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Theorem gsummpt2co 27637
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 3433 . . 3  |-  F/_ x [_ ( 2nd `  p
)  /  x ]_ C
2 gsummpt2co.b . . 3  |-  B  =  ( Base `  W
)
3 gsummpt2co.z . . 3  |-  .0.  =  ( 0g `  W )
4 csbeq1a 3426 . . 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 3505 . . . 4  |-  B  C_  B
87a1i 11 . . 3  |-  ( ph  ->  B  C_  B )
9 gsummpt2co.1 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  C  e.  B )
10 elcnv 5165 . . . . 5  |-  ( p  e.  `' F  <->  E. y E. x ( p  = 
<. y ,  x >.  /\  x F y ) )
11 vex 3096 . . . . . . . . 9  |-  y  e. 
_V
12 vex 3096 . . . . . . . . 9  |-  x  e. 
_V
1311, 12op2ndd 6792 . . . . . . . 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 5489 . . . . . . . . 9  |-  dom  F  C_  A
1712, 11breldm 5193 . . . . . . . . 9  |-  ( x F y  ->  x  e.  dom  F )
1816, 17sseldi 3484 . . . . . . . 8  |-  ( x F y  ->  x  e.  A )
1918adantl 466 . . . . . . 7  |-  ( ( p  =  <. y ,  x >.  /\  x F y )  ->  x  e.  A )
2014, 19eqeltrd 2529 . . . . . 6  |-  ( ( p  =  <. y ,  x >.  /\  x F y )  -> 
( 2nd `  p
)  e.  A )
2120exlimivv 1708 . . . . 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 5611 . . . . 5  |-  Fun  F
25 funcnvcnv 5632 . . . . 5  |-  ( Fun 
F  ->  Fun  `' `' F )
2624, 25mp1i 12 . . . 4  |-  ( (
ph  /\  x  e.  A )  ->  Fun  `' `' F )
27 dfdm4 5181 . . . . . . 7  |-  dom  F  =  ran  `' F
28 gsummpt2co.2 . . . . . . . 8  |-  ( (
ph  /\  x  e.  A )  ->  D  e.  E )
2915, 28dmmptd 5697 . . . . . . 7  |-  ( ph  ->  dom  F  =  A )
3027, 29syl5eqr 2496 . . . . . 6  |-  ( ph  ->  ran  `' F  =  A )
3130eleq2d 2511 . . . . 5  |-  ( ph  ->  ( x  e.  ran  `' F  <->  x  e.  A
) )
3231biimpar 485 . . . 4  |-  ( (
ph  /\  x  e.  A )  ->  x  e.  ran  `' F )
33 relcnv 5360 . . . . 5  |-  Rel  `' F
34 fcnvgreu 27379 . . . . 5  |-  ( ( ( Rel  `' F  /\  Fun  `' `' F
)  /\  x  e.  ran  `' F )  ->  E! p  e.  `'  F x  =  ( 2nd `  p ) )
3533, 34mpanl1 680 . . . 4  |-  ( ( Fun  `' `' F  /\  x  e.  ran  `' F )  ->  E! p  e.  `'  F x  =  ( 2nd `  p ) )
3626, 32, 35syl2anc 661 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  E! p  e.  `'  F x  =  ( 2nd `  p ) )
371, 2, 3, 4, 5, 6, 8, 9, 23, 36gsummptf1o 16859 . 2  |-  ( ph  ->  ( W  gsumg  ( x  e.  A  |->  C ) )  =  ( W  gsumg  ( p  e.  `' F  |->  [_ ( 2nd `  p
)  /  x ]_ C ) ) )
3828ralrimiva 2855 . . . . . 6  |-  ( ph  ->  A. x  e.  A  D  e.  E )
3915rnmptss 6041 . . . . . 6  |-  ( A. x  e.  A  D  e.  E  ->  ran  F  C_  E )
40 dfcnv2 27382 . . . . . 6  |-  ( ran 
F  C_  E  ->  `' F  =  U_ y  e.  E  ( {
