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Theorem gsummpt2co 28617
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  e.  V )
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,  .0. , y    x, A, y    x, B, y    y, C    x, E, y    x, F, y   
y, V    x, W, y    ph, x
Allowed substitution hints:    ph( y)    C( x)    D( x, y)    V( x)

Proof of Theorem gsummpt2co
Dummy variables  z  p are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfcsb1v 3365 . . . 4  |-  F/_ x [_ ( 2nd `  p
)  /  x ]_ C
2 gsummpt2co.b . . . 4  |-  B  =  ( Base `  W
)
3 gsummpt2co.z . . . 4  |-  .0.  =  ( 0g `  W )
4 csbeq1a 3358 . . . 4  |-  ( x  =  ( 2nd `  p
)  ->  C  =  [_ ( 2nd `  p
)  /  x ]_ C )
5 gsummpt2co.w . . . 4  |-  ( ph  ->  W  e. CMnd )
6 gsummpt2co.a . . . 4  |-  ( ph  ->  A  e.  Fin )
7 ssid 3437 . . . . 5  |-  B  C_  B
87a1i 11 . . . 4  |-  ( ph  ->  B  C_  B )
9 gsummpt2co.1 . . . 4  |-  ( (
ph  /\  x  e.  A )  ->  C  e.  B )
10 elcnv 5016 . . . . . 6  |-  ( p  e.  `' F  <->  E. z E. x ( p  = 
<. z ,  x >.  /\  x F z ) )
11 vex 3034 . . . . . . . . . 10  |-  z  e. 
_V
12 vex 3034 . . . . . . . . . 10  |-  x  e. 
_V
1311, 12op2ndd 6823 . . . . . . . . 9  |-  ( p  =  <. z ,  x >.  ->  ( 2nd `  p
)  =  x )
1413adantr 472 . . . . . . . 8  |-  ( ( p  =  <. z ,  x >.  /\  x F z )  -> 
( 2nd `  p
)  =  x )
15 gsummpt2co.3 . . . . . . . . . . 11  |-  F  =  ( x  e.  A  |->  D )
1615dmmptss 5338 . . . . . . . . . 10  |-  dom  F  C_  A
1712, 11breldm 5045 . . . . . . . . . 10  |-  ( x F z  ->  x  e.  dom  F )
1816, 17sseldi 3416 . . . . . . . . 9  |-  ( x F z  ->  x  e.  A )
1918adantl 473 . . . . . . . 8  |-  ( ( p  =  <. z ,  x >.  /\  x F z )  ->  x  e.  A )
2014, 19eqeltrd 2549 . . . . . . 7  |-  ( ( p  =  <. z ,  x >.  /\  x F z )  -> 
( 2nd `  p
)  e.  A )
2120exlimivv 1786 . . . . . 6  |-  ( E. z E. x ( p  =  <. z ,  x >.  /\  x F z )  -> 
( 2nd `  p
)  e.  A )
2210, 21sylbi 200 . . . . 5  |-  ( p  e.  `' F  -> 
( 2nd `  p
)  e.  A )
2322adantl 473 . . . 4  |-  ( (
ph  /\  p  e.  `' F )  ->  ( 2nd `  p )  e.  A )
2415funmpt2 5626 . . . . . . 7  |-  Fun  F
25 funcnvcnv 5651 . . . . . . 7  |-  ( Fun 
F  ->  Fun  `' `' F )
2624, 25ax-mp 5 . . . . . 6  |-  Fun  `' `' F
2726a1i 11 . . . . 5  |-  ( (
ph  /\  x  e.  A )  ->  Fun  `' `' F )
28 dfdm4 5032 . . . . . . . 8  |-  dom  F  =  ran  `' F
2915dmeqi 5041 . . . . . . . . 9  |-  dom  F  =  dom  ( x  e.  A  |->  D )
30 gsummpt2co.2 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  A )  ->  D  e.  E )
