Theorem gsumzres 16388

Description: Extend a finite group sum by padding outside with zeroes. (Contributed by Mario Carneiro, 24-Apr-2016.) (Revised by AV, 31-May-2019.)

Hypotheses

Ref	Expression
gsumzcl.b	|- B = ( Base ` G )
gsumzcl.0	|- .0. = ( 0g ` G )
gsumzcl.z	|- Z = (Cntz ` G )
gsumzcl.g	|- ( ph -> G e. Mnd )
gsumzcl.a	|- ( ph -> A e. V )
gsumzcl.f	|- ( ph -> F : A --> B )
gsumzcl.c	|- ( ph -> ran F C_ ( Z ` ran F ) )
gsumzres.s	|- ( ph -> ( F supp .0. ) C_ W )
gsumzres.w	|- ( ph -> F finSupp .0. )

Assertion

Ref	Expression
gsumzres	|- ( ph -> ( G gsumg ( F |` W ) ) = ( G gsumg F ) )

Proof of Theorem gsumzres

Dummy variables f k are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		gsumzcl.g	. . . . . . 7 |- ( ph -> G e. Mnd )
2		gsumzcl.a	. . . . . . . 8 |- ( ph -> A e. V )
3		inex1g 4435	. . . . . . . 8 |- ( A e. V -> ( A i^i W ) e. _V )
4	2, 3	syl 16	. . . . . . 7 |- ( ph -> ( A i^i W ) e. _V )
5		gsumzcl.0	. . . . . . . 8 |- .0. = ( 0g ` G )
6	5	gsumz 15511	. . . . . . 7 |- ( ( G e. Mnd /\ ( A i^i W ) e. _V ) -> ( G gsumg ( k e. ( A i^i W ) |-> .0. ) ) = .0. )
7	1, 4, 6	syl2anc 661	. . . . . 6 |- ( ph -> ( G gsumg ( k e. ( A i^i W ) |-> .0. ) ) = .0. )
8	5	gsumz 15511	. . . . . . 7 |- ( ( G e. Mnd /\ A e. V ) -> ( G gsumg ( k e. A |-> .0. ) ) = .0. )
9	1, 2, 8	syl2anc 661	. . . . . 6 |- ( ph -> ( G gsumg ( k e. A |-> .0. ) ) = .0. )
10	7, 9	eqtr4d 2478	. . . . 5 |- ( ph -> ( G gsumg ( k e. ( A i^i W ) |-> .0. ) ) = ( G gsumg ( k e. A |-> .0. ) ) )
11	10	adantr 465	. . . 4 |- ( ( ph /\ ( F supp .0. ) = (/) ) -> ( G gsumg ( k e. ( A i^i W ) |-> .0. ) ) = ( G gsumg ( k e. A |-> .0. ) ) )
12		resres 5123	. . . . . . . 8 |- ( ( F |` A ) |` W ) = ( F |` ( A i^i W ) )
13		gsumzcl.f	. . . . . . . . . 10 |- ( ph -> F : A --> B )
14		ffn 5559	. . . . . . . . . 10 |- ( F : A --> B -> F Fn A )
15		fnresdm 5520	. . . . . . . . . 10 |- ( F Fn A -> ( F |` A ) = F )
16	13, 14, 15	3syl 20	. . . . . . . . 9 |- ( ph -> ( F |` A ) = F )
17	16	reseq1d 5109	. . . . . . . 8 |- ( ph -> ( ( F |` A ) |` W ) = ( F |` W ) )
18	12, 17	syl5eqr 2489	. . . . . . 7 |- ( ph -> ( F |` ( A i^i W ) ) = ( F |` W ) )
19	18	adantr 465	. . . . . 6 |- ( ( ph /\ ( F supp .0. ) = (/) ) -> ( F |` ( A i^i W ) ) = ( F |` W ) )
20		fvex 5701	. . . . . . . . . . 11 |- ( 0g ` G ) e. _V
21	5, 20	eqeltri 2513	. . . . . . . . . 10 |- .0. e. _V
22	21	a1i 11	. . . . . . . . 9 |- ( ph -> .0. e. _V )
23		ssid 3375	. . . . . . . . . 10 |- ( F supp .0. ) C_ ( F supp .0. )
24	23	a1i 11	. . . . . . . . 9 |- ( ph -> ( F supp .0. ) C_ ( F supp .0. ) )
25	13, 2, 22, 24	gsumcllem 16386	. . . . . . . 8 |- ( ( ph /\ ( F supp .0. ) = (/) ) -> F = ( k e. A |-> .0. ) )
26	25	reseq1d 5109	. . . . . . 7 |- ( ( ph /\ ( F supp .0. ) = (/) ) -> ( F |` ( A i^i W ) ) = ( ( k e. A |-> .0. ) |` ( A i^i W ) ) )
27		inss1 3570	. . . . . . . 8 |- ( A i^i W ) C_ A
28		resmpt 5156	. . . . . . . 8 |- ( ( A i^i W ) C_ A -> ( ( k e. A |-> .0. ) |` ( A i^i W ) ) = ( k e. ( A i^i W ) |-> .0. ) )
29	27, 28	ax-mp 5	. . . . . . 7 |- ( ( k e. A |-> .0. ) |` ( A i^i W ) ) = ( k e. ( A i^i W ) |-> .0. )
30	26, 29	syl6eq 2491	. . . . . 6 |- ( ( ph /\ ( F supp .0. ) = (/) ) -> ( F |` ( A i^i W ) ) = ( k e. ( A i^i W ) |-> .0. ) )
31	19, 30	eqtr3d 2477	. . . . 5 |- ( ( ph /\ ( F supp .0. ) = (/) ) -> ( F |` W ) = ( k e. ( A i^i W ) |-> .0. ) )
