Theorem gsummpt2d 29781

Description: Express a finite sum over a two-dimensional range as a double sum. See also gsum2d 18371. (Contributed by Thierry Arnoux, 27-Apr-2020.)

Hypotheses

Ref	Expression
gsummpt2d.c	|- F/_ z C
gsummpt2d.0	|- F/ y ph
gsummpt2d.b	|- B = ( Base ` W )
gsummpt2d.1	|- ( x = <. y , z >. -> C = D )
gsummpt2d.r	|- ( ph -> Rel A )
gsummpt2d.2	|- ( ph -> A e. Fin )
gsummpt2d.m	|- ( ph -> W e. CMnd )
gsummpt2d.3	|- ( ( ph /\ x e. A ) -> C e. B )

Assertion

Ref	Expression
gsummpt2d	|- ( ph -> ( W gsumg ( x e. A |-> C ) ) = ( W gsumg ( y e. dom A |-> ( W gsumg ( z e. ( A " { y } ) |-> D ) ) ) ) )

Distinct variable groups:   x, A, y, z   x, B, y, z   y, C   x, D   x, W, y   ph, x, z

Allowed substitution hints:   ph( y)   C( x, z)   D( y, z)   W( z)

Proof of Theorem gsummpt2d

Step	Hyp	Ref	Expression
1		gsummpt2d.b	. . 3 |- B = ( Base ` W )
2		eqid 2622	. . 3 |- ( 0g ` W ) = ( 0g ` W )
3		gsummpt2d.m	. . 3 |- ( ph -> W e. CMnd )
4		gsummpt2d.2	. . 3 |- ( ph -> A e. Fin )
5		dmexg 7097	. . . 4 |- ( A e. Fin -> dom A e. _V )
6	4, 5	syl 17	. . 3 |- ( ph -> dom A e. _V )
7		gsummpt2d.3	. . 3 |- ( ( ph /\ x e. A ) -> C e. B )
8		gsummpt2d.r	. . . 4 |- ( ph -> Rel A )
9		1stdm 7215	. . . 4 |- ( ( Rel A /\ x e. A ) -> ( 1st ` x ) e. dom A )
10	8, 9	sylan 488	. . 3 |- ( ( ph /\ x e. A ) -> ( 1st ` x ) e. dom A )
11		fo1st 7188	. . . . . 6 |- 1st : _V -onto-> _V
12		fofn 6117	. . . . . 6 |- ( 1st : _V -onto-> _V -> 1st Fn _V )
13		dffn5 6241	. . . . . . 7 |- ( 1st Fn _V <-> 1st = ( x e. _V |-> ( 1st ` x ) ) )
14	13	biimpi 206	. . . . . 6 |- ( 1st Fn _V -> 1st = ( x e. _V |-> ( 1st ` x ) ) )
15	11, 12, 14	mp2b 10	. . . . 5 |- 1st = ( x e. _V |-> ( 1st ` x ) )
16	15	reseq1i 5392	. . . 4 |- ( 1st |` A ) = ( ( x e. _V |-> ( 1st ` x ) ) |` A )
17		ssv 3625	. . . . 5 |- A C_ _V
18		resmpt 5449	. . . . 5 |- ( A C_ _V -> ( ( x e. _V |-> ( 1st ` x ) ) |` A ) = ( x e. A |-> ( 1st ` x ) ) )
19	17, 18	ax-mp 5	. . . 4 |- ( ( x e. _V |-> ( 1st ` x ) ) |` A ) = ( x e. A |-> ( 1st ` x ) )
20	16, 19	eqtri 2644	. . 3 |- ( 1st |` A ) = ( x e. A |-> ( 1st ` x ) )
21	1, 2, 3, 4, 6, 7, 10, 20	gsummpt2co 29780	. 2 |- ( ph -> ( W gsumg ( x e. A |-> C ) ) = ( W gsumg ( y e. dom A |-> ( W gsumg ( x e. ( `' ( 1st |` A ) " { y } ) |-> C ) ) ) ) )
22		gsummpt2d.0	. . . 4 |- F/ y ph
23	3	adantr 481	. . . . . 6 |- ( ( ph /\ y e. dom A ) -> W e. CMnd )
24	4	adantr 481	. . . . . . 7 |- ( ( ph /\ y e. dom A ) -> A e. Fin )
25		imaexg 7103	. . . . . . 7 |- ( A e. Fin -> ( A " { y } ) e. _V )
26	24, 25	syl 17	. . . . . 6 |- ( ( ph /\ y e. dom A ) -> ( A " { y } ) e. _V )
