Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  gsummpt2co Structured version   Unicode version

Theorem gsummpt2co 26200
Description: Split a finite sum into a sum of a collection of sums over disjoint subsets. (Contributed by Thierry Arnoux, 27-Mar-2018.)
Hypotheses
Ref Expression
gsummpt2co.b  |-  B  =  ( Base `  W
)
gsummpt2co.z  |-  .0.  =  ( 0g `  W )
gsummpt2co.w  |-  ( ph  ->  W  e. CMnd )
gsummpt2co.a  |-  ( ph  ->  A  e.  Fin )
gsummpt2co.e  |-  ( ph  ->  E  ~<_  om )
gsummpt2co.1  |-  ( (
ph  /\  x  e.  A )  ->  C  e.  B )
gsummpt2co.2  |-  ( (
ph  /\  x  e.  A )  ->  D  e.  E )
gsummpt2co.3  |-  F  =  ( x  e.  A  |->  D )
Assertion
Ref Expression
gsummpt2co  |-  ( ph  ->  ( W  gsumg  ( x  e.  A  |->  C ) )  =  ( W  gsumg  ( y  e.  E  |->  ( W  gsumg  ( x  e.  ( `' F " { y } )  |->  C ) ) ) ) )
Distinct variable groups:    x, y, A    x, B, y    y, C    x, E, y    x, F, y    ph, x, y   
x,  .0. , y    x, W, y
Allowed substitution hints:    C( x)    D( x, y)

Proof of Theorem gsummpt2co
Dummy variable  p is distinct from all other variables.
StepHypRef Expression
1 nfcsb1v 3299 . . 3  |-  F/_ x [_ ( 2nd `  p
)  /  x ]_ C
2 gsummpt2co.b . . 3  |-  B  =  ( Base `  W
)
3 gsummpt2co.z . . 3  |-  .0.  =  ( 0g `  W )
4 csbeq1a 3292 . . 3  |-  ( x  =  ( 2nd `  p
)  ->  C  =  [_ ( 2nd `  p
)  /  x ]_ C )
5 gsummpt2co.w . . 3  |-  ( ph  ->  W  e. CMnd )
6 gsummpt2co.a . . 3  |-  ( ph  ->  A  e.  Fin )
7 ssid 3370 . . . 4  |-  B  C_  B
87a1i 11 . . 3  |-  ( ph  ->  B  C_  B )
9 gsummpt2co.1 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  C  e.  B )
10 elcnv 5011 . . . . 5  |-  ( p  e.  `' F  <->  E. y E. x ( p  = 
<. y ,  x >.  /\  x F y ) )
11 vex 2970 . . . . . . . . 9  |-  y  e. 
_V
12 vex 2970 . . . . . . . . 9  |-  x  e. 
_V
1311, 12op2ndd 6583 . . . . . . . 8  |-  ( p  =  <. y ,  x >.  ->  ( 2nd `  p
)  =  x )
1413adantr 465 . . . . . . 7  |-  ( ( p  =  <. y ,  x >.  /\  x F y )  -> 
( 2nd `  p
)  =  x )
15 gsummpt2co.3 . . . . . . . . . 10  |-  F  =  ( x  e.  A  |->  D )
1615dmmptss 5329 . . . . . . . . 9  |-  dom  F  C_  A
1712, 11breldm 5039 . . . . . . . . 9  |-  ( x F y  ->  x  e.  dom  F )
1816, 17sseldi 3349 . . . . . . . 8  |-  ( x F y  ->  x  e.  A )
1918adantl 466 . . . . . . 7  |-  ( ( p  =  <. y ,  x >.  /\  x F y )  ->  x  e.  A )
2014, 19eqeltrd 2512 . . . . . 6  |-  ( ( p  =  <. y ,  x >.  /\  x F y )  -> 
( 2nd `  p
)  e.  A )
2120exlimivv 1689 . . . . 5  |-  ( E. y E. x ( p  =  <. y ,  x >.  /\  x F y )  -> 
( 2nd `  p
)  e.  A )