32	31	oveq2d 6107	. . . 4 |- ( ( ph /\ ( F supp .0. ) = (/) ) -> ( G gsumg ( F |` W ) ) = ( G gsumg ( k e. ( A i^i W ) |-> .0. ) ) )
33	25	oveq2d 6107	. . . 4 |- ( ( ph /\ ( F supp .0. ) = (/) ) -> ( G gsumg F ) = ( G gsumg ( k e. A |-> .0. ) ) )
34	11, 32, 33	3eqtr4d 2485	. . 3 |- ( ( ph /\ ( F supp .0. ) = (/) ) -> ( G gsumg ( F |` W ) ) = ( G gsumg F ) )
35	34	ex 434	. 2 |- ( ph -> ( ( F supp .0. ) = (/) -> ( G gsumg ( F |` W ) ) = ( G gsumg F ) ) )
36		f1ofo 5648	. . . . . . . . . . . 12 |- ( f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) -> f : ( 1 ... ( # ` ( F supp .0. ) ) ) -onto-> ( F supp .0. ) )
37		forn 5623	. . . . . . . . . . . 12 |- ( f : ( 1 ... ( # ` ( F supp .0. ) ) ) -onto-> ( F supp .0. ) -> ran f = ( F supp .0. ) )
38	36, 37	syl 16	. . . . . . . . . . 11 |- ( f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) -> ran f = ( F supp .0. ) )
39	38	ad2antll 728	. . . . . . . . . 10 |- ( ( ph /\ ( ( # ` ( F supp .0. ) ) e. NN /\ f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) ) ) -> ran f = ( F supp .0. ) )
40		gsumzres.s	. . . . . . . . . . 11 |- ( ph -> ( F supp .0. ) C_ W )
41	40	adantr 465	. . . . . . . . . 10 |- ( ( ph /\ ( ( # ` ( F supp .0. ) ) e. NN /\ f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) ) ) -> ( F supp .0. ) C_ W )
42	39, 41	eqsstrd 3390	. . . . . . . . 9 |- ( ( ph /\ ( ( # ` ( F supp .0. ) ) e. NN /\ f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) ) ) -> ran f C_ W )
43		cores 5341	. . . . . . . . 9 |- ( ran f C_ W -> ( ( F |` W ) o. f ) = ( F o. f ) )
44	42, 43	syl 16	. . . . . . . 8 |- ( ( ph /\ ( ( # ` ( F supp .0. ) ) e. NN /\ f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) ) ) -> ( ( F |` W ) o. f ) = ( F o. f ) )
45	44	seqeq3d 11814	. . . . . . 7 |- ( ( ph /\ ( ( # ` ( F supp .0. ) ) e. NN /\ f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) ) ) -> seq 1 ( ( +g ` G ) , ( ( F |` W ) o. f ) ) = seq 1 ( ( +g ` G ) , ( F o. f ) ) )
46	45	fveq1d 5693	. . . . . 6 |- ( ( ph /\ ( ( # ` ( F supp .0. ) ) e. NN /\ f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) ) ) -> ( seq 1 ( ( +g ` G ) , ( ( F |` W ) o. f ) ) ` ( # ` ( F supp .0. ) ) ) = ( seq 1 ( ( +g ` G ) , ( F o. f ) ) ` ( # ` ( F supp .0. ) ) ) )
47		gsumzcl.b	. . . . . . 7 |- B = ( Base ` G )
48		eqid 2443	. . . . . . 7 |- ( +g ` G ) = ( +g ` G )
49		gsumzcl.z	. . . . . . 7 |- Z = (Cntz ` G )
50	1	adantr 465	. . . . . . 7 |- ( ( ph /\ ( ( # ` ( F supp .0. ) ) e. NN /\ f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) ) ) -> G e. Mnd )
51	4	adantr 465	. . . . . . 7 |- ( ( ph /\ ( ( # ` ( F supp .0. ) ) e. NN /\ f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) ) ) -> ( A i^i W ) e. _V )
52	13	adantr 465	. . . . . . . . 9 |- ( ( ph /\ ( ( # ` ( F supp .0. ) ) e. NN /\ f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) ) ) -> F : A --> B )
53		fssres 5578	. . . . . . . . 9 |- ( ( F : A --> B /\ ( A i^i W ) C_ A ) -> ( F |` ( A i^i W ) ) : ( A i^i W ) --> B )
54	52, 27, 53	sylancl 662	. . . . . . . 8 |- ( ( ph /\ ( ( # ` ( F supp .0. ) ) e. NN /\ f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) ) ) -> ( F |` ( A i^i W ) ) : ( A i^i W ) --> B )
55	18	feq1d 5546	. . . . . . . . 9 |- ( ph -> ( ( F |` ( A i^i W ) ) : ( A i^i W ) --> B <-> ( F |` W ) : ( A i^i W ) --> B ) )
56	55	biimpa 484	. . . . . . . 8 |- ( ( ph /\ ( F |` ( A i^i W ) ) : ( A i^i W ) --> B ) -> ( F |` W ) : ( A i^i W ) --> B )