27		gsummpt2d.1	. . . . . . . . . 10 |- ( x = <. y , z >. -> C = D )
28	27	adantl 482	. . . . . . . . 9 |- ( ( ( ( ( ph /\ y e. dom A ) /\ z e. ( A " { y } ) ) /\ x e. A ) /\ x = <. y , z >. ) -> C = D )
29		simp-4l 806	. . . . . . . . . 10 |- ( ( ( ( ( ph /\ y e. dom A ) /\ z e. ( A " { y } ) ) /\ x e. A ) /\ x = <. y , z >. ) -> ph )
30		simplr 792	. . . . . . . . . 10 |- ( ( ( ( ( ph /\ y e. dom A ) /\ z e. ( A " { y } ) ) /\ x e. A ) /\ x = <. y , z >. ) -> x e. A )
31	29, 30, 7	syl2anc 693	. . . . . . . . 9 |- ( ( ( ( ( ph /\ y e. dom A ) /\ z e. ( A " { y } ) ) /\ x e. A ) /\ x = <. y , z >. ) -> C e. B )
32	28, 31	eqeltrrd 2702	. . . . . . . 8 |- ( ( ( ( ( ph /\ y e. dom A ) /\ z e. ( A " { y } ) ) /\ x e. A ) /\ x = <. y , z >. ) -> D e. B )
33		vex 3203	. . . . . . . . . . . 12 |- y e. _V
34		vex 3203	. . . . . . . . . . . 12 |- z e. _V
35	33, 34	elimasn 5490	. . . . . . . . . . 11 |- ( z e. ( A " { y } ) <-> <. y , z >. e. A )
36	35	biimpi 206	. . . . . . . . . 10 |- ( z e. ( A " { y } ) -> <. y , z >. e. A )
37	36	adantl 482	. . . . . . . . 9 |- ( ( ( ph /\ y e. dom A ) /\ z e. ( A " { y } ) ) -> <. y , z >. e. A )
38		simpr 477	. . . . . . . . . 10 |- ( ( ( ( ph /\ y e. dom A ) /\ z e. ( A " { y } ) ) /\ x = <. y , z >. ) -> x = <. y , z >. )
39	38	eqeq1d 2624	. . . . . . . . 9 |- ( ( ( ( ph /\ y e. dom A ) /\ z e. ( A " { y } ) ) /\ x = <. y , z >. ) -> ( x = <. y , z >. <-> <. y , z >. = <. y , z >. ) )
40		eqidd 2623	. . . . . . . . 9 |- ( ( ( ph /\ y e. dom A ) /\ z e. ( A " { y } ) ) -> <. y , z >. = <. y , z >. )
41	37, 39, 40	rspcedvd 3317	. . . . . . . 8 |- ( ( ( ph /\ y e. dom A ) /\ z e. ( A " { y } ) ) -> E. x e. A x = <. y , z >. )
42	32, 41	r19.29a 3078	. . . . . . 7 |- ( ( ( ph /\ y e. dom A ) /\ z e. ( A " { y } ) ) -> D e. B )
43		eqid 2622	. . . . . . 7 |- ( z e. ( A " { y } ) |-> D ) = ( z e. ( A " { y } ) |-> D )
44	42, 43	fmptd 6385	. . . . . 6 |- ( ( ph /\ y e. dom A ) -> ( z e. ( A " { y } ) |-> D ) : ( A " { y } ) --> B )
45		imafi2 29489	. . . . . . . . 9 |- ( A e. Fin -> ( A " { y } ) e. Fin )
46	4, 45	syl 17	. . . . . . . 8 |- ( ph -> ( A " { y } ) e. Fin )
47	46	adantr 481	. . . . . . 7 |- ( ( ph /\ y e. dom A ) -> ( A " { y } ) e. Fin )
48		fvex 6201	. . . . . . . 8 |- ( 0g ` W ) e. _V
49	48	a1i 11	. . . . . . 7 |- ( ( ph /\ y e. dom A ) -> ( 0g ` W ) e. _V )
50	43, 47, 42, 49	fsuppmptdm 8286	. . . . . 6 |- ( ( ph /\ y e. dom A ) -> ( z e. ( A " { y } ) |-> D ) finSupp ( 0g ` W ) )
51		2ndconst 7266	. . . . . . . 8 |- ( y e. dom A -> ( 2nd |` ( { y } X. ( A " { y } ) ) ) : ( { y } X. ( A " { y } ) ) -1-1-onto-> ( A " { y } ) )
