Theorem gsum2d 16463

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 16462	. 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 6703	. . . . . 6 |- ( F supp .0. ) C_ dom F
12		fdm 5563	. . . . . . 7 |- ( F : A --> B -> dom F = A )
13	8, 12	syl 16	. . . . . 6 |- ( ph -> dom F = A )
14	11, 13	syl5sseq 3404	. . . . 5 |- ( ph -> ( F supp .0. ) C_ A )
15		relss 4927	. . . . . . 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 5358	. . . . . . 7 |- ( Rel ( F supp .0. ) -> ( F supp .0. ) C_ ( dom ( F supp .0. ) X. ran ( F supp .0. ) ) )
18		ssv 3376	. . . . . . . 8 |- ran ( F supp .0. ) C_ _V
19		xpss2 4949	. . . . . . . 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 3368	. . . . . 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 3574	. . . 4 |- ( ph -> ( F supp .0. ) C_ ( A i^i ( dom ( F supp .0. ) X. _V ) ) )
24		df-res 4852	. . . 4 |- ( A |` dom ( F supp .0. ) ) = ( A i^i ( dom ( F supp .0. ) X. _V ) )
25	23, 24	syl6sseqr 3403	. . 3 |- ( ph -> ( F supp .0. ) C_ ( A |` dom ( F supp .0. ) ) )
26	1, 2, 3, 4, 8, 25, 9	gsumres 16395	. 2 |- ( ph -> ( G gsumg ( F |` ( A |` dom ( F supp .0. ) ) ) ) = ( G gsumg F ) )
27		dmss 5039	. . . . . . 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 3366	. . . . 5 |- ( ph -> dom ( F supp .0. ) C_ D )
30		resmpt 5156	. . . . 5 |- ( dom ( F supp .0. ) C_ D -> ( ( 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	29, 30	syl 16	. . . 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 ) ) ) ) )
32	31	oveq2d 6107	. . 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 ) ) ) ) ) )
33	1, 2, 3, 4, 5, 6, 7, 8, 9	gsum2dlem1 16461	. . . . . 6 |- ( ph -> ( G gsumg ( k e. ( A " { j } ) |-> ( j F k ) ) ) e. B )
34	33	adantr 465	. . . . 5 |- ( ( ph /\ j e. D ) -> ( G gsumg ( k e. ( A " { j } ) |-> ( j F k ) ) ) e. B )
35		eqid 2443	. . . . 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 ) ) ) )
36	34, 35	fmptd 5867	. . . 4 |- ( ph -> ( j e. D |-> ( G gsumg ( k e. ( A " { j } ) |-> ( j F k ) ) ) ) : D --> B )
37		vex 2975	. . . . . . . . . . . . . 14 |- j e. _V
38		vex 2975	. . . . . . . . . . . . . 14 |- k e. _V
39	37, 38	elimasn 5194	. . . . . . . . . . . . 13 |- ( k e. ( A " { j } ) <-> <. j , k >. e. A )
40	39	biimpi 194	. . . . . . . . . . . 12 |- ( k e. ( A " { j } ) -> <. j , k >. e. A )
41	40	ad2antll 728	. . . . . . . . . . 11 |- ( ( ph /\ ( j e. ( D \ dom ( F supp .0. ) ) /\ k e. ( A " { j } ) ) ) -> <. j , k >. e. A )
42		eldifn 3479	. . . . . . . . . . . . 13 |- ( j e. ( D \ dom ( F supp .0. ) ) -> -. j e. dom ( F supp .0. ) )
43	42	ad2antrl 727	. . . . . . . . . . . 12 |- ( ( ph /\ ( j e. ( D \ dom ( F supp .0. ) ) /\ k e. ( A " { j } ) ) ) -> -. j e. dom ( F supp .0. ) )