57	54, 56	syldan 470	. . . . . . 7 |- ( ( ph /\ ( ( # ` ( F supp .0. ) ) e. NN /\ f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) ) ) -> ( F |` W ) : ( A i^i W ) --> B )
58		gsumzcl.c	. . . . . . . . 9 |- ( ph -> ran F C_ ( Z ` ran F ) )
59		resss 5134	. . . . . . . . . 10 |- ( F |` W ) C_ F
60		rnss 5068	. . . . . . . . . 10 |- ( ( F |` W ) C_ F -> ran ( F |` W ) C_ ran F )
61	59, 60	ax-mp 5	. . . . . . . . 9 |- ran ( F |` W ) C_ ran F
62	49	cntzidss 15855	. . . . . . . . 9 |- ( ( ran F C_ ( Z ` ran F ) /\ ran ( F |` W ) C_ ran F ) -> ran ( F |` W ) C_ ( Z ` ran ( F |` W ) ) )
63	58, 61, 62	sylancl 662	. . . . . . . 8 |- ( ph -> ran ( F |` W ) C_ ( Z ` ran ( F |` W ) ) )
64	63	adantr 465	. . . . . . 7 |- ( ( ph /\ ( ( # ` ( F supp .0. ) ) e. NN /\ f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) ) ) -> ran ( F |` W ) C_ ( Z ` ran ( F |` W ) ) )
65		simprl 755	. . . . . . 7 |- ( ( ph /\ ( ( # ` ( F supp .0. ) ) e. NN /\ f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) ) ) -> ( # ` ( F supp .0. ) ) e. NN )
66		f1of1 5640	. . . . . . . . 9 |- ( f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) -> f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-> ( F supp .0. ) )
67	66	ad2antll 728	. . . . . . . 8 |- ( ( ph /\ ( ( # ` ( F supp .0. ) ) e. NN /\ f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) ) ) -> f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-> ( F supp .0. ) )
68		suppssdm 6703	. . . . . . . . . . 11 |- ( F supp .0. ) C_ dom F
69		fdm 5563	. . . . . . . . . . . 12 |- ( F : A --> B -> dom F = A )
70	13, 69	syl 16	. . . . . . . . . . 11 |- ( ph -> dom F = A )
71	68, 70	syl5sseq 3404	. . . . . . . . . 10 |- ( ph -> ( F supp .0. ) C_ A )
72	71, 40	ssind 3574	. . . . . . . . 9 |- ( ph -> ( F supp .0. ) C_ ( A i^i W ) )
73	72	adantr 465	. . . . . . . 8 |- ( ( ph /\ ( ( # ` ( F supp .0. ) ) e. NN /\ f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) ) ) -> ( F supp .0. ) C_ ( A i^i W ) )
74		f1ss 5611	. . . . . . . 8 |- ( ( f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-> ( F supp .0. ) /\ ( F supp .0. ) C_ ( A i^i W ) ) -> f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-> ( A i^i W ) )
75	67, 73, 74	syl2anc 661	. . . . . . 7 |- ( ( ph /\ ( ( # ` ( F supp .0. ) ) e. NN /\ f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) ) ) -> f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-> ( A i^i W ) )
76		fex 5950	. . . . . . . . . . . . 13 |- ( ( F : A --> B /\ A e. V ) -> F e. _V )
77	13, 2, 76	syl2anc 661	. . . . . . . . . . . 12 |- ( ph -> F e. _V )
78		ressuppss 6708	. . . . . . . . . . . 12 |- ( ( F e. _V /\ .0. e. _V ) -> ( ( F |` W ) supp .0. ) C_ ( F supp .0. ) )
79	77, 21, 78	sylancl 662	. . . . . . . . . . 11 |- ( ph -> ( ( F |` W ) supp .0. ) C_ ( F supp .0. ) )
80		sseq2 3378	. . . . . . . . . . 11 |- ( ran f = ( F supp .0. ) -> ( ( ( F |` W ) supp .0. ) C_ ran f <-> ( ( F |` W ) supp .0. ) C_ ( F supp .0. ) ) )
81	79, 80	syl5ibr 221	. . . . . . . . . 10 |- ( ran f = ( F supp .0. ) -> ( ph -> ( ( F |` W ) supp .0. ) C_ ran f ) )
82	36, 37, 81	3syl 20	. . . . . . . . 9 |- ( f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) -> ( ph -> ( ( F |` W ) supp .0. ) C_ ran f ) )
83	82	adantl 466	. . . . . . . 8 |- ( ( ( # ` ( F supp .0. ) ) e. NN /\ f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) ) -> ( ph -> ( ( F |` W ) supp .0. ) C_ ran f ) )
