Theorem gsum2d 17198

Description: Write a sum over a two-dimensional region as a double sum. (Contributed by Mario Carneiro, 28-Dec-2014.) (Revised by AV, 8-Jun-2019.)

Hypotheses

Ref	Expression
gsum2d.b	|- B = ( Base ` G )
gsum2d.z	|- .0. = ( 0g ` G )
gsum2d.g	|- ( ph -> G e. CMnd )
gsum2d.a	|- ( ph -> A e. V )
gsum2d.r	|- ( ph -> Rel A )
gsum2d.d	|- ( ph -> D e. W )
gsum2d.s	|- ( ph -> dom A C_ D )
gsum2d.f	|- ( ph -> F : A --> B )
gsum2d.w	|- ( ph -> F finSupp .0. )

Assertion

Ref	Expression
gsum2d	|- ( ph -> ( G gsumg F ) = ( G gsumg ( j e. D |-> ( G gsumg ( k e. ( A " { j } ) |-> ( j F k ) ) ) ) ) )

Distinct variable groups:   j, k, A   j, F, k   j, G, k   ph, j, k   B, j, k   D, j, k   .0. , j, k

Allowed substitution hints:   V( j, k)   W( j, k)

Proof of Theorem gsum2d

Step	Hyp	Ref	Expression
1		gsum2d.b	. . 3 |- B = ( Base ` G )
2		gsum2d.z	. . 3 |- .0. = ( 0g ` G )
3		gsum2d.g	. . 3 |- ( ph -> G e. CMnd )
4		gsum2d.a	. . 3 |- ( ph -> A e. V )
5		gsum2d.r	. . 3 |- ( ph -> Rel A )
6		gsum2d.d	. . 3 |- ( ph -> D e. W )
7		gsum2d.s	. . 3 |- ( ph -> dom A C_ D )
8		gsum2d.f	. . 3 |- ( ph -> F : A --> B )
9		gsum2d.w	. . 3 |- ( ph -> F finSupp .0. )
10	1, 2, 3, 4, 5, 6, 7, 8, 9	gsum2dlem2 17197	. 2 |- ( ph -> ( G gsumg ( F |` ( A |` dom ( F supp .0. ) ) ) ) = ( G gsumg ( j e. dom ( F supp .0. ) |-> ( G gsumg ( k e. ( A " { j } ) |-> ( j F k ) ) ) ) ) )
11		suppssdm 6904	. . . . . 6 |- ( F supp .0. ) C_ dom F
12		fdm 5717	. . . . . . 7 |- ( F : A --> B -> dom F = A )
13	8, 12	syl 16	. . . . . 6 |- ( ph -> dom F = A )
14	11, 13	syl5sseq 3537	. . . . 5 |- ( ph -> ( F supp .0. ) C_ A )
15		relss 5078	. . . . . . 7 |- ( ( F supp .0. ) C_ A -> ( Rel A -> Rel ( F supp .0. ) ) )
16	14, 5, 15	sylc 60	. . . . . 6 |- ( ph -> Rel ( F supp .0. ) )
17		relssdmrn 5511	. . . . . . 7 |- ( Rel ( F supp .0. ) -> ( F supp .0. ) C_ ( dom ( F supp .0. ) X. ran ( F supp .0. ) ) )
18		ssv 3509	. . . . . . . 8 |- ran ( F supp .0. ) C_ _V
19		xpss2 5100	. . . . . . . 8 |- ( ran ( F supp .0. ) C_ _V -> ( dom ( F supp .0. ) X. ran ( F supp .0. ) ) C_ ( dom ( F supp .0. ) X. _V ) )
20	18, 19	ax-mp 5	. . . . . . 7 |- ( dom ( F supp .0. ) X. ran ( F supp .0. ) ) C_ ( dom ( F supp .0. ) X. _V )
21	17, 20	syl6ss 3501	. . . . . 6 |- ( Rel ( F supp .0. ) -> ( F supp .0. ) C_ ( dom ( F supp .0. ) X. _V ) )
22	16, 21	syl 16	. . . . 5 |- ( ph -> ( F supp .0. ) C_ ( dom ( F supp .0. ) X. _V ) )
23	14, 22	ssind 3708	. . . 4 |- ( ph -> ( F supp .0. ) C_ ( A i^i ( dom ( F supp .0. ) X. _V ) ) )
24		df-res 5000	. . . 4 |- ( A |` dom ( F supp .0. ) ) = ( A i^i ( dom ( F supp .0. ) X. _V ) )
25	23, 24	syl6sseqr 3536	. . 3 |- ( ph -> ( F supp .0. ) C_ ( A |` dom ( F supp .0. ) ) )
