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Theorem gsummpt2co 26387
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 3405 . . 3  |-  F/_ x [_ ( 2nd `  p
)  /  x ]_ C
2 gsummpt2co.b . . 3  |-  B  =  ( Base `  W
)
3 gsummpt2co.z . . 3  |-  .0.  =  ( 0g `  W )
4 csbeq1a 3398 . . 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 3476 . . . 4  |-  B  C_  B
87a1i 11 . . 3  |-  ( ph  ->  B  C_  B )
9 gsummpt2co.1 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  C  e.  B )
10 elcnv 5117 . . . . 5  |-  ( p  e.  `' F  <->  E. y E. x ( p  = 
<. y ,  x >.  /\  x F y ) )
11 vex 3074 . . . . . . . . 9  |-  y  e. 
_V
12 vex 3074 . . . . . . . . 9  |-  x  e. 
_V
1311, 12op2ndd 6691 . . . . . . . 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 5435 . . . . . . . . 9  |-  dom  F  C_  A
1712, 11breldm 5145 . . . . . . . . 9  |-  ( x F y  ->  x  e.  dom  F )
1816, 17sseldi 3455 . . . . . . . 8  |-  ( x F y  ->  x  e.  A )
1918adantl 466 . . . . . . 7  |-  ( ( p  =  <. y ,  x >.  /\  x F y )  ->  x  e.  A )
2014, 19eqeltrd 2539 . . . . . 6  |-  ( ( p  =  <. y ,  x >.  /\  x F y )  -> 
( 2nd `  p
)  e.  A )
2120exlimivv 1690 . . . . 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 5556 . . . . . 6  |-  Fun  F
25 funcnvcnv 5577 . . . . . 6  |-  ( Fun 
F  ->  Fun  `' `' F )
2624, 25ax-mp 5 . . . . 5  |-  Fun  `' `' F
2726a1i 11 . . . 4  |-  ( (
ph  /\  x  e.  A )  ->  Fun  `' `' F )
28 dfdm4 5133 . . . . . . 7  |-  dom  F  =  ran  `' F
2915dmeqi 5142 . . . . . . . 8  |-  dom  F  =  dom  ( x  e.  A  |->  D )
30 gsummpt2co.2 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  A )  ->  D  e.  E )
3130ralrimiva 2825 . . . . . . . . 9  |-  ( ph  ->  A. x  e.  A  D  e.  E )
32 dmmptg 5436 . . . . . . . . 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 2504 . . . . . . 7  |-  ( ph  ->  dom  F  =  A )
3528, 34syl5eqr 2506 . . . . . 6  |-  ( ph  ->  ran  `' F  =  A )
3635eleq2d 2521 . . . . 5  |-  ( ph  ->  ( x  e.  ran  `' F  <->  x  e.  A
) )
3736biimpar 485 . . . 4  |-  ( (
ph  /\  x  e.  A )  ->  x  e.  ran  `' F )
38 relcnv 5307 . . . . 5  |-  Rel  `' F
39 fcnvgreu 26135 . . . . 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 26385 . 2  |-  ( ph  ->  ( W  gsumg  ( x  e.  A  |->  C ) )  =  ( W  gsumg  ( p  e.  `' F  |->  [_ ( 2nd `  p
)  /  x ]_ C ) ) )
4315rnmptss 5974 . . . . . . 7  |-  ( A. x  e.  A  D  e.  E  ->  ran  F  C_  E )
4431, 43syl 16 . . . . . 6  |-  ( ph  ->  ran  F  C_  E
)
45 dfcnv2 26138 . . . . . 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 4474 . . . 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 2613 . . . . 5  |-  F/_ y [_ ( 2nd `  p
)  /  x ]_ C
49 csbeq1 3392 . . . . . . 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 3397 . . . . . 6  |-  [_ x  /  x ]_ C  =  C
5250, 51syl6eq 2508 . . . . 5  |-  ( p  =  <. y ,  x >.  ->  [_ ( 2nd `  p
)  /  x ]_ C  =  C )
5348, 1, 52mpt2mptxf 26139 . . . 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 2508 . . 3  |-  ( ph  ->  ( p  e.  `' F  |->  [_ ( 2nd `  p
)  /  x ]_ C )  =  ( y  e.  E ,  x  e.  ( `' F " { y } )  |->  C ) )