84	83	impcom 430	. . . . . . 7 |- ( ( ph /\ ( ( # ` ( F supp .0. ) ) e. NN /\ f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) ) ) -> ( ( F |` W ) supp .0. ) C_ ran f )
85		eqid 2443	. . . . . . 7 |- ( ( ( F |` W ) o. f ) supp .0. ) = ( ( ( F |` W ) o. f ) supp .0. )
86	47, 5, 48, 49, 50, 51, 57, 64, 65, 75, 84, 85	gsumval3 16385	. . . . . 6 |- ( ( ph /\ ( ( # ` ( F supp .0. ) ) e. NN /\ f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) ) ) -> ( G gsumg ( F |` W ) ) = ( seq 1 ( ( +g ` G ) , ( ( F |` W ) o. f ) ) ` ( # ` ( F supp .0. ) ) ) )
87	2	adantr 465	. . . . . . 7 |- ( ( ph /\ ( ( # ` ( F supp .0. ) ) e. NN /\ f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) ) ) -> A e. V )
88	58	adantr 465	. . . . . . 7 |- ( ( ph /\ ( ( # ` ( F supp .0. ) ) e. NN /\ f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) ) ) -> ran F C_ ( Z ` ran F ) )
89	71	adantr 465	. . . . . . . 8 |- ( ( ph /\ ( ( # ` ( F supp .0. ) ) e. NN /\ f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) ) ) -> ( F supp .0. ) C_ A )
90		f1ss 5611	. . . . . . . 8 |- ( ( f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-> ( F supp .0. ) /\ ( F supp .0. ) C_ A ) -> f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-> A )
91	67, 89, 90	syl2anc 661	. . . . . . 7 |- ( ( ph /\ ( ( # ` ( F supp .0. ) ) e. NN /\ f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) ) ) -> f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-> A )
92	23, 39	syl5sseqr 3405	. . . . . . 7 |- ( ( ph /\ ( ( # ` ( F supp .0. ) ) e. NN /\ f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) ) ) -> ( F supp .0. ) C_ ran f )
93		eqid 2443	. . . . . . 7 |- ( ( F o. f ) supp .0. ) = ( ( F o. f ) supp .0. )
94	47, 5, 48, 49, 50, 87, 52, 88, 65, 91, 92, 93	gsumval3 16385	. . . . . 6 |- ( ( ph /\ ( ( # ` ( F supp .0. ) ) e. NN /\ f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) ) ) -> ( G gsumg F ) = ( seq 1 ( ( +g ` G ) , ( F o. f ) ) ` ( # ` ( F supp .0. ) ) ) )
95	46, 86, 94	3eqtr4d 2485	. . . . 5 |- ( ( ph /\ ( ( # ` ( F supp .0. ) ) e. NN /\ f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) ) ) -> ( G gsumg ( F |` W ) ) = ( G gsumg F ) )
96	95	expr 615	. . . 4 |- ( ( ph /\ ( # ` ( F supp .0. ) ) e. NN ) -> ( f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) -> ( G gsumg ( F |` W ) ) = ( G gsumg F ) ) )
97	96	exlimdv 1690	. . 3 |- ( ( ph /\ ( # ` ( F supp .0. ) ) e. NN ) -> ( E. f f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) -> ( G gsumg ( F |` W ) ) = ( G gsumg F ) ) )
98	97	expimpd 603	. 2 |- ( ph -> ( ( ( # ` ( F supp .0. ) ) e. NN /\ E. f f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) ) -> ( G gsumg ( F |` W ) ) = ( G gsumg F ) ) )
99		gsumzres.w	. . 3 |- ( ph -> F finSupp .0. )
100		fsuppimp 7626	. . . 4 |- ( F finSupp .0. -> ( Fun F /\ ( F supp .0. ) e. Fin ) )
101	100	simprd 463	. . 3 |- ( F finSupp .0. -> ( F supp .0. ) e. Fin )
102		fz1f1o 13187	. . 3 |- ( ( F supp .0. ) e. Fin -> ( ( F supp .0. ) = (/) \/ ( ( # ` ( F supp .0. ) ) e. NN /\ E. f f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) ) ) )
103	99, 101, 102	3syl 20	. 2 |- ( ph -> ( ( F supp .0. ) = (/) \/ ( ( # ` ( F supp .0. ) ) e. NN /\ E. f f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) ) ) )
104	35, 98, 103	mpjaod 381	1 |- ( ph -> ( G gsumg ( F |` W ) ) = ( G gsumg F ) )

