel  ( F supp  .0.  )  ->  ( F supp  .0.  )  C_  ( dom  ( F supp 
.0.  )  X.  ran  ( F supp  .0.  ) ) )
18 ssv 3376 . . . . . . . 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 3368 . . . . . 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 3574 . . . 4  |-  ( ph  ->  ( F supp  .0.  )  C_  ( A  i^i  ( dom  ( F supp  .0.  )  X.  _V ) ) )
24 df-res 4852 . . . 4  |-  ( A  |`  dom  ( F supp  .0.  ) )  =  ( A  i^i  ( dom  ( F supp  .0.  )  X.  _V ) )
2523, 24syl6sseqr 3403 . . 3  |-  ( ph  ->  ( F supp  .0.  )  C_  ( A  |`  dom  ( F supp  .0.  ) ) )
261, 2, 3, 4, 8, 25, 9gsumres 16395 . 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 16 . . . . . 6  |-  ( ph  ->  dom  ( F supp  .0.  )  C_  dom  A )
2928, 7sstrd 3366 . . . . 5  |-  ( ph  ->  dom  ( F supp  .0.  )  C_  D )
30 resmpt 5156 . . . . 5  |-  ( dom  ( F supp  .0.  )  C_  D  ->  ( (
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 ) ) ) ) )
3129, 30syl 16 . . . 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 ) ) ) ) )
3231oveq2d 6107 . . 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 ) ) ) ) ) )
331, 2, 3, 4, 5, 6, 7, 8, 9gsum2dlem1 16461 . . . . . 6  |-  ( ph  ->  ( G  gsumg  ( k  e.  ( A " { j } )  |->  ( j F k ) ) )  e.  B )
3433adantr 465 . . . . 5  |-  ( (
ph  /\  j  e.  D )  ->  ( G  gsumg  ( k  e.  ( A " { j } )  |->  ( j F k ) ) )  e.  B )
35 eqid 2443 . . . . 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 ) ) ) )
3634, 35fmptd 5867 . . . 4  |-  ( ph  ->  ( j  e.  D  |->  ( G  gsumg  ( k  e.  ( A " { j } )  |->  ( j F k ) ) ) ) : D --> B )
37 vex 2975 . . . . . . . . . . . . . 14  |-  j  e. 
_V
38 vex 2975 . . . . . . . . . . . . . 14  |-  k  e. 
_V
3937, 38elimasn 5194 . . . . . . . . . . . . 13  |-  ( k  e.  ( A " { j } )  <->  <. j ,  k >.  e.  A )
4039biimpi 194 . . . . . . . . . . . 12  |-  ( k  e.  ( A " { j } )  ->  <. j ,  k
>.  e.  A )
4140ad2antll 728 . . . . . . . . . . 11  |-  ( (
ph  /\  ( j  e.  ( D  \  dom  ( F supp  .0.  ) )  /\  k  e.  ( A " { j } ) ) )  ->  <. j ,  k
>.  e.  A )
42 eldifn 3479 . . . . . . . . . . . . 13  |-  ( j  e.  ( D  \  dom  ( F supp  .0.  )
)  ->  -.  j  e.  dom  ( F supp  .0.  ) )
4342ad2antrl 727 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( j  e.  ( D  \  dom  ( F supp  .0.  ) )  /\  k  e.  ( A " { j } ) ) )  ->  -.  j  e.  dom  ( F supp  .0.  )
)
4437, 38opeldm 5043 . . . . . . . . . . . 12  |-  ( <.