y }  X.  ( `' F " { y } ) ) )
4138, 39, 403syl 20 . . . . 5  |-  ( ph  ->  `' F  =  U_ y  e.  E  ( { y }  X.  ( `' F " { y } ) ) )
4241mpteq1d 4514 . . . 4  |-  ( ph  ->  ( p  e.  `' F  |->  [_ ( 2nd `  p
)  /  x ]_ C )  =  ( p  e.  U_ y  e.  E  ( {
y }  X.  ( `' F " { y } ) )  |->  [_ ( 2nd `  p )  /  x ]_ C
) )
43 nfcv 2603 . . . . 5  |-  F/_ y [_ ( 2nd `  p
)  /  x ]_ C
4413csbeq1d 3424 . . . . . 6  |-  ( p  =  <. y ,  x >.  ->  [_ ( 2nd `  p
)  /  x ]_ C  =  [_ x  /  x ]_ C )
45 csbid 3425 . . . . . 6  |-  [_ x  /  x ]_ C  =  C
4644, 45syl6eq 2498 . . . . 5  |-  ( p  =  <. y ,  x >.  ->  [_ ( 2nd `  p
)  /  x ]_ C  =  C )
4743, 1, 46mpt2mptxf 27383 . . . 4  |-  ( p  e.  U_ y  e.  E  ( { y }  X.  ( `' F " { y } ) )  |->  [_ ( 2nd `  p )  /  x ]_ C
)  =  ( y  e.  E ,  x  e.  ( `' F " { y } ) 
|->  C )
4842, 47syl6eq 2498 . . 3  |-  ( ph  ->  ( p  e.  `' F  |->  [_ ( 2nd `  p
)  /  x ]_ C )  =  ( y  e.  E ,  x  e.  ( `' F " { y } )  |->  C ) )
4948oveq2d 6293 . 2  |-  ( ph  ->  ( W  gsumg  ( p  e.  `' F  |->  [_ ( 2nd `  p
)  /  x ]_ C ) )  =  ( W  gsumg  ( y  e.  E ,  x  e.  ( `' F " { y } )  |->  C ) ) )
50 gsummpt2co.e . . . 4  |-  ( ph  ->  E  ~<_  om )
51 ctex 27396 . . . 4  |-  ( E  ~<_  om  ->  E  e.  _V )
5250, 51syl 16 . . 3  |-  ( ph  ->  E  e.  _V )
53 mptfi 7817 . . . . . . 7  |-  ( A  e.  Fin  ->  (
x  e.  A  |->  D )  e.  Fin )
5415, 53syl5eqel 2533 . . . . . 6  |-  ( A  e.  Fin  ->  F  e.  Fin )
55 cnvfi 7802 . . . . . 6  |-  ( F  e.  Fin  ->  `' F  e.  Fin )
566, 54, 553syl 20 . . . . 5  |-  ( ph  ->  `' F  e.  Fin )
57 imaexg 6718 . . . . 5  |-  ( `' F  e.  Fin  ->  ( `' F " { y } )  e.  _V )
5856, 57syl 16 . . . 4  |-  ( ph  ->  ( `' F " { y } )  e.  _V )
5958adantr 465 . . 3  |-  ( (
ph  /\  y  e.  E )  ->  ( `' F " { y } )  e.  _V )
60 simpll 753 . . . . 5  |-  ( ( ( ph  /\  y  e.  E )  /\  x  e.  ( `' F " { y } ) )  ->  ph )
61 cnvimass 5343 . . . . . . 7  |-  ( `' F " { y } )  C_  dom  F
6261, 16sstri 3495 . . . . . 6  |-  ( `' F " { y } )  C_  A
63 simpr 461 . . . . . 6  |-  ( ( ( ph  /\  y  e.  E )  /\  x  e.  ( `' F " { y } ) )  ->  x  e.  ( `' F " { y } ) )
6462, 63sseldi 3484 . . . . 5  |-  ( ( ( ph  /\  y  e.  E )  /\  x  e.  ( `' F " { y } ) )  ->  x  e.  A )
6560, 64, 9syl2anc 661 . . . 4  |-  ( ( ( ph  /\  y  e.  E )  /\  x  e.  ( `' F " { y } ) )  ->  C  e.  B )
6665anasss 647 . . 3  |-  ( (
ph  /\  ( y  e.  E  /\  x  e.  ( `' F " { y } ) ) )  ->  C  e.  B )
6711, 12elimasn 5348 . . . . . . . . 9  |-  ( x  e.  ( `' F " { y } )  <->  <. y ,  x >.  e.  `' F )