26	1, 2, 3, 4, 8, 25, 9	gsumres 17123	. 2 |- ( ph -> ( G gsumg ( F |` ( A |` dom ( F supp .0. ) ) ) ) = ( G gsumg F ) )
27		dmss 5191	. . . . . . 7 |- ( ( F supp .0. ) C_ A -> dom ( F supp .0. ) C_ dom A )
28	14, 27	syl 16	. . . . . 6 |- ( ph -> dom ( F supp .0. ) C_ dom A )
29	28, 7	sstrd 3499	. . . . 5 |- ( ph -> dom ( F supp .0. ) C_ D )
30	29	resmptd 5313	. . . 4 |- ( ph -> ( ( j e. D |-> ( G gsumg ( k e. ( A " { j } ) |-> ( j F k ) ) ) ) |` dom ( F supp .0. ) ) = ( j e. dom ( F supp .0. ) |-> ( G gsumg ( k e. ( A " { j } ) |-> ( j F k ) ) ) ) )
31	30	oveq2d 6286	. . 3 |- ( ph -> ( G gsumg ( ( j e. D |-> ( G gsumg ( k e. ( A " { j } ) |-> ( j F k ) ) ) ) |` dom ( F supp .0. ) ) ) = ( G gsumg ( j e. dom ( F supp .0. ) |-> ( G gsumg ( k e. ( A " { j } ) |-> ( j F k ) ) ) ) ) )
32	1, 2, 3, 4, 5, 6, 7, 8, 9	gsum2dlem1 17196	. . . . . 6 |- ( ph -> ( G gsumg ( k e. ( A " { j } ) |-> ( j F k ) ) ) e. B )
33	32	adantr 463	. . . . 5 |- ( ( ph /\ j e. D ) -> ( G gsumg ( k e. ( A " { j } ) |-> ( j F k ) ) ) e. B )
34		eqid 2454	. . . . 5 |- ( j e. D |-> ( G gsumg ( k e. ( A " { j } ) |-> ( j F k ) ) ) ) = ( j e. D |-> ( G gsumg ( k e. ( A " { j } ) |-> ( j F k ) ) ) )
35	33, 34	fmptd 6031	. . . 4 |- ( ph -> ( j e. D |-> ( G gsumg ( k e. ( A " { j } ) |-> ( j F k ) ) ) ) : D --> B )
36		vex 3109	. . . . . . . . . . . . . 14 |- j e. _V
37		vex 3109	. . . . . . . . . . . . . 14 |- k e. _V
38	36, 37	elimasn 5350	. . . . . . . . . . . . 13 |- ( k e. ( A " { j } ) <-> <. j , k >. e. A )
39	38	biimpi 194	. . . . . . . . . . . 12 |- ( k e. ( A " { j } ) -> <. j , k >. e. A )
40	39	ad2antll 726	. . . . . . . . . . 11 |- ( ( ph /\ ( j e. ( D \ dom ( F supp .0. ) ) /\ k e. ( A " { j } ) ) ) -> <. j , k >. e. A )
41		eldifn 3613	. . . . . . . . . . . . 13 |- ( j e. ( D \ dom ( F supp .0. ) ) -> -. j e. dom ( F supp .0. ) )
42	41	ad2antrl 725	. . . . . . . . . . . 12 |- ( ( ph /\ ( j e. ( D \ dom ( F supp .0. ) ) /\ k e. ( A " { j } ) ) ) -> -. j e. dom ( F supp .0. ) )
43	36, 37	opeldm 5195	. . . . . . . . . . . 12 |- ( <. j , k >. e. ( F supp .0. ) -> j e. dom ( F supp .0. ) )
44	42, 43	nsyl 121	. . . . . . . . . . 11 |- ( ( ph /\ ( j e. ( D \ dom ( F supp .0. ) ) /\ k e. ( A " { j } ) ) ) -> -. <. j , k >. e. ( F supp .0. ) )
45	40, 44	eldifd 3472	. . . . . . . . . 10 |- ( ( ph /\ ( j e. ( D \ dom ( F supp .0. ) ) /\ k e. ( A " { j } ) ) ) -> <. j , k >. e. ( A \ ( F supp .0. ) ) )
46		df-ov 6273	. . . . . . . . . . 11 |- ( j F k ) = ( F ` <. j , k >. )
47		ssid 3508	. . . . . . . . . . . . 13 |- ( F supp .0. ) C_ ( F supp .0. )
48	47	a1i 11	. . . . . . . . . . . 12 |- ( ph -> ( F supp .0. ) C_ ( F supp .0. ) )
49		fvex 5858	. . . . . . . . . . . . . 14 |- ( 0g ` G ) e. _V
50	2, 49	eqeltri 2538	. . . . . . . . . . . . 13 |- .0. e. _V
51	50	a1i 11	. . . . . . . . . . . 12 |- ( ph -> .0. e. _V )
52	8, 48, 4, 51	suppssr 6923	. . . . . . . . . . 11 |- ( ( ph /\ <. j , k >. e. ( A \ ( F supp .0. ) ) ) -> ( F ` <. j , k >. ) = .0. )
53	46, 52	syl5eq 2507	. . . . . . . . . 10 |- ( ( ph /\ <. j , k >. e. ( A \ ( F supp .0. ) ) ) -> ( j F k ) = .0. )
54	45, 53	syldan 468	. . . . . . . . 9 |- ( ( ph /\ ( j e. ( D \ dom ( F supp .0. ) ) /\ k e. ( A " { j } ) ) ) -> ( j F k ) = .0. )
55	54	anassrs 646	. . . . . . . 8 |- ( ( ( ph /\ j e. ( D \ dom ( F supp .0. ) ) ) /\ k e. ( A " { j } ) ) -> ( j F k ) = .0. )
56	55	mpteq2dva 4525	. . . . . . 7 |- ( ( ph /\ j e. ( D \ dom ( F supp .0. ) ) ) -> ( k e. ( A " { j } ) |-> ( j F k ) ) = ( k e. ( A " { j } ) |-> .0. ) )
57	56	oveq2d 6286	. . . . . 6 |- ( ( ph /\ j e. ( D \ dom ( F supp .0. ) ) ) -> ( G gsumg ( k e. ( A " { j } ) |-> ( j F k ) ) ) = ( G gsumg ( k e. ( A " { j } ) |-> .0. ) ) )
58		cmnmnd 17015	. . . . . . . . 9 |- ( G e. CMnd -> G e. Mnd )
59	3, 58	syl 16	. . . . . . . 8 |- ( ph -> G e. Mnd )
60		imaexg 6710	. . . . . . . . 9 |- ( A e. V -> ( A " { j } ) e. _V )
61	4, 60	syl 16	. . . . . . . 8 |- ( ph -> ( A " { j } ) e. _V )
62	2	gsumz 16207	. . . . . . . 8 |- ( ( G e. Mnd /\ ( A " { j } ) e. _V ) -> ( G gsumg ( k e. ( A " { j } ) |-> .0. ) ) = .0. )
63	59, 61, 62	syl2anc 659	. . . . . . 7 |- ( ph -> ( G gsumg ( k e. ( A " { j } ) |-> .0. ) ) = .0. )
64	63	adantr 463	. . . . . 6 |- ( ( ph /\ j e. ( D \ dom ( F supp .0. ) ) ) -> ( G gsumg ( k e. ( A " { j } ) |-> .0. ) ) = .0. )
65	57, 64	eqtrd 2495	. . . . 5 |- ( ( ph /\ j e. ( D \ dom ( F supp .0. ) ) ) -> ( G gsumg ( k e. ( A " { j } ) |-> ( j F k ) ) ) = .0. )
66	65, 6	suppss2 6926	. . . 4 |- ( ph -> ( ( j e. D |-> ( G gsumg ( k e. ( A " { j } ) |-> ( j F k ) ) ) ) supp .0. ) C_ dom ( F supp .0. ) )
67		funmpt 5606	. . . . . 6 |- Fun ( j e. D |-> ( G gsumg ( k e. ( A " { j } ) |-> ( j F k ) ) ) )
68	67	a1i 11	. . . . 5 |- ( ph -> Fun ( j e. D |-> ( G gsumg ( k e. ( A " { j } ) |-> ( j F k ) ) ) ) )
69	9	fsuppimpd 7828	. . . . . . 7 |- ( ph -> ( F supp .0. ) e. Fin )
70		dmfi 7795	. . . . . . 7 |- ( ( F supp .0. ) e. Fin -> dom ( F supp .0. ) e. Fin )
71	69, 70	syl 16	. . . . . 6 |- ( ph -> dom ( F supp .0. ) e. Fin )
72		ssfi 7733	. . . . . 6 |- ( ( dom ( F supp .0. ) e. Fin /\ ( ( j e. D |-> ( G gsumg ( k e. ( A " { j } ) |-> ( j F k ) ) ) ) supp .0. ) C_ dom ( F supp .0. ) ) -> ( ( j e. D |-> ( G gsumg ( k e. ( A " { j } ) |-> ( j F k ) ) ) ) supp .0. ) e. Fin )
73	71, 66, 72	syl2anc 659	. . . . 5 |- ( ph -> ( ( j e. D |-> ( G gsumg ( k e. ( A " { j } ) |-> ( j F k ) ) ) ) supp .0. ) e. Fin )
74		mptexg 6117	. . . . . . 7 |- ( D e. W -> ( j e. D |-> ( G gsumg ( k e. ( A " { j } ) |-> ( j F k ) ) ) ) e. _V )