j ,  k >.  e.  ( F supp  .0.  )  ->  j  e.  dom  ( F supp  .0.  ) )
4543, 44nsyl 121 . . . . . . . . . . 11  |-  ( (
ph  /\  ( j  e.  ( D  \  dom  ( F supp  .0.  ) )  /\  k  e.  ( A " { j } ) ) )  ->  -.  <. j ,  k >.  e.  ( F supp  .0.  ) )
4641, 45eldifd 3339 . . . . . . . . . 10  |-  ( (
ph  /\  ( j  e.  ( D  \  dom  ( F supp  .0.  ) )  /\  k  e.  ( A " { j } ) ) )  ->  <. j ,  k
>.  e.  ( A  \ 
( F supp  .0.  )
) )
47 df-ov 6094 . . . . . . . . . . 11  |-  ( j F k )  =  ( F `  <. j ,  k >. )
48 ssid 3375 . . . . . . . . . . . . 13  |-  ( F supp 
.0.  )  C_  ( F supp  .0.  )
4948a1i 11 . . . . . . . . . . . 12  |-  ( ph  ->  ( F supp  .0.  )  C_  ( F supp  .0.  )
)
50 fvex 5701 . . . . . . . . . . . . . 14  |-  ( 0g
`  G )  e. 
_V
512, 50eqeltri 2513 . . . . . . . . . . . . 13  |-  .0.  e.  _V
5251a1i 11 . . . . . . . . . . . 12  |-  ( ph  ->  .0.  e.  _V )
538, 49, 4, 52suppssr 6720 . . . . . . . . . . 11  |-  ( (
ph  /\  <. j ,  k >.  e.  ( A  \  ( F supp  .0.  ) ) )  -> 
( F `  <. j ,  k >. )  =  .0.  )
5447, 53syl5eq 2487 . . . . . . . . . 10  |-  ( (
ph  /\  <. j ,  k >.  e.  ( A  \  ( F supp  .0.  ) ) )  -> 
( j F k )  =  .0.  )
5546, 54syldan 470 . . . . . . . . 9  |-  ( (
ph  /\  ( j  e.  ( D  \  dom  ( F supp  .0.  ) )  /\  k  e.  ( A " { j } ) ) )  ->  ( j F k )  =  .0.  )
5655anassrs 648 . . . . . . . 8  |-  ( ( ( ph  /\  j  e.  ( D  \  dom  ( F supp  .0.  ) ) )  /\  k  e.  ( A " {
j } ) )  ->  ( j F k )  =  .0.  )
5756mpteq2dva 4378 . . . . . . 7  |-  ( (
ph  /\  j  e.  ( D  \  dom  ( F supp  .0.  ) ) )  ->  ( k  e.  ( A " {
j } )  |->  ( j F k ) )  =  ( k  e.  ( A " { j } ) 
|->  .0.  ) )
5857oveq2d 6107 . . . . . 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.  )
) )
59 cmnmnd 16292 . . . . . . . . 9  |-  ( G  e. CMnd  ->  G  e.  Mnd )
603, 59syl 16 . . . . . . . 8  |-  ( ph  ->  G  e.  Mnd )
61 imaexg 6515 . . . . . . . . 9  |-  ( A  e.  V  ->  ( A " { j } )  e.  _V )
624, 61syl 16 . . . . . . . 8  |-  ( ph  ->  ( A " {
j } )  e. 
