Theorem gsummpt2co 29780

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.

Step	Hyp	Ref	Expression
1		nfcsb1v 3549	. . . 4 |- F/_ x [_ ( 2nd ` p ) / x ]_ C
2		gsummpt2co.b	. . . 4 |- B = ( Base ` W )
3		gsummpt2co.z	. . . 4 |- .0. = ( 0g ` W )
4		csbeq1a 3542	. . . 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 3624	. . . . 5 |- B C_ B
8	7	a1i 11	. . . 4 |- ( ph -> B C_ B )
9		gsummpt2co.1	. . . 4 |- ( ( ph /\ x e. A ) -> C e. B )
10		elcnv 5299	. . . . . 6 |- ( p e. `' F <-> E. z E. x ( p = <. z , x >. /\ x F z ) )
11		vex 3203	. . . . . . . . . 10 |- z e. _V
12		vex 3203	. . . . . . . . . 10 |- x e. _V
13	11, 12	op2ndd 7179	. . . . . . . . 9 |- ( p = <. z , x >. -> ( 2nd ` p ) = x )
14	13	adantr 481	. . . . . . . 8 |- ( ( p = <. z , x >. /\ x F z ) -> ( 2nd ` p ) = x )
15		gsummpt2co.3	. . . . . . . . . . 11 |- F = ( x e. A |-> D )
16	15	dmmptss 5631	. . . . . . . . . 10 |- dom F C_ A
17	12, 11	breldm 5329	. . . . . . . . . 10 |- ( x F z -> x e. dom F )
18	16, 17	sseldi 3601	. . . . . . . . 9 |- ( x F z -> x e. A )
19	18	adantl 482	. . . . . . . 8 |- ( ( p = <. z , x >. /\ x F z ) -> x e. A )
20	14, 19	eqeltrd 2701	. . . . . . 7 |- ( ( p = <. z , x >. /\ x F z ) -> ( 2nd ` p ) e. A )
21	20	exlimivv 1860	. . . . . 6 |- ( E. z E. x ( p = <. z , x >. /\ x F z ) -> ( 2nd ` p ) e. A )
22	10, 21	sylbi 207	. . . . 5 |- ( p e. `' F -> ( 2nd ` p ) e. A )
23	22	adantl 482	. . . 4 |- ( ( ph /\ p e. `' F ) -> ( 2nd ` p ) e. A )
24	15	funmpt2 5927	. . . . . . 7 |- Fun F
25		funcnvcnv 5956	. . . . . . 7 |- ( Fun F -> Fun `' `' F )
26	24, 25	ax-mp 5	. . . . . 6 |- Fun `' `' F
27	26	a1i 11	. . . . 5 |- ( ( ph /\ x e. A ) -> Fun `' `' F )
28		dfdm4 5316	. . . . . . . 8 |- dom F = ran `' F
29	15	dmeqi 5325	. . . . . . . . 9 |- dom F = dom ( x e. A |-> D )
30		gsummpt2co.2	. . . . . . . . . . 11 |- ( ( ph /\ x e. A ) -> D e. E )
31	30	ralrimiva 2966	. . . . . . . . . 10 |- ( ph -> A. x e. A D e. E )
32		dmmptg 5632	. . . . . . . . . 10 |- ( A. x e. A D e. E -> dom ( x e. A |-> D ) = A )
33	31, 32	syl 17	. . . . . . . . 9 |- ( ph -> dom ( x e. A |-> D ) = A )
34	29, 33	syl5eq 2668	. . . . . . . 8 |- ( ph -> dom F = A )
35	28, 34	syl5eqr 2670	. . . . . . 7 |- ( ph -> ran `' F = A )
36	35	eleq2d 2687	. . . . . 6 |- ( ph -> ( x e. ran `' F <-> x e. A ) )
37	36	biimpar 502	. . . . 5 |- ( ( ph /\ x e. A ) -> x e. ran `' F )
38		relcnv 5503	. . . . . 6 |- Rel `' F
39		fcnvgreu 29472	. . . . . 6 |- ( ( ( Rel `' F /\ Fun `' `' F ) /\ x e. ran `' F ) -> E! p e. `' F x = ( 2nd ` p ) )
40	38, 39	mpanl1 716	. . . . 5 |- ( ( Fun `' `' F /\ x e. ran `' F ) -> E! p e. `' F x = ( 2nd ` p ) )
41	27, 37, 40	syl2anc 693	. . . 4 |- ( ( ph /\ x e. A ) -> E! p e. `' F x = ( 2nd ` p ) )