75	6, 74	syl 16	. . . . . 6 |- ( ph -> ( j e. D |-> ( G gsumg ( k e. ( A " { j } ) |-> ( j F k ) ) ) ) e. _V )
76		isfsupp 7825	. . . . . 6 |- ( ( ( j e. D |-> ( G gsumg ( k e. ( A " { j } ) |-> ( j F k ) ) ) ) e. _V /\ .0. e. _V ) -> ( ( j e. D |-> ( G gsumg ( k e. ( A " { j } ) |-> ( j F k ) ) ) ) finSupp .0. <-> ( Fun ( j e. D |-> ( G gsumg ( k e. ( A " { j } ) |-> ( j F k ) ) ) ) /\ ( ( j e. D |-> ( G gsumg ( k e. ( A " { j } ) |-> ( j F k ) ) ) ) supp .0. ) e. Fin ) ) )
77	75, 51, 76	syl2anc 659	. . . . 5 |- ( ph -> ( ( j e. D |-> ( G gsumg ( k e. ( A " { j } ) |-> ( j F k ) ) ) ) finSupp .0. <-> ( Fun ( j e. D |-> ( G gsumg ( k e. ( A " { j } ) |-> ( j F k ) ) ) ) /\ ( ( j e. D |-> ( G gsumg ( k e. ( A " { j } ) |-> ( j F k ) ) ) ) supp .0. ) e. Fin ) ) )
78	68, 73, 77	mpbir2and 920	. . . 4 |- ( ph -> ( j e. D |-> ( G gsumg ( k e. ( A " { j } ) |-> ( j F k ) ) ) ) finSupp .0. )
79	1, 2, 3, 6, 35, 66, 78	gsumres 17123	. . 3 |- ( ph -> ( G gsumg ( ( j e. D |-> ( G gsumg ( k e. ( A " { j } ) |-> ( j F k ) ) ) ) |` dom ( F supp .0. ) ) ) = ( G gsumg ( j e. D |-> ( G gsumg ( k e. ( A " { j } ) |-> ( j F k ) ) ) ) ) )
80	31, 79	eqtr3d 2497	. 2 |- ( ph -> ( G gsumg ( j e. dom ( F supp .0. ) |-> ( G gsumg ( k e. ( A " { j } ) |-> ( j F k ) ) ) ) ) = ( G gsumg ( j e. D |-> ( G gsumg ( k e. ( A " { j } ) |-> ( j F k ) ) ) ) ) )
81	10, 26, 80	3eqtr3d 2503	1 |- ( ph -> ( G gsumg F ) = ( G gsumg ( j e. D |-> ( G gsumg ( k e. ( A " { j } ) |-> ( j F k ) ) ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   /\ wa 367   = wceq 1398   e. wcel 1823   _Vcvv 3106   \ cdif 3458   i^i cin 3460   C_ wss 3461   {csn 4016   <.cop 4022   class class class wbr 4439   |-> cmpt 4497   X. cxp 4986   dom cdm 4988   ran crn 4989   |` cres 4990   "cima 4991   Rel wrel 4993   Fun wfun 5564   -->wf 5566   ` cfv 5570  (class class class)co 6270   supp csupp 6891   Fincfn 7509   finSupp cfsupp 7821   Basecbs 14719   0gc0g 14932   gsum_g cgsu 14933   Mndcmnd 16121  CMndccmn 17000

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-inf2 8049  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-iin 4318  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-of 6513  df-om 6674  df-1st 6773  df-2nd 6774  df-supp 6892  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-fsupp 7822  df-oi 7927  df-card 8311  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-n0 10792  df-z 10861  df-uz 11083  df-fz 11676  df-fzo 11800  df-seq 12093  df-hash 12391  df-ndx 14722  df-slot 14723  df-base 14724  df-sets 14725  df-ress 14726  df-plusg 14800  df-0g 14934  df-gsum 14935  df-mre 15078  df-mrc 15079  df-acs 15081  df-mgm 16074  df-sgrp 16113  df-mnd 16123  df-submnd 16169  df-mulg 16262  df-cntz 16557  df-cmn 17002

This theorem is referenced by:  gsum2d2  17201  gsumxp  17203