52	51	adantl 482	. . . . . . 7 |- ( ( ph /\ y e. dom A ) -> ( 2nd |` ( { y } X. ( A " { y } ) ) ) : ( { y } X. ( A " { y } ) ) -1-1-onto-> ( A " { y } ) )
53		1stpreimas 29483	. . . . . . . . . 10 |- ( ( Rel A /\ y e. dom A ) -> ( `' ( 1st |` A ) " { y } ) = ( { y } X. ( A " { y } ) ) )
54	8, 53	sylan 488	. . . . . . . . 9 |- ( ( ph /\ y e. dom A ) -> ( `' ( 1st |` A ) " { y } ) = ( { y } X. ( A " { y } ) ) )
55	54	reseq2d 5396	. . . . . . . 8 |- ( ( ph /\ y e. dom A ) -> ( 2nd |` ( `' ( 1st |` A ) " { y } ) ) = ( 2nd |` ( { y } X. ( A " { y } ) ) ) )
56		f1oeq1 6127	. . . . . . . 8 |- ( ( 2nd |` ( `' ( 1st |` A ) " { y } ) ) = ( 2nd |` ( { y } X. ( A " { y } ) ) ) -> ( ( 2nd |` ( `' ( 1st |` A ) " { y } ) ) : ( { y } X. ( A " { y } ) ) -1-1-onto-> ( A " { y } ) <-> ( 2nd |` ( { y } X. ( A " { y } ) ) ) : ( { y } X. ( A " { y } ) ) -1-1-onto-> ( A " { y } ) ) )
57	55, 56	syl 17	. . . . . . 7 |- ( ( ph /\ y e. dom A ) -> ( ( 2nd |` ( `' ( 1st |` A ) " { y } ) ) : ( { y } X. ( A " { y } ) ) -1-1-onto-> ( A " { y } ) <-> ( 2nd |` ( { y } X. ( A " { y } ) ) ) : ( { y } X. ( A " { y } ) ) -1-1-onto-> ( A " { y } ) ) )
58	52, 57	mpbird 247	. . . . . 6 |- ( ( ph /\ y e. dom A ) -> ( 2nd |` ( `' ( 1st |` A ) " { y } ) ) : ( { y } X. ( A " { y } ) ) -1-1-onto-> ( A " { y } ) )
59	1, 2, 23, 26, 44, 50, 58	gsumf1o 18317	. . . . 5 |- ( ( ph /\ y e. dom A ) -> ( W gsumg ( z e. ( A " { y } ) |-> D ) ) = ( W gsumg ( ( z e. ( A " { y } ) |-> D ) o. ( 2nd |` ( `' ( 1st |` A ) " { y } ) ) ) ) )
60		simpr 477	. . . . . . . . . . 11 |- ( ( ( ph /\ y e. dom A ) /\ x e. ( `' ( 1st |` A ) " { y } ) ) -> x e. ( `' ( 1st |` A ) " { y } ) )
61	54	adantr 481	. . . . . . . . . . 11 |- ( ( ( ph /\ y e. dom A ) /\ x e. ( `' ( 1st |` A ) " { y } ) ) -> ( `' ( 1st |` A ) " { y } ) = ( { y } X. ( A " { y } ) ) )
62	60, 61	eleqtrd 2703	. . . . . . . . . 10 |- ( ( ( ph /\ y e. dom A ) /\ x e. ( `' ( 1st |` A ) " { y } ) ) -> x e. ( { y } X. ( A " { y } ) ) )
63		xp2nd 7199	. . . . . . . . . 10 |- ( x e. ( { y } X. ( A " { y } ) ) -> ( 2nd ` x ) e. ( A " { y } ) )
64	62, 63	syl 17	. . . . . . . . 9 |- ( ( ( ph /\ y e. dom A ) /\ x e. ( `' ( 1st |` A ) " { y } ) ) -> ( 2nd ` x ) e. ( A " { y } ) )
65	64	ralrimiva 2966	. . . . . . . 8 |- ( ( ph /\ y e. dom A ) -> A. x e. ( `' ( 1st |` A ) " { y } ) ( 2nd ` x ) e. ( A " { y } ) )
66		fo2nd 7189	. . . . . . . . . . . 12 |- 2nd : _V -onto-> _V
67		fofn 6117	. . . . . . . . . . . 12 |- ( 2nd : _V -onto-> _V -> 2nd Fn _V )
68		dffn5 6241	. . . . . . . . . . . . 13 |- ( 2nd Fn _V <-> 2nd = ( x e. _V |-> ( 2nd ` x ) ) )