2210, 21sylbi 195 . . . 4  |-  ( p  e.  `' F  -> 
( 2nd `  p
)  e.  A )
2322adantl 466 . . 3  |-  ( (
ph  /\  p  e.  `' F )  ->  ( 2nd `  p )  e.  A )
2415funmpt2 5450 . . . . . 6  |-  Fun  F
25 funcnvcnv 5471 . . . . . 6  |-  ( Fun 
F  ->  Fun  `' `' F )
2624, 25ax-mp 5 . . . . 5  |-  Fun  `' `' F
2726a1i 11 . . . 4  |-  ( (
ph  /\  x  e.  A )  ->  Fun  `' `' F )
28 dfdm4 5027 . . . . . . 7  |-  dom  F  =  ran  `' F
2915dmeqi 5036 . . . . . . . 8  |-  dom  F  =  dom  ( x  e.  A  |->  D )
30 gsummpt2co.2 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  A )  ->  D  e.  E )
3130ralrimiva 2794 . . . . . . . . 9  |-  ( ph  ->  A. x  e.  A  D  e.  E )
32 dmmptg 5330 . . . . . . . . 9  |-  ( A. x  e.  A  D  e.  E  ->  dom  (
x  e.  A  |->  D )  =  A )
3331, 32syl 16 . . . . . . . 8  |-  ( ph  ->  dom  ( x  e.  A  |->  D )  =  A )
3429, 33syl5eq 2482 . . . . . . 7  |-  ( ph  ->  dom  F  =  A )
3528, 34syl5eqr 2484 . . . . . 6  |-  ( ph  ->  ran  `' F  =  A )
3635eleq2d 2505 . . . . 5  |-  ( ph  ->  ( x  e.  ran  `' F  <->  x  e.  A
) )
3736biimpar 485 . . . 4  |-  ( (
ph  /\  x  e.  A )  ->  x  e.  ran  `' F )
38 relcnv 5201 . . . . 5  |-  Rel  `' F
39 fcnvgreu 25942 . . . . 5  |-  ( ( ( Rel  `' F  /\  Fun  `' `' F
)  /\  x  e.  ran  `' F )  ->  E! p  e.  `'  F x  =  ( 2nd `  p ) )
4038, 39mpanl1 680 . . . 4  |-  ( ( Fun  `' `' F  /\  x  e.  ran  `' F )  ->  E! p  e.  `'  F x  =  ( 2nd `  p ) )
4127, 37, 40syl2anc 661 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  E! p  e.  `'  F x  =  ( 2nd `  p ) )
421, 2, 3, 4, 5, 6, 8, 9, 23, 41gsummptf1o 26198 . 2  |-  ( ph  ->  ( W  gsumg  ( x  e.  A  |->  C ) )  =  ( W  gsumg  ( p  e.  `' F  |->  [_ ( 2nd `  p
)  /  x ]_ C ) ) )
4315rnmptss 5867 . . . . . . 7  |-  ( A. x  e.  A  D  e.  E  ->  ran  F  C_  E )
4431, 43syl 16 . . . . . 6  |-  ( ph  ->  ran  F  C_  E
)
45 dfcnv2 25945 . . . . . 6  |-  ( ran 
F  C_  E  ->  `' F  =  U_ y  e.  E  ( {
y }  X.  ( `' F " { y } ) ) )
4644, 45syl 16 . . . . 5  |-  ( ph  ->  `' F  =  U_ y  e.  E  ( { y }  X.  ( `' F " { y } ) ) )
4746mpteq1d 4368 . . . 4  |-  ( ph  ->  ( p  e.  `' F  |->  [_ ( 2nd `  p
)  /  x ]_ C )  =  ( p  e.  U_ y  e.  E  ( {
y }  X.  ( `' F " { y } ) )  |->  [_ ( 2nd `  p )  /  x ]_ C
) )
48 nfcv 2574 . . . . 5  |-  F/_ y [_ ( 2nd `  p
)  /  x ]_ C
49 csbeq1 3286 . . . . . . 7  |-  ( ( 2nd `  p )  =  x  ->  [_ ( 2nd `  p )  /  x ]_ C  =  [_ x  /  x ]_ C
)
5013, 49syl 16 . . . . . 6  |-  ( p  =  <. y ,  x >.  ->  [_ ( 2nd `  p
)  /  x ]_ C  =  [_ x  /  x ]_ C )
