Theorem gsum2d 18371

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 18370	. 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 7308	. . . . . 6 |- ( F supp .0. ) C_ dom F
12		fdm 6051	. . . . . . 7 |- ( F : A --> B -> dom F = A )
13	8, 12	syl 17	. . . . . 6 |- ( ph -> dom F = A )
14	11, 13	syl5sseq 3653	. . . . 5 |- ( ph -> ( F supp .0. ) C_ A )
15		relss 5206	. . . . . . 7 |- ( ( F supp .0. ) C_ A -> ( Rel A -> Rel ( F supp .0. ) ) )
16	14, 5, 15	sylc 65	. . . . . 6 |- ( ph -> Rel ( F supp .0. ) )
17		relssdmrn 5656	. . . . . . 7 |- ( Rel ( F supp .0. ) -> ( F supp .0. ) C_ ( dom ( F supp .0. ) X. ran ( F supp .0. ) ) )
18		ssv 3625	. . . . . . . 8 |- ran ( F supp .0. ) C_ _V
19		xpss2 5229	. . . . . . . 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 3615	. . . . . 6 |- ( Rel ( F supp .0. ) -> ( F supp .0. ) C_ ( dom ( F supp .0. ) X. _V ) )
22	16, 21	syl 17	. . . . 5 |- ( ph -> ( F supp .0. ) C_ ( dom ( F supp .0. ) X. _V ) )
23	14, 22	ssind 3837	. . . 4 |- ( ph -> ( F supp .0. ) C_ ( A i^i ( dom ( F supp .0. ) X. _V ) ) )
24		df-res 5126	. . . 4 |- ( A |` dom ( F supp .0. ) ) = ( A i^i ( dom ( F supp .0. ) X. _V ) )
25	23, 24	syl6sseqr 3652	. . 3 |- ( ph -> ( F supp .0. ) C_ ( A |` dom ( F supp .0. ) ) )
26	1, 2, 3, 4, 8, 25, 9	gsumres 18314	. 2 |- ( ph -> ( G gsumg ( F |` ( A |` dom ( F supp .0. ) ) ) ) = ( G gsumg F ) )
27		dmss 5323	. . . . . . 7 |- ( ( F supp .0. ) C_ A -> dom ( F supp .0. ) C_ dom A )
28	14, 27	syl 17	. . . . . 6 |- ( ph -> dom ( F supp .0. ) C_ dom A )
29	28, 7	sstrd 3613	. . . . 5 |- ( ph -> dom ( F supp .0. ) C_ D )
30	29	resmptd 5452	. . . 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 6666	. . 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 18369	. . . . . 6 |- ( ph -> ( G gsumg ( k e. ( A " { j } ) |-> ( j F k ) ) ) e. B )
33	32	adantr 481	. . . . 5 |- ( ( ph /\ j e. D ) -> ( G gsumg ( k e. ( A " { j } ) |-> ( j F k ) ) ) e. B )
34		eqid 2622	. . . . 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 6385	. . . 4 |- ( ph -> ( j e. D |-> ( G gsumg ( k e. ( A " { j } ) |-> ( j F k ) ) ) ) : D --> B )
36		vex 3203	. . . . . . . . . . . . . 14 |- j e. _V
37		vex 3203	. . . . . . . . . . . . . 14 |- k e. _V
38	36, 37	elimasn 5490	. . . . . . . . . . . . 13 |- ( k e. ( A " { j } ) <-> <. j , k >. e. A )
39	38	biimpi 206	. . . . . . . . . . . 12 |- ( k e. ( A " { j } ) -> <. j , k >. e. A )
40	39	ad2antll 765	. . . . . . . . . . 11 |- ( ( ph /\ ( j e. ( D \ dom ( F supp .0. ) ) /\ k e. ( A " { j } ) ) ) -> <. j , k >. e. A )
41		eldifn 3733	. . . . . . . . . . . . 13 |- ( j e. ( D \ dom ( F supp .0. ) ) -> -. j e. dom ( F supp .0. ) )
42	41	ad2antrl 764	. . . . . . . . . . . 12 |- ( ( ph /\ ( j e. ( D \ dom ( F supp .0. ) ) /\ k e. ( A " { j } ) ) ) -> -. j e. dom ( F supp .0. ) )
43	36, 37	opeldm 5328	. . . . . . . . . . . 12 |- ( <. j , k >. e. ( F supp .0. ) -> j e. dom ( F supp .0. ) )
44	42, 43	nsyl 135	. . . . . . . . . . 11 |- ( ( ph /\ ( j e. ( D \ dom ( F supp .0. ) ) /\ k e. ( A " { j } ) ) ) -> -. <. j , k >. e. ( F supp .0. ) )
45	40, 44	eldifd 3585	. . . . . . . . . 10 |- ( ( ph /\ ( j e. ( D \ dom ( F supp .0. ) ) /\ k e. ( A " { j } ) ) ) -> <. j , k >. e. ( A \ ( F supp .0. ) ) )
46		df-ov 6653	. . . . . . . . . . 11 |- ( j F k ) = ( F ` <. j , k >. )
47		ssid 3624	. . . . . . . . . . . . 13 |- ( F supp .0. ) C_ ( F supp .0. )
48	47	a1i 11	. . . . . . . . . . . 12 |- ( ph -> ( F supp .0. ) C_ ( F supp .0. ) )