3130ralrimiva 2809 . . . . . . . . . 10  |-  ( ph  ->  A. x  e.  A  D  e.  E )
32 dmmptg 5339 . . . . . . . . . 10  |-  ( A. x  e.  A  D  e.  E  ->  dom  (
x  e.  A  |->  D )  =  A )
3331, 32syl 17 . . . . . . . . 9  |-  ( ph  ->  dom  ( x  e.  A  |->  D )  =  A )
3429, 33syl5eq 2517 . . . . . . . 8  |-  ( ph  ->  dom  F  =  A )
3528, 34syl5eqr 2519 . . . . . . 7  |-  ( ph  ->  ran  `' F  =  A )
3635eleq2d 2534 . . . . . 6  |-  ( ph  ->  ( x  e.  ran  `' F  <->  x  e.  A
) )
3736biimpar 493 . . . . 5  |-  ( (
ph  /\  x  e.  A )  ->  x  e.  ran  `' F )
38 relcnv 5213 . . . . . 6  |-  Rel  `' F
39 fcnvgreu 28350 . . . . . 6  |-  ( ( ( Rel  `' F  /\  Fun  `' `' F
)  /\  x  e.  ran  `' F )  ->  E! p  e.  `'  F x  =  ( 2nd `  p ) )
4038, 39mpanl1 694 . . . . 5  |-  ( ( Fun  `' `' F  /\  x  e.  ran  `' F )  ->  E! p  e.  `'  F x  =  ( 2nd `  p ) )
4127, 37, 40syl2anc 673 . . . 4  |-  ( (
ph  /\  x  e.  A )  ->  E! p  e.  `'  F x  =  ( 2nd `  p ) )
421, 2, 3, 4, 5, 6, 8, 9, 23, 41gsummptf1o 17673 . . 3  |-  ( ph  ->  ( W  gsumg  ( x  e.  A  |->  C ) )  =  ( W  gsumg  ( p  e.  `' F  |->  [_ ( 2nd `  p
)  /  x ]_ C ) ) )
4315rnmptss 6068 . . . . . . . 8  |-  ( A. x  e.  A  D  e.  E  ->  ran  F  C_  E )
4431, 43syl 17 . . . . . . 7  |-  ( ph  ->  ran  F  C_  E
)
45 dfcnv2 28354 . . . . . . 7  |-  ( ran 
F  C_  E  ->  `' F  =  U_ z  e.  E  ( {
z }  X.  ( `' F " { z } ) ) )
4644, 45syl 17 . . . . . 6  |-  ( ph  ->  `' F  =  U_ z  e.  E  ( { z }  X.  ( `' F " { z } ) ) )
4746mpteq1d 4477 . . . . 5  |-  ( ph  ->  ( p  e.  `' F  |->  [_ ( 2nd `  p
)  /  x ]_ C )  =  ( p  e.  U_ z  e.  E  ( {
z }  X.  ( `' F " { z } ) )  |->  [_ ( 2nd `  p )  /  x ]_ C
) )
48 nfcv 2612 . . . . . 6  |-  F/_ z [_ ( 2nd `  p
)  /  x ]_ C
49 csbeq1 3352 . . . . . . . 8  |-  ( ( 2nd `  p )  =  x  ->  [_ ( 2nd `  p )  /  x ]_ C  =  [_ x  /  x ]_ C
)
5013, 49syl 17 . . . . . . 7  |-  ( p  =  <. z ,  x >.  ->  [_ ( 2nd `  p
)  /  x ]_ C  =  [_ x  /  x ]_ C )
51 csbid 3357 . . . . . . 7  |-  [_ x  /  x ]_ C  =  C
5250, 51syl6eq 2521 . . . . . 6  |-  ( p  =  <. z ,  x >.  ->  [_ ( 2nd `  p
)  /  x ]_ C  =  C )
5348, 1, 52mpt2mptxf 28355 . . . . 5  |-  ( p  e.  U_ z  e.  E  ( { z }  X.  ( `' F " { z } ) )  |->  [_ ( 2nd `  p )  /  x ]_ C
)  =  ( z  e.  E ,  x  e.  ( `' F " { z } ) 
|->  C )
5447, 53syl6eq 2521 . . . 4  |-  ( ph  ->  ( p  e.  `' F  |->  [_ ( 2nd `  p
)  /  x ]_ C )  =  ( z  e.  E ,  x  e.  ( `' F " { z } )  |->  C ) )
5554oveq2d 6324 . . 3  |-  ( ph  ->  ( W  gsumg  ( p  e.  `' F  |->  [_ ( 2nd `  p
)  /  x ]_ C ) )  =  ( W  gsumg  ( z  e.  E ,  x  e.  ( `' F " { z } )  |->  C ) ) )