51 csbid 3291 . . . . . 6  |-  [_ x  /  x ]_ C  =  C
5250, 51syl6eq 2486 . . . . 5  |-  ( p  =  <. y ,  x >.  ->  [_ ( 2nd `  p
)  /  x ]_ C  =  C )
5348, 1, 52mpt2mptxf 25946 . . . 4  |-  ( p  e.  U_ y  e.  E  ( { y }  X.  ( `' F " { y } ) )  |->  [_ ( 2nd `  p )  /  x ]_ C
)  =  ( y  e.  E ,  x  e.  ( `' F " { y } ) 
|->  C )
5447, 53syl6eq 2486 . . 3  |-  ( ph  ->  ( p  e.  `' F  |->  [_ ( 2nd `  p
)  /  x ]_ C )  =  ( y  e.  E ,  x  e.  ( `' F " { y } )  |->  C ) )
5554oveq2d 6102 . 2  |-  ( ph  ->  ( W  gsumg  ( p  e.  `' F  |->  [_ ( 2nd `  p
)  /  x ]_ C ) )  =  ( W  gsumg  ( y  e.  E ,  x  e.  ( `' F " { y } )  |->  C ) ) )
56 gsummpt2co.e . . . 4  |-  ( ph  ->  E  ~<_  om )
57 ctex 25959 . . . 4  |-  ( E  ~<_  om  ->  E  e.  _V )
5856, 57syl 16 . . 3  |-  ( ph  ->  E  e.  _V )
59 mptfi 7602 . . . . . . 7  |-  ( A  e.  Fin  ->  (
x  e.  A  |->  D )  e.  Fin )
6015, 59syl5eqel 2522 . . . . . 6  |-  ( A  e.  Fin  ->  F  e.  Fin )
61 cnvfi 7587 . . . . . 6  |-  ( F  e.  Fin  ->  `' F  e.  Fin )
626, 60, 613syl 20 . . . . 5  |-  ( ph  ->  `' F  e.  Fin )
63 imaexg 6510 . . . . 5  |-  ( `' F  e.  Fin  ->  ( `' F " { y } )  e.  _V )
6462, 63syl 16 . . . 4  |-  ( ph  ->  ( `' F " { y } )  e.  _V )
6564adantr 465 . . 3  |-  ( (
ph  /\  y  e.  E )  ->  ( `' F " { y } )  e.  _V )
66 simpll 753 . . . . 5  |-  ( ( ( ph  /\  y  e.  E )  /\  x  e.  ( `' F " { y } ) )  ->  ph )
67 imassrn 5175 . . . . . . . 8  |-  ( `' F " { y } )  C_  ran  `' F
6867, 28sseqtr4i 3384 . . . . . . 7  |-  ( `' F " { y } )  C_  dom  F
6968, 16sstri 3360 . . . . . 6  |-  ( `' F " { y } )  C_  A
7011, 12elimasn 5189 . . . . . . . . 9  |-  ( x  e.  ( `' F " { y } )  <->  <. y ,  x >.  e.  `' F )
7170biimpi 194 . . . . . . . 8  |-  ( x  e.  ( `' F " { y } )  ->  <. y ,  x >.  e.  `' F )
7271adantl 466 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  E )  /\  x  e.  ( `' F " { y } ) )  ->  <. y ,  x >.  e.  `' F )
7372, 70sylibr 212 . . . . . 6  |-  ( ( ( ph  /\  y  e.  E )  /\  x  e.  ( `' F " { y } ) )  ->  x  e.  ( `' F " { y } ) )
7469, 73sseldi 3349 . . . . 5  |-  ( ( ( ph  /\  y  e.  E )  /\  x  e.  ( `' F " { y } ) )  ->  x  e.  A )
7566, 74, 9syl2anc 661 . . . 4  |-  ( ( ( ph  /\  y  e.  E )  /\  x  e.  ( `' F " { y } ) )  ->  C  e.  B )
7675anasss 647 . . 3  |-  ( (
ph  /\  ( y  e.  E  /\  x  e.  ( `' F " { y } ) ) )  ->  C  e.  B )
77 df-br 4288 . . . . . . . 8  |-  ( y `' F x  <->  <. y ,  x >.  e.  `' F )
7872, 77sylibr 212 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  E )  /\  x  e.  ( `' F " { y } ) )  ->  y `' F x )