69	68	biimpi 206	. . . . . . . . . . . 12 |- ( 2nd Fn _V -> 2nd = ( x e. _V |-> ( 2nd ` x ) ) )
70	66, 67, 69	mp2b 10	. . . . . . . . . . 11 |- 2nd = ( x e. _V |-> ( 2nd ` x ) )
71	70	reseq1i 5392	. . . . . . . . . 10 |- ( 2nd |` ( `' ( 1st |` A ) " { y } ) ) = ( ( x e. _V |-> ( 2nd ` x ) ) |` ( `' ( 1st |` A ) " { y } ) )
72		ssv 3625	. . . . . . . . . . 11 |- ( `' ( 1st |` A ) " { y } ) C_ _V
73		resmpt 5449	. . . . . . . . . . 11 |- ( ( `' ( 1st |` A ) " { y } ) C_ _V -> ( ( x e. _V |-> ( 2nd ` x ) ) |` ( `' ( 1st |` A ) " { y } ) ) = ( x e. ( `' ( 1st |` A ) " { y } ) |-> ( 2nd ` x ) ) )
74	72, 73	ax-mp 5	. . . . . . . . . 10 |- ( ( x e. _V |-> ( 2nd ` x ) ) |` ( `' ( 1st |` A ) " { y } ) ) = ( x e. ( `' ( 1st |` A ) " { y } ) |-> ( 2nd ` x ) )
75	71, 74	eqtri 2644	. . . . . . . . 9 |- ( 2nd |` ( `' ( 1st |` A ) " { y } ) ) = ( x e. ( `' ( 1st |` A ) " { y } ) |-> ( 2nd ` x ) )
76	75	a1i 11	. . . . . . . 8 |- ( ( ph /\ y e. dom A ) -> ( 2nd |` ( `' ( 1st |` A ) " { y } ) ) = ( x e. ( `' ( 1st |` A ) " { y } ) |-> ( 2nd ` x ) ) )
77		eqidd 2623	. . . . . . . 8 |- ( ( ph /\ y e. dom A ) -> ( z e. ( A " { y } ) |-> D ) = ( z e. ( A " { y } ) |-> D ) )
78	65, 76, 77	fmptcos 6398	. . . . . . 7 |- ( ( ph /\ y e. dom A ) -> ( ( z e. ( A " { y } ) |-> D ) o. ( 2nd |` ( `' ( 1st |` A ) " { y } ) ) ) = ( x e. ( `' ( 1st |` A ) " { y } ) |-> [_ ( 2nd ` x ) / z ]_ D ) )
79		nfv 1843	. . . . . . . . 9 |- F/ z ( ( ph /\ y e. dom A ) /\ x e. ( `' ( 1st |` A ) " { y } ) )
80		gsummpt2d.c	. . . . . . . . . 10 |- F/_ z C
81	80	a1i 11	. . . . . . . . 9 |- ( ( ( ph /\ y e. dom A ) /\ x e. ( `' ( 1st |` A ) " { y } ) ) -> F/_ z C )
82	62	adantr 481	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ y e. dom A ) /\ x e. ( `' ( 1st |` A ) " { y } ) ) /\ z = ( 2nd ` x ) ) -> x e. ( { y } X. ( A " { y } ) ) )
83		xp1st 7198	. . . . . . . . . . . . . 14 |- ( x e. ( { y } X. ( A " { y } ) ) -> ( 1st ` x ) e. { y } )
84	82, 83	syl 17	. . . . . . . . . . . . 13 |- ( ( ( ( ph /\ y e. dom A ) /\ x e. ( `' ( 1st |` A ) " { y } ) ) /\ z = ( 2nd ` x ) ) -> ( 1st ` x ) e. { y } )
85		fvex 6201	. . . . . . . . . . . . . 14 |- ( 1st ` x ) e. _V
86	85	elsn 4192	. . . . . . . . . . . . 13 |- ( ( 1st ` x ) e. { y } <-> ( 1st ` x ) = y )
87	84, 86	sylib 208	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ y e. dom A ) /\ x e. ( `' ( 1st |` A ) " { y } ) ) /\ z = ( 2nd ` x ) ) -> ( 1st ` x ) = y )
88		simpr 477	. . . . . . . . . . . . 13 |- ( ( ( ( ph /\ y e. dom A ) /\ x e. ( `' ( 1st |` A ) " { y } ) ) /\ z = ( 2nd ` x ) ) -> z = ( 2nd ` x ) )