5554oveq2d 6209 . 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 26152 . . . 4  |-  ( E  ~<_  om  ->  E  e.  _V )
5856, 57syl 16 . . 3  |-  ( ph  ->  E  e.  _V )
59 mptfi 7714 . . . . . . 7  |-  ( A  e.  Fin  ->  (
x  e.  A  |->  D )  e.  Fin )
6015, 59syl5eqel 2543 . . . . . 6  |-  ( A  e.  Fin  ->  F  e.  Fin )
61 cnvfi 7699 . . . . . 6  |-  ( F  e.  Fin  ->  `' F  e.  Fin )
626, 60, 613syl 20 . . . . 5  |-  ( ph  ->  `' F  e.  Fin )
63 imaexg 6618 . . . . 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 5281 . . . . . . . 8  |-  ( `' F " { y } )  C_  ran  `' F
6867, 28sseqtr4i 3490 . . . . . . 7  |-  ( `' F " { y } )  C_  dom  F
6968, 16sstri 3466 . . . . . 6  |-  ( `' F " { y } )  C_  A
7011, 12elimasn 5295 . . . . . . . . 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 3455 . . . . 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 4394 . . . . . . . 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 16580 . 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 2496 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 1370   E.wex 1587    e. wcel 1758   A.wral 2795   E!wreu 2797   _Vcvv 3071   [_csb 3389    C_ wss 3429   {csn 3978   <.cop 3984   U_ciun 4272   class class class wbr 4393    |-> cmpt 4451    X. cxp 4939   `'ccnv 4940   dom cdm 4941   ran crn 4942   "cima 4944   Rel wrel 4946   Fun wfun 5513   ` cfv 5519  (class class class)co 6193    |-> cmpt2 6195   omcom 6579   2ndc2nd 6679    ~<_ cdom 7411   Fincfn 7413   Basecbs 14285   0gc0g 14489    gsumg cgsu 14490  CMndccmn 16390
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-inf2 7951  ax-cnex 9442  ax-resscn 9443  ax-1cn 9444  ax-icn 9445  ax-addcl 9446  ax-addrcl 9447  ax-mulcl 9448  ax-mulrcl 9449  ax-mulcom 9450  ax-addass 9451  ax-mulass 9452  ax-distr 9453  ax-i2m1 9454  ax-1ne0 9455  ax-1rid 9456  ax-rnegex 9457  ax-rrecex 9458  ax-cnre 9459  ax-pre-lttri 9460  ax-pre-lttrn 9461  ax-pre-ltadd 9462  ax-pre-mulgt0 9463
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-uni 4193  df-int 4230  df-iun 4274  df-iin 4275  df-br 4394  df-opab 4452  df-mpt 4453  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-se 4781  df-we 4782  df-ord 4823  df-on 4824  df-lim 4825  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-isom 5528  df-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-of 6423  df-om 6580  df-1st 6680  df-2nd 6681  df-supp 6794  df-recs 6935  df-rdg 6969  df-1o 7023  df-oadd 7027  df-er 7204  df-en 7414  df-dom 7415  df-sdom 7416  df-fin 7417  df-fsupp 7725  df-oi 7828  df-card 8213  df-pnf 9524  df-mnf 9525  df-xr 9526  df-ltxr 9527  df-le 9528  df-sub 9701  df-neg 9702  df-nn 10427  df-2 10484  df-n0 10684  df-z 10751  df-uz 10966  df-fz 11548  df-fzo 11659  df-seq 11917  df-hash 12214  df-ndx 14288  df-slot 14289  df-base 14290  df-sets 14291  df-ress 14292  df-plusg 14362  df-0g 14491  df-gsum 14492  df-mre 14635  df-mrc 14636  df-acs 14638  df-mnd 15526  df-submnd 15576  df-mulg 15659  df-cntz 15946  df-cmn 16392
This theorem is referenced by: (None)
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