7978anasss 647 . . . . . 6  |-  ( (
ph  /\  ( y  e.  E  /\  x  e.  ( `' F " { y } ) ) )  ->  y `' F x )
8079pm2.24d 143 . . . . 5  |-  ( (
ph  /\  ( y  e.  E  /\  x  e.  ( `' F " { y } ) ) )  ->  ( -.  y `' F x  ->  C  =  .0.  ) )
8180imp 429 . . . 4  |-  ( ( ( ph  /\  (
y  e.  E  /\  x  e.  ( `' F " { y } ) ) )  /\  -.  y `' F x )  ->  C  =  .0.  )
8281anasss 647 . . 3  |-  ( (
ph  /\  ( (
y  e.  E  /\  x  e.  ( `' F " { y } ) )  /\  -.  y `' F x ) )  ->  C  =  .0.  )
832, 3, 5, 58, 65, 76, 62, 82gsum2d2 16454 . 2  |-  ( ph  ->  ( W  gsumg  ( y  e.  E ,  x  e.  ( `' F " { y } )  |->  C ) )  =  ( W 
gsumg  ( y  e.  E  |->  ( W  gsumg  ( x  e.  ( `' F " { y } )  |->  C ) ) ) ) )
8442, 55, 833eqtrd 2474 1  |-  ( ph  ->  ( W  gsumg  ( x  e.  A  |->  C ) )  =  ( W  gsumg  ( y  e.  E  |->  ( W  gsumg  ( x  e.  ( `' F " { y } )  |->  C ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1369   E.wex 1586    e. wcel 1756   A.wral 2710   E!wreu 2712   _Vcvv 2967   [_csb 3283    C_ wss 3323   {csn 3872   <.cop 3878   U_ciun 4166   class class class wbr 4287    e. cmpt 4345    X. cxp 4833   `'ccnv 4834   dom cdm 4835   ran crn 4836   "cima 4838   Rel wrel 4840   Fun wfun 5407   ` cfv 5413  (class class class)co 6086    e. cmpt2 6088   omcom 6471   2ndc2nd 6571    ~<_ cdom 7300   Fincfn 7302   Basecbs 14166   0gc0g 14370    gsumg cgsu 14371  CMndccmn 16268
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 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-inf2 7839  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351
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 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-iin 4169  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-se 4675  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-of 6315  df-om 6472  df-1st 6572  df-2nd 6573  df-supp 6686  df-recs 6824  df-rdg 6858  df-1o 6912  df-oadd 6916  df-er 7093  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-fsupp 7613  df-oi 7716  df-card 8101  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-nn 10315  df-2 10372  df-n0 10572  df-z 10639  df-uz 10854  df-fz 11430  df-fzo 11541  df-seq 11799  df-hash 12096  df-ndx 14169  df-slot 14170  df-base 14171  df-sets 14172  df-ress 14173  df-plusg 14243  df-0g 14372  df-gsum 14373  df-mre 14516  df-mrc 14517  df-acs 14519  df-mnd 15407  df-submnd 15457  df-mulg 15539  df-cntz 15826  df-cmn 16270
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator