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Theorem gsummpt2co 28543
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 3379 . . . 4  |-  F/_ x [_ ( 2nd `  p
)  /  x ]_ C
2 gsummpt2co.b . . . 4  |-  B  =  ( Base `  W
)
3 gsummpt2co.z . . . 4  |-  .0.  =  ( 0g `  W )
4 csbeq1a 3372 . . . 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 3451 . . . . 5  |-  B  C_  B
87a1i 11 . . . 4  |-  ( ph  ->  B  C_  B )
9 gsummpt2co.1 . . . 4  |-  ( (
ph  /\  x  e.  A )  ->  C  e.  B )
10 elcnv 5011 . . . . . 6  |-  ( p  e.  `' F  <->  E. z E. x ( p  = 
<. z ,  x >.  /\  x F z ) )
11 vex 3048 . . . . . . . . . 10  |-  z  e. 
_V
12 vex 3048 . . . . . . . . . 10  |-  x  e. 
_V
1311, 12op2ndd 6804 . . . . . . . . 9  |-  ( p  =  <. z ,  x >.  ->  ( 2nd `  p
)  =  x )
1413adantr 467 . . . . . . . 8  |-  ( ( p  =  <. z ,  x >.  /\  x F z )  -> 
( 2nd `  p
)  =  x )
15 gsummpt2co.3 . . . . . . . . . . 11  |-  F  =  ( x  e.  A  |->  D )
1615dmmptss 5331 . . . . . . . . . 10  |-  dom  F  C_  A
1712, 11breldm 5039 . . . . . . . . . 10  |-  ( x F z  ->  x  e.  dom  F )
1816, 17sseldi 3430 . . . . . . . . 9  |-  ( x F z  ->  x  e.  A )
1918adantl 468 . . . . . . . 8  |-  ( ( p  =  <. z ,  x >.  /\  x F z )  ->  x  e.  A )
2014, 19eqeltrd 2529 . . . . . . 7  |-  ( ( p  =  <. z ,  x >.  /\  x F z )  -> 
( 2nd `  p
)  e.  A )
2120exlimivv 1778 . . . . . 6  |-  ( E. z E. x ( p  =  <. z ,  x >.  /\  x F z )  -> 
( 2nd `  p
)  e.  A )
2210, 21sylbi 199 . . . . 5  |-  ( p  e.  `' F  -> 
( 2nd `  p
)  e.  A )
2322adantl 468 . . . 4  |-  ( (
ph  /\  p  e.  `' F )  ->  ( 2nd `  p )  e.  A )
2415funmpt2 5619 . . . . . . 7  |-  Fun  F
25 funcnvcnv 5641 . . . . . . 7  |-  ( Fun 
F  ->  Fun  `' `' F )
2624, 25ax-mp 5 . . . . . 6  |-  Fun  `' `' F
2726a1i 11 . . . . 5  |-  ( (
ph  /\  x  e.  A )  ->  Fun  `' `' F )
28 dfdm4 5027 . . . . . . . 8  |-  dom  F  =  ran  `' F
2915dmeqi 5036 . . . . . . . . 9  |-  dom  F  =  dom  ( x  e.  A  |->  D )
30 gsummpt2co.2 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  A )  ->  D  e.  E )
3130ralrimiva 2802 . . . . . . . . . 10  |-  ( ph  ->  A. x  e.  A  D  e.  E )
32 dmmptg 5332 . . . . . . . . . 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 2497 . . . . . . . 8  |-  ( ph  ->  dom  F  =  A )
3528, 34syl5eqr 2499 . . . . . . 7  |-  ( ph  ->  ran  `' F  =  A )
3635eleq2d 2514 . . . . . 6  |-  ( ph  ->  ( x  e.  ran  `' F  <->  x  e.  A
) )
3736biimpar 488 . . . . 5  |-  ( (
ph  /\  x  e.  A )  ->  x  e.  ran  `' F )
38 relcnv 5207 . . . . . 6  |-  Rel  `' F
39 fcnvgreu 28275 . . . . . 6  |-  ( ( ( Rel  `' F  /\  Fun  `' `' F
)  /\  x  e.  ran  `' F )  ->  E! p  e.  `'  F x  =  ( 2nd `  p ) )
4038, 39mpanl1 686 . . . . 5  |-  ( ( Fun  `' `' F  /\  x  e.  ran  `' F )  ->  E! p  e.  `'  F x  =  ( 2nd `  p ) )
4127, 37, 40syl2anc 667 . . . 4  |-  ( (
ph  /\  x  e.  A )  ->  E! p  e.  `'  F x  =  ( 2nd `  p ) )
421, 2, 3, 4, 5, 6, 8, 9, 23, 41gsummptf1o 17595 . . 3  |-  ( ph  ->  ( W  gsumg  ( x  e.  A  |->  C ) )  =  ( W  gsumg  ( p  e.  `' F  |->  [_ ( 2nd `  p
)  /  x ]_ C ) ) )
4315rnmptss 6052 . . . . . . . 8  |-  ( A. x  e.  A  D  e.  E  ->  ran  F  C_  E )
4431, 43syl 17 . . . . . . 7  |-  ( ph  ->  ran  F  C_  E
)
45 dfcnv2 28279 . . . . . . 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 4484 . . . . 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 2592 . . . . . 6  |-  F/_ z [_ ( 2nd `  p
)  /  x ]_ C
49 csbeq1 3366 . . . . . . . 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 3371 . . . . . . 7  |-  [_ x  /  x ]_ C  =  C
5250, 51syl6eq 2501 . . . . . 6  |-  ( p  =  <. z ,  x >.  ->  [_ ( 2nd `  p
)  /  x ]_ C  =  C )
5348, 1, 52mpt2mptxf 28280 . . . . 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 2501 . . . 4  |-  ( ph  ->  ( p  e.  `' F  |->  [_ ( 2nd `  p
)  /  x ]_ C )  =  ( z  e.  E ,  x  e.  ( `' F " { z } )  |->  C ) )
5554oveq2d 6306 . . 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 7873 . . . . . . . 8  |-  ( A  e.  Fin  ->  (
x  e.  A  |->  D )  e.  Fin )
5815, 57syl5eqel 2533 . . . . . . 7  |-  ( A  e.  Fin  ->  F  e.  Fin )
59 cnvfi 7856 . . . . . . 7  |-  ( F  e.  Fin  ->  `' F  e.  Fin )
606, 58, 593syl 18 . . . . . 6  |-  ( ph  ->  `' F  e.  Fin )
61 imaexg 6730 . . . . . 6  |-  ( `' F  e.  Fin  ->  ( `' F " { z } )  e.  _V )
6260, 61syl 17 . . . . 5  |-  ( ph  ->  ( `' F " { z } )  e.  _V )
6362adantr 467 . . . 4  |-  ( (
ph  /\  z  e.  E )  ->  ( `' F " { z } )  e.  _V )
64 simpll 760 . . . . . 6  |-  ( ( ( ph  /\  z  e.  E )  /\  x  e.  ( `' F " { z } ) )  ->  ph )
65 imassrn 5179 . . . . . . . . 9  |-  ( `' F " { z } )  C_  ran  `' F
6665, 28sseqtr4i 3465 . . . . . . . 8  |-  ( `' F " { z } )  C_  dom  F
6766, 16sstri 3441 . . . . . . 7  |-  ( `' F " { z } )  C_  A
6811, 12elimasn 5193 . . . . . . . . . 10  |-  ( x  e.  ( `' F " { z } )  <->  <. z ,  x >.  e.  `' F )
6968biimpi 198 . . . . . . . . 9  |-  ( x  e.  ( `' F " { z } )  ->  <. z ,  x >.  e.  `' F )
7069adantl 468 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  E )  /\  x  e.  ( `' F " { z } ) )  ->  <. z ,  x >.  e.  `' F )
7170, 68sylibr 216 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  E )  /\  x  e.  ( `' F " { z } ) )  ->  x  e.  ( `' F " { z } ) )
7267, 71sseldi 3430 . . . . . 6  |-  ( ( ( ph  /\  z  e.  E )  /\  x  e.  ( `' F " { z } ) )  ->  x  e.  A )
7364, 72, 9syl2anc 667 . . . . 5  |-  ( ( ( ph  /\  z  e.  E )  /\  x  e.  ( `' F " { z } ) )  ->  C  e.  B )
7473anasss 653 . . . 4  |-  ( (
ph  /\  ( z  e.  E  /\  x  e.  ( `' F " { z } ) ) )  ->  C  e.  B )
75 df-br 4403 . . . . . . . . 9  |-  ( z `' F x  <->  <. z ,  x >.  e.  `' F )
7670, 75sylibr 216 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  E )  /\  x  e.  ( `' F " { z } ) )  ->  z `' F x )
7776anasss 653 . . . . . . 7  |-  ( (
ph  /\  ( z  e.  E  /\  x  e.  ( `' F " { z } ) ) )  ->  z `' F x )
7877pm2.24d 138 . . . . . 6  |-  ( (
ph  /\  ( z  e.  E  /\  x  e.  ( `' F " { z } ) ) )  ->  ( -.  z `' F x  ->  C  =  .0.  ) )