44	37, 38	opeldm 5043	. . . . . . . . . . . 12 |- ( <. j , k >. e. ( F supp .0. ) -> j e. dom ( F supp .0. ) )
45	43, 44	nsyl 121	. . . . . . . . . . 11 |- ( ( ph /\ ( j e. ( D \ dom ( F supp .0. ) ) /\ k e. ( A " { j } ) ) ) -> -. <. j , k >. e. ( F supp .0. ) )
46	41, 45	eldifd 3339	. . . . . . . . . 10 |- ( ( ph /\ ( j e. ( D \ dom ( F supp .0. ) ) /\ k e. ( A " { j } ) ) ) -> <. j , k >. e. ( A \ ( F supp .0. ) ) )
47		df-ov 6094	. . . . . . . . . . 11 |- ( j F k ) = ( F ` <. j , k >. )
48		ssid 3375	. . . . . . . . . . . . 13 |- ( F supp .0. ) C_ ( F supp .0. )
49	48	a1i 11	. . . . . . . . . . . 12 |- ( ph -> ( F supp .0. ) C_ ( F supp .0. ) )
50		fvex 5701	. . . . . . . . . . . . . 14 |- ( 0g ` G ) e. _V
51	2, 50	eqeltri 2513	. . . . . . . . . . . . 13 |- .0. e. _V
52	51	a1i 11	. . . . . . . . . . . 12 |- ( ph -> .0. e. _V )
53	8, 49, 4, 52	suppssr 6720	. . . . . . . . . . 11 |- ( ( ph /\ <. j , k >. e. ( A \ ( F supp .0. ) ) ) -> ( F ` <. j , k >. ) = .0. )
54	47, 53	syl5eq 2487	. . . . . . . . . 10 |- ( ( ph /\ <. j , k >. e. ( A \ ( F supp .0. ) ) ) -> ( j F k ) = .0. )
55	46, 54	syldan 470	. . . . . . . . 9 |- ( ( ph /\ ( j e. ( D \ dom ( F supp .0. ) ) /\ k e. ( A " { j } ) ) ) -> ( j F k ) = .0. )
56	55	anassrs 648	. . . . . . . 8 |- ( ( ( ph /\ j e. ( D \ dom ( F supp .0. ) ) ) /\ k e. ( A " { j } ) ) -> ( j F k ) = .0. )
57	56	mpteq2dva 4378	. . . . . . 7 |- ( ( ph /\ j e. ( D \ dom ( F supp .0. ) ) ) -> ( k e. ( A " { j } ) |-> ( j F k ) ) = ( k e. ( A " { j } ) |-> .0. ) )
58	57	oveq2d 6107	. . . . . 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. ) ) )
59		cmnmnd 16292	. . . . . . . . 9 |- ( G e. CMnd -> G e. Mnd )
60	3, 59	syl 16	. . . . . . . 8 |- ( ph -> G e. Mnd )
61		imaexg 6515	. . . . . . . . 9 |- ( A e. V -> ( A " { j } ) e. _V )
62	4, 61	syl 16	. . . . . . . 8 |- ( ph -> ( A " { j } ) e. _V )
63	2	gsumz 15511	. . . . . . . 8 |- ( ( G e. Mnd /\ ( A " { j } ) e. _V ) -> ( G gsumg ( k e. ( A " { j } ) |-> .0. ) ) = .0. )
64	60, 62, 63	syl2anc 661	. . . . . . 7 |- ( ph -> ( G gsumg ( k e. ( A " { j } ) |-> .0. ) ) = .0. )
65	64	adantr 465	. . . . . 6 |- ( ( ph /\ j e. ( D \ dom ( F supp .0. ) ) ) -> ( G gsumg ( k e. ( A " { j } ) |-> .0. ) ) = .0. )
66	58, 65	eqtrd 2475	. . . . 5 |- ( ( ph /\ j e. ( D \ dom ( F supp .0. ) ) ) -> ( G gsumg ( k e. ( A " { j } ) |-> ( j F k ) ) ) = .0. )
67	66, 6	suppss2 6723	. . . 4 |- ( ph -> ( ( j e. D |-> ( G gsumg ( k e. ( A " { j } ) |-> ( j F k ) ) ) ) supp .0. ) C_ dom ( F supp .0. ) )
68		funmpt 5454	. . . . . 6 |- Fun ( j e. D |-> ( G gsumg ( k e. ( A " { j } ) |-> ( j F k ) ) ) )
69	68	a1i 11	. . . . 5 |- ( ph -> Fun ( j e. D |-> ( G gsumg ( k e. ( A " { j } ) |-> ( j F k ) ) ) ) )
70	9	fsuppimpd 7627	. . . . . . 7 |- ( ph -> ( F supp .0. ) e. Fin )
71		dmfi 7594	. . . . . . 7 |- ( ( F supp .0. ) e. Fin -> dom ( F supp .0. ) e. Fin )
72	70, 71	syl 16	. . . . . 6 |- ( ph -> dom ( F supp .0. ) e. Fin )
73		ssfi 7533	. . . . . 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 )
74	72, 67, 73	syl2anc 661	. . . . 5 |- ( ph -> ( ( j e. D |-> ( G gsumg ( k e. ( A " { j } ) |-> ( j F k ) ) ) ) supp .0. ) e. Fin )
75		mptexg 5947	. . . . . . 7 |- ( D e. W -> ( j e. D |-> ( G gsumg ( k e. ( A " { j } ) |-> ( j F k ) ) ) ) e. _V )
76	6, 75	syl 16	. . . . . 6 |- ( ph -> ( j e. D |-> ( G gsumg ( k e. ( A " { j } ) |-> ( j F k ) ) ) ) e. _V )
77		isfsupp 7624	. . . . . 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 ) ) )
78	76, 52, 77	syl2anc 661	. . . . 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 ) ) )
79	69, 74, 78	mpbir2and 913	. . . 4 |- ( ph -> ( j e. D |-> ( G gsumg ( k e. ( A " { j } ) |-> ( j F k ) ) ) ) finSupp .0. )
80	1, 2, 3, 6, 36, 67, 79	gsumres 16395	. . 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 ) ) ) ) ) )
81	32, 80	eqtr3d 2477	. 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 ) ) ) ) ) )
82	10, 26, 81	3eqtr3d 2483	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 369   = wceq 1369   e. wcel 1756   _Vcvv 2972   \ cdif 3325   i^i cin 3327   C_ wss 3328   {csn 3877   <.cop 3883   class class class wbr 4292   e. cmpt 4350   X. cxp 4838   dom cdm 4840   ran crn 4841   |` cres 4842   "cima 4843   Rel wrel 4845   Fun wfun 5412   -->wf 5414   ` cfv 5418  (class class class)co 6091   supp csupp 6690   Fincfn 7310   finSupp cfsupp 7620   Basecbs 14174   0gc0g 14378   gsum_g cgsu 14379   Mndcmnd 15409  CMndccmn 16277

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-inf2 7847  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-iin 4174  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-se 4680  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-isom 5427  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-of 6320  df-om 6477  df-1st 6577  df-2nd 6578  df-supp 6691  df-recs 6832  df-rdg 6866  df-1o 6920  df-oadd 6924  df-er 7101  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-fsupp 7621  df-oi 7724  df-card 8109  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-nn 10323  df-2 10380  df-n0 10580  df-z 10647  df-uz 10862  df-fz 11438  df-fzo 11549  df-seq 11807  df-hash 12104  df-ndx 14177  df-slot 14178  df-base 14179  df-sets 14180  df-ress 14181  df-plusg 14251  df-0g 14380  df-gsum 14381  df-mre 14524  df-mrc 14525  df-acs 14527  df-mnd 15415  df-submnd 15465  df-mulg 15548  df-cntz 15835  df-cmn 16279

This theorem is referenced by:  gsum2d2  16466  gsumxp  16468