49		fvex 6201	. . . . . . . . . . . . . 14 |- ( 0g ` G ) e. _V
50	2, 49	eqeltri 2697	. . . . . . . . . . . . 13 |- .0. e. _V
51	50	a1i 11	. . . . . . . . . . . 12 |- ( ph -> .0. e. _V )
52	8, 48, 4, 51	suppssr 7326	. . . . . . . . . . 11 |- ( ( ph /\ <. j , k >. e. ( A \ ( F supp .0. ) ) ) -> ( F ` <. j , k >. ) = .0. )
53	46, 52	syl5eq 2668	. . . . . . . . . 10 |- ( ( ph /\ <. j , k >. e. ( A \ ( F supp .0. ) ) ) -> ( j F k ) = .0. )
54	45, 53	syldan 487	. . . . . . . . 9 |- ( ( ph /\ ( j e. ( D \ dom ( F supp .0. ) ) /\ k e. ( A " { j } ) ) ) -> ( j F k ) = .0. )
55	54	anassrs 680	. . . . . . . 8 |- ( ( ( ph /\ j e. ( D \ dom ( F supp .0. ) ) ) /\ k e. ( A " { j } ) ) -> ( j F k ) = .0. )
56	55	mpteq2dva 4744	. . . . . . 7 |- ( ( ph /\ j e. ( D \ dom ( F supp .0. ) ) ) -> ( k e. ( A " { j } ) |-> ( j F k ) ) = ( k e. ( A " { j } ) |-> .0. ) )
57	56	oveq2d 6666	. . . . . 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 18208	. . . . . . . . 9 |- ( G e. CMnd -> G e. Mnd )
59	3, 58	syl 17	. . . . . . . 8 |- ( ph -> G e. Mnd )
60		imaexg 7103	. . . . . . . . 9 |- ( A e. V -> ( A " { j } ) e. _V )
61	4, 60	syl 17	. . . . . . . 8 |- ( ph -> ( A " { j } ) e. _V )
62	2	gsumz 17374	. . . . . . . 8 |- ( ( G e. Mnd /\ ( A " { j } ) e. _V ) -> ( G gsumg ( k e. ( A " { j } ) |-> .0. ) ) = .0. )
63	59, 61, 62	syl2anc 693	. . . . . . 7 |- ( ph -> ( G gsumg ( k e. ( A " { j } ) |-> .0. ) ) = .0. )
64	63	adantr 481	. . . . . 6 |- ( ( ph /\ j e. ( D \ dom ( F supp .0. ) ) ) -> ( G gsumg ( k e. ( A " { j } ) |-> .0. ) ) = .0. )
65	57, 64	eqtrd 2656	. . . . 5 |- ( ( ph /\ j e. ( D \ dom ( F supp .0. ) ) ) -> ( G gsumg ( k e. ( A " { j } ) |-> ( j F k ) ) ) = .0. )
66	65, 6	suppss2 7329	. . . 4 |- ( ph -> ( ( j e. D |-> ( G gsumg ( k e. ( A " { j } ) |-> ( j F k ) ) ) ) supp .0. ) C_ dom ( F supp .0. ) )
67		funmpt 5926	. . . . . 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 8282	. . . . . . 7 |- ( ph -> ( F supp .0. ) e. Fin )
70		dmfi 8244	. . . . . . 7 |- ( ( F supp .0. ) e. Fin -> dom ( F supp .0. ) e. Fin )
71	69, 70	syl 17	. . . . . 6 |- ( ph -> dom ( F supp .0. ) e. Fin )
72		ssfi 8180	. . . . . 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 693	. . . . 5 |- ( ph -> ( ( j e. D |-> ( G gsumg ( k e. ( A " { j } ) |-> ( j F k ) ) ) ) supp .0. ) e. Fin )
74		mptexg 6484	. . . . . . 7 |- ( D e. W -> ( j e. D |-> ( G gsumg ( k e. ( A " { j } ) |-> ( j F k ) ) ) ) e. _V )
75	6, 74	syl 17	. . . . . 6 |- ( ph -> ( j e. D |-> ( G gsumg ( k e. ( A " { j } ) |-> ( j F k ) ) ) ) e. _V )
76		isfsupp 8279	. . . . . 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 693	. . . . 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 957	. . . 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 18314	. . 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 2658	. 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 2664	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 196   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   \ cdif 3571   i^i cin 3573   C_ wss 3574   {csn 4177   <.cop 4183   class class class wbr 4653   |-> cmpt 4729   X. cxp 5112   dom cdm 5114   ran crn 5115   |` cres 5116   "cima 5117   Rel wrel 5119   Fun wfun 5882   -->wf 5884   ` cfv 5888  (class class class)co 6650   supp csupp 7295   Fincfn 7955   finSupp cfsupp 8275   Basecbs 15857   0gc0g 16100   gsum_g cgsu 16101   Mndcmnd 17294  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:  gsum2d2  18373  gsumxp  18375