7978imp 431 . . . . 5  |-  ( ( ( ph  /\  (
z  e.  E  /\  x  e.  ( `' F " { z } ) ) )  /\  -.  z `' F x )  ->  C  =  .0.  )
8079anasss 653 . . . 4  |-  ( (
ph  /\  ( (
z  e.  E  /\  x  e.  ( `' F " { z } ) )  /\  -.  z `' F x ) )  ->  C  =  .0.  )
812, 3, 5, 56, 63, 74, 60, 80gsum2d2 17606 . . 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 2489 . 2  |-  ( ph  ->  ( W  gsumg  ( x  e.  A  |->  C ) )  =  ( W  gsumg  ( z  e.  E  |->  ( W  gsumg  ( x  e.  ( `' F " { z } )  |->  C ) ) ) ) )
83 nfcv 2592 . . . 4  |-  F/_ z
( W  gsumg  ( x  e.  ( `' F " { y } )  |->  C ) )
84 nfcv 2592 . . . 4  |-  F/_ y
( W  gsumg  ( x  e.  ( `' F " { z } )  |->  C ) )
85 sneq 3978 . . . . . . 7  |-  ( y  =  z  ->  { y }  =  { z } )
8685imaeq2d 5168 . . . . . 6  |-  ( y  =  z  ->  ( `' F " { y } )  =  ( `' F " { z } ) )
8786mpteq1d 4484 . . . . 5  |-  ( y  =  z  ->  (
x  e.  ( `' F " { y } )  |->  C )  =  ( x  e.  ( `' F " { z } ) 
|->  C ) )
8887oveq2d 6306 . . . 4  |-  ( y  =  z  ->  ( W  gsumg  ( x  e.  ( `' F " { y } )  |->  C ) )  =  ( W 
gsumg  ( x  e.  ( `' F " { z } )  |->  C ) ) )
8983, 84, 88cbvmpt 4494 . . 3  |-  ( y  e.  E  |->  ( W 
gsumg  ( x  e.  ( `' F " { y } )  |->  C ) ) )  =  ( z  e.  E  |->  ( W  gsumg  ( x  e.  ( `' F " { z } )  |->  C ) ) )
9089oveq2i 6301 . 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 2503 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 371    = wceq 1444   E.wex 1663    e. wcel 1887   A.wral 2737   E!wreu 2739   _Vcvv 3045   [_csb 3363    C_ wss 3404   {csn 3968   <.cop 3974   U_ciun 4278   class class class wbr 4402    |-> cmpt 4461    X. cxp 4832   `'ccnv 4833   dom cdm 4834   ran crn 4835   "cima 4837   Rel wrel 4839   Fun wfun 5576   ` cfv 5582  (class class class)co 6290    |-> cmpt2 6292   2ndc2nd 6792   Fincfn 7569   Basecbs 15121   0gc0g 15338    gsumg cgsu 15339  CMndccmn 17430
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-inf2 8146  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-iin 4281  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-se 4794  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-isom 5591  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-of 6531  df-om 6693  df-1st 6793  df-2nd 6794  df-supp 6915  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-oadd 7186  df-er 7363  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-fsupp 7884  df-oi 8025  df-card 8373  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-nn 10610  df-2 10668  df-n0 10870  df-z 10938  df-uz 11160  df-fz 11785  df-fzo 11916  df-seq 12214  df-hash 12516  df-ndx 15124  df-slot 15125  df-base 15126  df-sets 15127  df-ress 15128  df-plusg 15203  df-0g 15340  df-gsum 15341  df-mre 15492  df-mrc 15493  df-acs 15495  df-mgm 16488  df-sgrp 16527  df-mnd 16537  df-submnd 16583  df-mulg 16676  df-cntz 16971  df-cmn 17432
This theorem is referenced by:  gsummpt2d  28544
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