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Theorem gsummpt2co 27284
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 3444 . . 3  |-  F/_ x [_ ( 2nd `  p
)  /  x ]_ C
2 gsummpt2co.b . . 3  |-  B  =  ( Base `  W
)
3 gsummpt2co.z . . 3  |-  .0.  =  ( 0g `  W )
4 csbeq1a 3437 . . 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 3516 . . . 4  |-  B  C_  B
87a1i 11 . . 3  |-  ( ph  ->  B  C_  B )
9 gsummpt2co.1 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  C  e.  B )
10 elcnv 5170 . . . . 5  |-  ( p  e.  `' F  <->  E. y E. x ( p  = 
<. y ,  x >.  /\  x F y ) )
11 vex 3109 . . . . . . . . 9  |-  y  e. 
_V
12 vex 3109 . . . . . . . . 9  |-  x  e. 
_V
1311, 12op2ndd 6785 . . . . . . . 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 5494 . . . . . . . . 9  |-  dom  F  C_  A
1712, 11breldm 5198 . . . . . . . . 9  |-  ( x F y  ->  x  e.  dom  F )
1816, 17sseldi 3495 . . . . . . . 8  |-  ( x F y  ->  x  e.  A )
1918adantl 466 . . . . . . 7  |-  ( ( p  =  <. y ,  x >.  /\  x F y )  ->  x  e.  A )
2014, 19eqeltrd 2548 . . . . . 6  |-  ( ( p  =  <. y ,  x >.  /\  x F y )  -> 
( 2nd `  p
)  e.  A )
2120exlimivv 1694 . . . . 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 5616 . . . . . 6  |-  Fun  F
25 funcnvcnv 5637 . . . . . 6  |-  ( Fun 
F  ->  Fun  `' `' F )
2624, 25ax-mp 5 . . . . 5  |-  Fun  `' `' F
2726a1i 11 . . . 4  |-  ( (
ph  /\  x  e.  A )  ->  Fun  `' `' F )
28 dfdm4 5186 . . . . . . 7  |-  dom  F  =  ran  `' F
2915dmeqi 5195 . . . . . . . 8  |-  dom  F  =  dom  ( x  e.  A  |->  D )
30 gsummpt2co.2 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  A )  ->  D  e.  E )
3130ralrimiva 2871 . . . . . . . . 9  |-  ( ph  ->  A. x  e.  A  D  e.  E )
32 dmmptg 5495 . . . . . . . . 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 2513 . . . . . . 7  |-  ( ph  ->  dom  F  =  A )
3528, 34syl5eqr 2515 . . . . . 6  |-  ( ph  ->  ran  `' F  =  A )
3635eleq2d 2530 . . . . 5  |-  ( ph  ->  ( x  e.  ran  `' F  <->  x  e.  A
) )
3736biimpar 485 . . . 4  |-  ( (
ph  /\  x  e.  A )  ->  x  e.  ran  `' F )
38 relcnv 5365 . . . . 5  |-  Rel  `' F
39 fcnvgreu 27036 . . . . 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 16774 . 2  |-  ( ph  ->  ( W  gsumg  ( x  e.  A  |->  C ) )  =  ( W  gsumg  ( p  e.  `' F  |->  [_ ( 2nd `  p
)  /  x ]_ C ) ) )
4315rnmptss 6041 . . . . . . 7  |-  ( A. x  e.  A  D  e.  E  ->  ran  F  C_  E )
4431, 43syl 16 . . . . . 6  |-  ( ph  ->  ran  F  C_  E
)
45 dfcnv2 27039 . . . . . 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 4521 . . . 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 2622 . . . . 5  |-  F/_ y [_ ( 2nd `  p
)  /  x ]_ C
49 csbeq1 3431 . . . . . . 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 3436 . . . . . 6  |-  [_ x  /  x ]_ C  =  C
5250, 51syl6eq 2517 . . . . 5  |-  ( p  =  <. y ,  x >.  ->  [_ ( 2nd `  p
)  /  x ]_ C  =  C )
5348, 1, 52mpt2mptxf 27040 . . . 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 2517 . . 3  |-  ( ph  ->  ( p  e.  `' F  |->  [_ ( 2nd `  p
)  /  x ]_ C )  =  ( y  e.  E ,  x  e.  ( `' F " { y } )  |->  C ) )
5554oveq2d 6291 . 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 27053 . . . 4  |-  ( E  ~<_  om  ->  E  e.  _V )
5856, 57syl 16 . . 3  |-  ( ph  ->  E  e.  _V )
59 mptfi 7808 . . . . . . 7  |-  ( A  e.  Fin  ->  (
x  e.  A  |->  D )  e.  Fin )
6015, 59syl5eqel 2552 . . . . . 6  |-  ( A  e.  Fin  ->  F  e.  Fin )
61 cnvfi 7793 . . . . . 6  |-  ( F  e.  Fin  ->  `' F  e.  Fin )
626, 60, 613syl 20 . . . . 5  |-  ( ph  ->  `' F  e.  Fin )
63 imaexg 6711 . . . . 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 5339 . . . . . . . 8  |-  ( `' F " { y } )  C_  ran  `' F
6867, 28sseqtr4i 3530 . . . . . . 7  |-  ( `' F " { y } )  C_  dom  F
6968, 16sstri 3506 . . . . . 6  |-  ( `' F " { y } )  C_  A
7011, 12elimasn 5353 . . . . . . . . 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 3495 . . . . 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 4441 . . . . . . . 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 16786 . 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 2505 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 1374   E.wex 1591    e. wcel 1762   A.wral 2807   E!wreu 2809   _Vcvv 3106   [_csb 3428    C_ wss 3469   {csn 4020   <.cop 4026   U_ciun 4318   class class class wbr 4440    |-> cmpt 4498    X. cxp 4990   `'ccnv 4991   dom cdm 4992   ran crn 4993   "cima 4995   Rel wrel 4997   Fun wfun 5573   ` cfv 5579  (class class class)co 6275    |-> cmpt2 6277   omcom 6671   2ndc2nd 6773    ~<_ cdom 7504   Fincfn 7506   Basecbs 14479   0gc0g 14684    gsumg cgsu 14685  CMndccmn 16587
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-inf2 8047  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 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-iin 4321  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-se 4832  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-isom 5588  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-of 6515  df-om 6672  df-1st 6774  df-2nd 6775  df-supp 6892  df-recs 7032  df-rdg 7066  df-1o 7120  df-oadd 7124  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-fsupp 7819  df-oi 7924  df-card 8309  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-nn 10526  df-2 10583  df-n0 10785  df-z 10854  df-uz 11072  df-fz 11662  df-fzo 11782  df-seq 12064  df-hash 12361  df-ndx 14482  df-slot 14483  df-base 14484  df-sets 14485  df-ress 14486  df-plusg 14557  df-0g 14686  df-gsum 14687  df-mre 14830  df-mrc 14831  df-acs 14833  df-mnd 15721  df-submnd 15771  df-mulg 15854  df-cntz 16143  df-cmn 16589
This theorem is referenced by: (None)
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