56 gsummpt2co.e . . . 4  |-  ( ph  ->  E  e.  V )
57 mptfi 7891 . . . . . . . 8  |-  ( A  e.  Fin  ->  (
x  e.  A  |->  D )  e.  Fin )
5815, 57syl5eqel 2553 . . . . . . 7  |-  ( A  e.  Fin  ->  F  e.  Fin )
59 cnvfi 7874 . . . . . . 7  |-  ( F  e.  Fin  ->  `' F  e.  Fin )
606, 58, 593syl 18 . . . . . 6  |-  ( ph  ->  `' F  e.  Fin )
61 imaexg 6749 . . . . . 6  |-  ( `' F  e.  Fin  ->  ( `' F " { z } )  e.  _V )
6260, 61syl 17 . . . . 5  |-  ( ph  ->  ( `' F " { z } )  e.  _V )
6362adantr 472 . . . 4  |-  ( (
ph  /\  z  e.  E )  ->  ( `' F " { z } )  e.  _V )
64 simpll 768 . . . . . 6  |-  ( ( ( ph  /\  z  e.  E )  /\  x  e.  ( `' F " { z } ) )  ->  ph )
65 imassrn 5185 . . . . . . . . 9  |-  ( `' F " { z } )  C_  ran  `' F
6665, 28sseqtr4i 3451 . . . . . . . 8  |-  ( `' F " { z } )  C_  dom  F
6766, 16sstri 3427 . . . . . . 7  |-  ( `' F " { z } )  C_  A
6811, 12elimasn 5199 . . . . . . . . . 10  |-  ( x  e.  ( `' F " { z } )  <->  <. z ,  x >.  e.  `' F )
6968biimpi 199 . . . . . . . . 9  |-  ( x  e.  ( `' F " { z } )  ->  <. z ,  x >.  e.  `' F )
7069adantl 473 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  E )  /\  x  e.  ( `' F " { z } ) )  ->  <. z ,  x >.  e.  `' F )
7170, 68sylibr 217 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  E )  /\  x  e.  ( `' F " { z } ) )  ->  x  e.  ( `' F " { z } ) )
7267, 71sseldi 3416 . . . . . 6  |-  ( ( ( ph  /\  z  e.  E )  /\  x  e.  ( `' F " { z } ) )  ->  x  e.  A )
7364, 72, 9syl2anc 673 . . . . 5  |-  ( ( ( ph  /\  z  e.  E )  /\  x  e.  ( `' F " { z } ) )  ->  C  e.  B )
7473anasss 659 . . . 4  |-  ( (
ph  /\  ( z  e.  E  /\  x  e.  ( `' F " { z } ) ) )  ->  C  e.  B )
75 df-br 4396 . . . . . . . . 9  |-  ( z `' F x  <->  <. z ,  x >.  e.  `' F )
7670, 75sylibr 217 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  E )  /\  x  e.  ( `' F " { z } ) )  ->  z `' F x )
7776anasss 659 . . . . . . 7  |-  ( (
ph  /\  ( z  e.  E  /\  x  e.  ( `' F " { z } ) ) )  ->  z `' F x )
7877pm2.24d 139 . . . . . 6  |-  ( (
ph  /\  ( z  e.  E  /\  x  e.  ( `' F " { z } ) ) )  ->  ( -.  z `' F x  ->  C  =  .0.  ) )
7978imp 436 . . . . 5  |-  ( ( ( ph  /\  (
z  e.  E  /\  x  e.  ( `' F " { z } ) ) )  /\  -.  z `' F x )  ->  C  =  .0.  )
8079anasss 659 . . . 4  |-  ( (
ph  /\  ( (
z  e.  E  /\  x  e.  ( `' F " { z } ) )  /\  -.  z `' F x ) )  ->  C  =  .0.  )
812, 3, 5, 56, 63, 74, 60, 80gsum2d2 17684 . . 3  |-  ( ph  ->  ( W  gsumg  ( z  e.  E ,  x  e.  ( `' F " { z } )  |->  C ) )  =  ( W 