Syntax hints:   -> wi 4   \/ wo 368   /\ wa 369   = wceq 1369   E.wex 1586   e. wcel 1756   _Vcvv 2972   i^i cin 3327   C_ wss 3328   (/)c0 3637   class class class wbr 4292   e. cmpt 4350   dom cdm 4840   ran crn 4841   |` cres 4842   o. ccom 4844   Fun wfun 5412   Fn wfn 5413   -->wf 5414   -1-1->wf1 5415   -onto->wfo 5416   -1-1-onto->wf1o 5417   ` cfv 5418  (class class class)co 6091   supp csupp 6690   Fincfn 7310   finSupp cfsupp 7620   1c1 9283   NNcn 10322   ...cfz 11437   seqcseq 11806   #chash 12103   Basecbs 14174   +g cplusg 14238   0gc0g 14378   gsum_g cgsu 14379   Mndcmnd 15409  Cntzccntz 15833

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-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-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-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-n0 10580  df-z 10647  df-uz 10862  df-fz 11438  df-fzo 11549  df-seq 11807  df-hash 12104  df-0g 14380  df-gsum 14381  df-mnd 15415  df-cntz 15835

This theorem is referenced by:  gsumres  16395  gsumzsplit  16418  gsumpt  16454  dmdprdsplitlem  16534  dpjidcl  16557  mplcoe5  17548