_V )
632gsumz 15511 . . . . . . . 8  |-  ( ( G  e.  Mnd  /\  ( A " { j } )  e.  _V )  ->  ( G  gsumg  ( k  e.  ( A " { j } ) 
|->  .0.  ) )  =  .0.  )
6460, 62, 63syl2anc 661 . . . . . . 7  |-  ( ph  ->  ( G  gsumg  ( k  e.  ( A " { j } )  |->  .0.  )
)  =  .0.  )
6564adantr 465 . . . . . 6  |-  ( (
ph  /\  j  e.  ( D  \  dom  ( F supp  .0.  ) ) )  ->  ( G  gsumg  ( k  e.  ( A " { j } ) 
|->  .0.  ) )  =  .0.  )
6658, 65eqtrd 2475 . . . . 5  |-  ( (
ph  /\  j  e.  ( D  \  dom  ( F supp  .0.  ) ) )  ->  ( G  gsumg  ( k  e.  ( A " { j } ) 
|->  ( j F k ) ) )  =  .0.  )
6766, 6suppss2 6723 . . . 4  |-  ( ph  ->  ( ( j  e.  D  |->  ( G  gsumg  ( k  e.  ( A " { j } ) 
|->  ( j F k ) ) ) ) supp 
.0.  )  C_  dom  ( F supp  .0.  ) )
68 funmpt 5454 . . . . . 6  |-  Fun  (
j  e.  D  |->  ( G  gsumg  ( k  e.  ( A " { j } )  |->  ( j F k ) ) ) )
6968a1i 11 . . . . 5  |-  ( ph  ->  Fun  ( j  e.  D  |->  ( G  gsumg  ( k  e.  ( A " { j } ) 
|->  ( j F k ) ) ) ) )
709fsuppimpd 7627 . . . . . . 7  |-  ( ph  ->  ( F supp  .0.  )  e.  Fin )
71 dmfi 7594 . . . . . . 7  |-  ( ( F supp  .0.  )  e.  Fin  ->  dom  ( F supp  .0.  )  e.  Fin )
7270, 71syl 16 . . . . . 6  |-  ( ph  ->  dom  ( F supp  .0.  )  e.  Fin )
73 ssfi 7533 . . . . . 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 )
7472, 67, 73syl2anc 661 . . . . 5  |-  ( ph  ->  ( ( j  e.  D  |->  ( G  gsumg  ( k  e.  ( A " { j } ) 
|->  ( j F k ) ) ) ) supp 
.0.  )  e.  Fin )
75 mptexg 5947 . . . . . . 7  |-  ( D  e.  W  ->  (
j  e.  D  |->  ( G  gsumg  ( k  e.  ( A " { j } )  |->  ( j F k ) ) ) )  e.  _V )
766, 75syl 16 . . . . . 6  |-  ( ph  ->  ( j  e.  D  |->  ( G  gsumg  ( k  e.  ( A " { j } )  |->  ( j F k ) ) ) )  e.  _V )
77 isfsupp 7624 . . . . . 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 ) ) )
7876, 52, 77syl2anc 661 . . . . 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 ) ) )
7969, 74, 78mpbir2and 913 . . . 4  |-  ( ph  ->  ( j  e.  D  |->  ( G  gsumg  ( k  e.  ( A " { j } )  |->  ( j F k ) ) ) ) finSupp  .0.  )
801, 2, 3, 6, 36, 67, 79gsumres 16395 . . 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 ) ) ) ) ) )
8132, 80eqtr3d 2477 . 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 ) ) ) ) ) )
8210, 26, 813eqtr3d 2483 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 369    = wceq 1369    e. wcel 1756   _Vcvv 2972    \ cdif 3325    i^i cin 3327    C_ wss 3328   {csn 3877   <.cop 3883   class class class wbr 4292    e. cmpt 4350    X. cxp 4838   dom cdm 4840   ran crn 4841    |` cres 4842   "cima 4843   Rel wrel 4845   Fun wfun 5412   -->wf 5414   ` cfv 5418  (class class class)co 6091   supp csupp 6690   Fincfn 7310   finSupp cfsupp 7620   Basecbs 14174   0gc0g 14378    gsumg cgsu 14379   Mndcmnd 15409  CMndccmn 16277
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-inf2 7847  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-iin 4174  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-se 4680  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-isom 5427  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-of 6320  df-om 6477  df-1st 6577  df-2nd 6578  df-supp 6691  df-recs 6832  df-rdg 6866  df-1o 6920  df-oadd 6924  df-er 7101  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-fsupp 7621  df-oi 7724  df-card 8109  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-nn 10323  df-2 10380  df-n0 10580  df-z 10647  df-uz 10862  df-fz 11438  df-fzo 11549  df-seq 11807  df-hash 12104  df-ndx 14177  df-slot 14178  df-base 14179  df-sets 14180  df-ress 14181  df-plusg 14251  df-0g 14380  df-gsum 14381  df-mre 14524  df-mrc 14525  df-acs 14527  df-mnd 15415  df-submnd 15465  df-mulg 15548  df-cntz 15835  df-cmn 16279
This theorem is referenced by:  gsum2d2  16466  gsumxp  16468
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