6867biimpi 194 . . . . . . . 8  |-  ( x  e.  ( `' F " { y } )  ->  <. y ,  x >.  e.  `' F )
6968adantl 466 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  E )  /\  x  e.  ( `' F " { y } ) )  ->  <. y ,  x >.  e.  `' F )
70 df-br 4434 . . . . . . 7  |-  ( y `' F x  <->  <. y ,  x >.  e.  `' F )
7169, 70sylibr 212 . . . . . 6  |-  ( ( ( ph  /\  y  e.  E )  /\  x  e.  ( `' F " { y } ) )  ->  y `' F x )
7271anasss 647 . . . . 5  |-  ( (
ph  /\  ( y  e.  E  /\  x  e.  ( `' F " { y } ) ) )  ->  y `' F x )
7372pm2.24d 143 . . . 4  |-  ( (
ph  /\  ( y  e.  E  /\  x  e.  ( `' F " { y } ) ) )  ->  ( -.  y `' F x  ->  C  =  .0.  ) )
7473impr 619 . . 3  |-  ( (
ph  /\  ( (
y  e.  E  /\  x  e.  ( `' F " { y } ) )  /\  -.  y `' F x ) )  ->  C  =  .0.  )
752, 3, 5, 52, 59, 66, 56, 74gsum2d2 16871 . 2  |-  ( ph  ->  ( W  gsumg  ( y  e.  E ,  x  e.  ( `' F " { y } )  |->  C ) )  =  ( W 
gsumg  ( y  e.  E  |->  ( W  gsumg  ( x  e.  ( `' F " { y } )  |->  C ) ) ) ) )
7637, 49, 753eqtrd 2486 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 1381   E.wex 1597    e. wcel 1802   A.wral 2791   E!wreu 2793   _Vcvv 3093   [_csb 3417    C_ wss 3458   {csn 4010   <.cop 4016   U_ciun 4311   class class class wbr 4433    |-> cmpt 4491    X. cxp 4983   `'ccnv 4984   dom cdm 4985   ran crn 4986   "cima 4988   Rel wrel 4990   Fun wfun 5568   ` cfv 5574  (class class class)co 6277    |-> cmpt2 6279   omcom 6681   2ndc2nd 6780    ~<_ cdom 7512   Fincfn 7514   Basecbs 14504   0gc0g 14709    gsumg cgsu 14710  CMndccmn 16667
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573  ax-inf2 8056  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-int 4268  df-iun 4313  df-iin 4314  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-se 4825  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-isom 5583  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-of 6521  df-om 6682  df-1st 6781  df-2nd 6782  df-supp 6900  df-recs 7040  df-rdg 7074  df-1o 7128  df-oadd 7132  df-er 7309  df-en 7515  df-dom 7516  df-sdom 7517  df-fin 7518  df-fsupp 7828  df-oi 7933  df-card 8318  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9807  df-neg 9808  df-nn 10538  df-2 10595  df-n0 10797  df-z 10866  df-uz 11086  df-fz 11677  df-fzo 11799  df-seq 12082  df-hash 12380  df-ndx 14507  df-slot 14508  df-base 14509  df-sets 14510  df-ress 14511  df-plusg 14582  df-0g 14711  df-gsum 14712  df-mre 14855  df-mrc 14856  df-acs 14858  df-mgm 15741  df-sgrp 15780  df-mnd 15790  df-submnd 15836  df-mulg 15929  df-cntz 16224  df-cmn 16669
This theorem is referenced by: (None)
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