42	1, 2, 3, 4, 5, 6, 8, 9, 23, 41	gsummptf1o 18362	. . 3 |- ( ph -> ( W gsumg ( x e. A |-> C ) ) = ( W gsumg ( p e. `' F |-> [_ ( 2nd ` p ) / x ]_ C ) ) )
43	15	rnmptss 6392	. . . . . . . 8 |- ( A. x e. A D e. E -> ran F C_ E )
44	31, 43	syl 17	. . . . . . 7 |- ( ph -> ran F C_ E )
45		dfcnv2 29476	. . . . . . 7 |- ( ran F C_ E -> `' F = U_ z e. E ( { z } X. ( `' F " { z } ) ) )
46	44, 45	syl 17	. . . . . 6 |- ( ph -> `' F = U_ z e. E ( { z } X. ( `' F " { z } ) ) )
47	46	mpteq1d 4738	. . . . 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 2764	. . . . . 6 |- F/_ z [_ ( 2nd ` p ) / x ]_ C
49		csbeq1 3536	. . . . . . . 8 |- ( ( 2nd ` p ) = x -> [_ ( 2nd ` p ) / x ]_ C = [_ x / x ]_ C )
50	13, 49	syl 17	. . . . . . 7 |- ( p = <. z , x >. -> [_ ( 2nd ` p ) / x ]_ C = [_ x / x ]_ C )
51		csbid 3541	. . . . . . 7 |- [_ x / x ]_ C = C
52	50, 51	syl6eq 2672	. . . . . 6 |- ( p = <. z , x >. -> [_ ( 2nd ` p ) / x ]_ C = C )
53	48, 1, 52	mpt2mptxf 29477	. . . . 5 |- ( p e. U_ z e. E ( { z } X. ( `' F " { z } ) ) |-> [_ ( 2nd ` p ) / x ]_ C ) = ( z e. E , x e. ( `' F " { z } ) |-> C )
54	47, 53	syl6eq 2672	. . . 4 |- ( ph -> ( p e. `' F |-> [_ ( 2nd ` p ) / x ]_ C ) = ( z e. E , x e. ( `' F " { z } ) |-> C ) )
55	54	oveq2d 6666	. . 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 8265	. . . . . . . 8 |- ( A e. Fin -> ( x e. A |-> D ) e. Fin )
58	15, 57	syl5eqel 2705	. . . . . . 7 |- ( A e. Fin -> F e. Fin )
59		cnvfi 8248	. . . . . . 7 |- ( F e. Fin -> `' F e. Fin )
60	6, 58, 59	3syl 18	. . . . . 6 |- ( ph -> `' F e. Fin )
61		imaexg 7103	. . . . . 6 |- ( `' F e. Fin -> ( `' F " { z } ) e. _V )
62	60, 61	syl 17	. . . . 5 |- ( ph -> ( `' F " { z } ) e. _V )
63	62	adantr 481	. . . 4 |- ( ( ph /\ z e. E ) -> ( `' F " { z } ) e. _V )
64		simpll 790	. . . . . 6 |- ( ( ( ph /\ z e. E ) /\ x e. ( `' F " { z } ) ) -> ph )
65		imassrn 5477	. . . . . . . . 9 |- ( `' F " { z } ) C_ ran `' F
66	65, 28	sseqtr4i 3638	. . . . . . . 8 |- ( `' F " { z } ) C_ dom F
67	66, 16	sstri 3612	. . . . . . 7 |- ( `' F " { z } ) C_ A
68	11, 12	elimasn 5490	. . . . . . . . . 10 |- ( x e. ( `' F " { z } ) <-> <. z , x >. e. `' F )
69	68	biimpi 206	. . . . . . . . 9 |- ( x e. ( `' F " { z } ) -> <. z , x >. e. `' F )
70	69	adantl 482	. . . . . . . 8 |- ( ( ( ph /\ z e. E ) /\ x e. ( `' F " { z } ) ) -> <. z , x >. e. `' F )
71	70, 68	sylibr 224	. . . . . . 7 |- ( ( ( ph /\ z e. E ) /\ x e. ( `' F " { z } ) ) -> x e. ( `' F " { z } ) )
72	67, 71	sseldi 3601	. . . . . 6 |- ( ( ( ph /\ z e. E ) /\ x e. ( `' F " { z } ) ) -> x e. A )
73	64, 72, 9	syl2anc 693	. . . . 5 |- ( ( ( ph /\ z e. E ) /\ x e. ( `' F " { z } ) ) -> C e. B )