gsumg  ( z  e.  E  |->  ( W  gsumg  ( x  e.  ( `' F " { z } )  |->  C ) ) ) ) )
8242, 55, 813eqtrd 2509 . 2  |-  ( ph  ->  ( W  gsumg  ( x  e.  A  |->  C ) )  =  ( W  gsumg  ( z  e.  E  |->  ( W  gsumg  ( x  e.  ( `' F " { z } )  |->  C ) ) ) ) )
83 nfcv 2612 . . . 4  |-  F/_ z
( W  gsumg  ( x  e.  ( `' F " { y } )  |->  C ) )
84 nfcv 2612 . . . 4  |-  F/_ y
( W  gsumg  ( x  e.  ( `' F " { z } )  |->  C ) )
85 sneq 3969 . . . . . . 7  |-  ( y  =  z  ->  { y }  =  { z } )
8685imaeq2d 5174 . . . . . 6  |-  ( y  =  z  ->  ( `' F " { y } )  =  ( `' F " { z } ) )
8786mpteq1d 4477 . . . . 5  |-  ( y  =  z  ->  (
x  e.  ( `' F " { y } )  |->  C )  =  ( x  e.  ( `' F " { z } ) 
|->  C ) )
8887oveq2d 6324 . . . 4  |-  ( y  =  z  ->  ( W  gsumg  ( x  e.  ( `' F " { y } )  |->  C ) )  =  ( W 
gsumg  ( x  e.  ( `' F " { z } )  |->  C ) ) )
8983, 84, 88cbvmpt 4487 . . 3  |-  ( y  e.  E  |->  ( W 
gsumg  ( x  e.  ( `' F " { y } )  |->  C ) ) )  =  ( z  e.  E  |->  ( W  gsumg  ( x  e.  ( `' F " { z } )  |->  C ) ) )
9089oveq2i 6319 . 2  |-  ( W 
gsumg  ( y  e.  E  |->  ( W  gsumg  ( x  e.  ( `' F " { y } )  |->  C ) ) ) )  =  ( W  gsumg  ( z  e.  E  |->  ( W  gsumg  ( x  e.  ( `' F " { z } )  |->  C ) ) ) )
9182, 90syl6eqr 2523 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 376    = wceq 1452   E.wex 1671    e. wcel 1904   A.wral 2756   E!wreu 2758   _Vcvv 3031   [_csb 3349    C_ wss 3390   {csn 3959   <.cop 3965   U_ciun 4269   class class class wbr 4395    |-> cmpt 4454    X. cxp 4837   `'ccnv 4838   dom cdm 4839   ran crn 4840   "cima 4842   Rel wrel 4844   Fun wfun 5583   ` cfv 5589  (class class class)co 6308    |-> cmpt2 6310   2ndc2nd 6811   Fincfn 7587   Basecbs 15199   0gc0g 15416    gsumg cgsu 15417  CMndccmn 17508
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-iin 4272  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-of 6550  df-om 6712  df-1st 6812  df-2nd 6813  df-supp 6934  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-fsupp 7902  df-oi 8043  df-card 8391  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-2 10690  df-n0 10894  df-z 10962  df-uz 11183  df-fz 11811  df-fzo 11943  df-seq 12252  df-hash 12554  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-ress 15206  df-plusg 15281  df-0g 15418  df-gsum 15419  df-mre 15570  df-mrc 15571  df-acs 15573  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-submnd 16661  df-mulg 16754  df-cntz 17049  df-cmn 17510
This theorem is referenced by:  gsummpt2d  28618
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