89	88	eqcomd 2628	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ y e. dom A ) /\ x e. ( `' ( 1st |` A ) " { y } ) ) /\ z = ( 2nd ` x ) ) -> ( 2nd ` x ) = z )
90		eqopi 7202	. . . . . . . . . . . 12 |- ( ( x e. ( { y } X. ( A " { y } ) ) /\ ( ( 1st ` x ) = y /\ ( 2nd ` x ) = z ) ) -> x = <. y , z >. )
91	82, 87, 89, 90	syl12anc 1324	. . . . . . . . . . 11 |- ( ( ( ( ph /\ y e. dom A ) /\ x e. ( `' ( 1st |` A ) " { y } ) ) /\ z = ( 2nd ` x ) ) -> x = <. y , z >. )
92	91, 27	syl 17	. . . . . . . . . 10 |- ( ( ( ( ph /\ y e. dom A ) /\ x e. ( `' ( 1st |` A ) " { y } ) ) /\ z = ( 2nd ` x ) ) -> C = D )
93	92	eqcomd 2628	. . . . . . . . 9 |- ( ( ( ( ph /\ y e. dom A ) /\ x e. ( `' ( 1st |` A ) " { y } ) ) /\ z = ( 2nd ` x ) ) -> D = C )
94	79, 81, 64, 93	csbiedf 3554	. . . . . . . 8 |- ( ( ( ph /\ y e. dom A ) /\ x e. ( `' ( 1st |` A ) " { y } ) ) -> [_ ( 2nd ` x ) / z ]_ D = C )
95	94	mpteq2dva 4744	. . . . . . 7 |- ( ( ph /\ y e. dom A ) -> ( x e. ( `' ( 1st |` A ) " { y } ) |-> [_ ( 2nd ` x ) / z ]_ D ) = ( x e. ( `' ( 1st |` A ) " { y } ) |-> C ) )
96	78, 95	eqtrd 2656	. . . . . 6 |- ( ( ph /\ y e. dom A ) -> ( ( z e. ( A " { y } ) |-> D ) o. ( 2nd |` ( `' ( 1st |` A ) " { y } ) ) ) = ( x e. ( `' ( 1st |` A ) " { y } ) |-> C ) )
97	96	oveq2d 6666	. . . . 5 |- ( ( ph /\ y e. dom A ) -> ( W gsumg ( ( z e. ( A " { y } ) |-> D ) o. ( 2nd |` ( `' ( 1st |` A ) " { y } ) ) ) ) = ( W gsumg ( x e. ( `' ( 1st |` A ) " { y } ) |-> C ) ) )
98	59, 97	eqtr2d 2657	. . . 4 |- ( ( ph /\ y e. dom A ) -> ( W gsumg ( x e. ( `' ( 1st |` A ) " { y } ) |-> C ) ) = ( W gsumg ( z e. ( A " { y } ) |-> D ) ) )
99	22, 98	mpteq2da 4743	. . 3 |- ( ph -> ( y e. dom A |-> ( W gsumg ( x e. ( `' ( 1st |` A ) " { y } ) |-> C ) ) ) = ( y e. dom A |-> ( W gsumg ( z e. ( A " { y } ) |-> D ) ) ) )
100	99	oveq2d 6666	. 2 |- ( ph -> ( W gsumg ( y e. dom A |-> ( W gsumg ( x e. ( `' ( 1st |` A ) " { y } ) |-> C ) ) ) ) = ( W gsumg ( y e. dom A |-> ( W gsumg ( z e. ( A " { y } ) |-> D ) ) ) ) )
101	21, 100	eqtrd 2656	1 |- ( ph -> ( W gsumg ( x e. A |-> C ) ) = ( W gsumg ( y e. dom A |-> ( W gsumg ( z e. ( A " { y } ) |-> D ) ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   F/wnf 1708   e. wcel 1990   F/_wnfc 2751   _Vcvv 3200   [_csb 3533   C_ wss 3574   {csn 4177   <.cop 4183   |-> cmpt 4729   X. cxp 5112   `'ccnv 5113   dom cdm 5114   |` cres 5116   "cima 5117   o. ccom 5118   Rel wrel 5119   Fn wfn 5883   -onto->wfo 5886   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   1stc1st 7166   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:  esum2d  30155