74	73	anasss 679	. . . 4 |- ( ( ph /\ ( z e. E /\ x e. ( `' F " { z } ) ) ) -> C e. B )
75		df-br 4654	. . . . . . . . 9 |- ( z `' F x <-> <. z , x >. e. `' F )
76	70, 75	sylibr 224	. . . . . . . 8 |- ( ( ( ph /\ z e. E ) /\ x e. ( `' F " { z } ) ) -> z `' F x )
77	76	anasss 679	. . . . . . 7 |- ( ( ph /\ ( z e. E /\ x e. ( `' F " { z } ) ) ) -> z `' F x )
78	77	pm2.24d 147	. . . . . 6 |- ( ( ph /\ ( z e. E /\ x e. ( `' F " { z } ) ) ) -> ( -. z `' F x -> C = .0. ) )
79	78	imp 445	. . . . 5 |- ( ( ( ph /\ ( z e. E /\ x e. ( `' F " { z } ) ) ) /\ -. z `' F x ) -> C = .0. )
80	79	anasss 679	. . . 4 |- ( ( ph /\ ( ( z e. E /\ x e. ( `' F " { z } ) ) /\ -. z `' F x ) ) -> C = .0. )
81	2, 3, 5, 56, 63, 74, 60, 80	gsum2d2 18373	. . 3 |- ( ph -> ( W gsumg ( z e. E , x e. ( `' F " { z } ) |-> C ) ) = ( W gsumg ( z e. E |-> ( W gsumg ( x e. ( `' F " { z } ) |-> C ) ) ) ) )
82	42, 55, 81	3eqtrd 2660	. 2 |- ( ph -> ( W gsumg ( x e. A |-> C ) ) = ( W gsumg ( z e. E |-> ( W gsumg ( x e. ( `' F " { z } ) |-> C ) ) ) ) )
83		nfcv 2764	. . . 4 |- F/_ z ( W gsumg ( x e. ( `' F " { y } ) |-> C ) )
84		nfcv 2764	. . . 4 |- F/_ y ( W gsumg ( x e. ( `' F " { z } ) |-> C ) )
85		sneq 4187	. . . . . . 7 |- ( y = z -> { y } = { z } )
86	85	imaeq2d 5466	. . . . . 6 |- ( y = z -> ( `' F " { y } ) = ( `' F " { z } ) )
87	86	mpteq1d 4738	. . . . 5 |- ( y = z -> ( x e. ( `' F " { y } ) |-> C ) = ( x e. ( `' F " { z } ) |-> C ) )
88	87	oveq2d 6666	. . . 4 |- ( y = z -> ( W gsumg ( x e. ( `' F " { y } ) |-> C ) ) = ( W gsumg ( x e. ( `' F " { z } ) |-> C ) ) )
89	83, 84, 88	cbvmpt 4749	. . 3 |- ( y e. E |-> ( W gsumg ( x e. ( `' F " { y } ) |-> C ) ) ) = ( z e. E |-> ( W gsumg ( x e. ( `' F " { z } ) |-> C ) ) )
90	89	oveq2i 6661	. 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 ) ) ) )
91	82, 90	syl6eqr 2674	1 |- ( ph -> ( W gsumg ( x e. A |-> C ) ) = ( W gsumg ( y e. E |-> ( W gsumg ( x e. ( `' F " { y } ) |-> C ) ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   A.wral 2912   E!wreu 2914   _Vcvv 3200   [_csb 3533   C_ wss 3574   {csn 4177   <.cop 4183   U_ciun 4520   class class class wbr 4653   |-> cmpt 4729   X. cxp 5112   `'ccnv 5113   dom cdm 5114   ran crn 5115   "cima 5117   Rel wrel 5119   Fun wfun 5882   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   2ndc2nd 7167   Fincfn 7955   Basecbs 15857   0gc0g 16100   gsum_g cgsu 16101  CMndccmn 18193

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-seq 12802  df-hash 13118  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-0g 16102  df-gsum 16103  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-mulg 17541  df-cntz 17750  df-cmn 18195

This theorem is referenced by:  gsummpt2d  29781