Theorem gsumzres 18310

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 4801	. . . . . . . 8 |- ( A e. V -> ( A i^i W ) e. _V )
4	2, 3	syl 17	. . . . . . 7 |- ( ph -> ( A i^i W ) e. _V )
5		gsumzcl.0	. . . . . . . 8 |- .0. = ( 0g ` G )
6	5	gsumz 17374	. . . . . . 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 693	. . . . . 6 |- ( ph -> ( G gsumg ( k e. ( A i^i W ) |-> .0. ) ) = .0. )
8	5	gsumz 17374	. . . . . . 7 |- ( ( G e. Mnd /\ A e. V ) -> ( G gsumg ( k e. A |-> .0. ) ) = .0. )
9	1, 2, 8	syl2anc 693	. . . . . 6 |- ( ph -> ( G gsumg ( k e. A |-> .0. ) ) = .0. )
10	7, 9	eqtr4d 2659	. . . . 5 |- ( ph -> ( G gsumg ( k e. ( A i^i W ) |-> .0. ) ) = ( G gsumg ( k e. A |-> .0. ) ) )
11	10	adantr 481	. . . 4 |- ( ( ph /\ ( F supp .0. ) = (/) ) -> ( G gsumg ( k e. ( A i^i W ) |-> .0. ) ) = ( G gsumg ( k e. A |-> .0. ) ) )
12		resres 5409	. . . . . . . 8 |- ( ( F |` A ) |` W ) = ( F |` ( A i^i W ) )
13		gsumzcl.f	. . . . . . . . . 10 |- ( ph -> F : A --> B )
14		ffn 6045	. . . . . . . . . 10 |- ( F : A --> B -> F Fn A )
15		fnresdm 6000	. . . . . . . . . 10 |- ( F Fn A -> ( F |` A ) = F )
16	13, 14, 15	3syl 18	. . . . . . . . 9 |- ( ph -> ( F |` A ) = F )
17	16	reseq1d 5395	. . . . . . . 8 |- ( ph -> ( ( F |` A ) |` W ) = ( F |` W ) )
18	12, 17	syl5eqr 2670	. . . . . . 7 |- ( ph -> ( F |` ( A i^i W ) ) = ( F |` W ) )
19	18	adantr 481	. . . . . 6 |- ( ( ph /\ ( F supp .0. ) = (/) ) -> ( F |` ( A i^i W ) ) = ( F |` W ) )
20		fvex 6201	. . . . . . . . . . 11 |- ( 0g ` G ) e. _V
21	5, 20	eqeltri 2697	. . . . . . . . . 10 |- .0. e. _V
22	21	a1i 11	. . . . . . . . 9 |- ( ph -> .0. e. _V )
23		ssid 3624	. . . . . . . . . 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 18309	. . . . . . . 8 |- ( ( ph /\ ( F supp .0. ) = (/) ) -> F = ( k e. A |-> .0. ) )
26	25	reseq1d 5395	. . . . . . 7 |- ( ( ph /\ ( F supp .0. ) = (/) ) -> ( F |` ( A i^i W ) ) = ( ( k e. A |-> .0. ) |` ( A i^i W ) ) )
27		inss1 3833	. . . . . . . 8 |- ( A i^i W ) C_ A
28		resmpt 5449	. . . . . . . 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 2672	. . . . . 6 |- ( ( ph /\ ( F supp .0. ) = (/) ) -> ( F |` ( A i^i W ) ) = ( k e. ( A i^i W ) |-> .0. ) )
31	19, 30	eqtr3d 2658	. . . . 5 |- ( ( ph /\ ( F supp .0. ) = (/) ) -> ( F |` W ) = ( k e. ( A i^i W ) |-> .0. ) )
32	31	oveq2d 6666	. . . 4 |- ( ( ph /\ ( F supp .0. ) = (/) ) -> ( G gsumg ( F |` W ) ) = ( G gsumg ( k e. ( A i^i W ) |-> .0. ) ) )
33	25	oveq2d 6666	. . . 4 |- ( ( ph /\ ( F supp .0. ) = (/) ) -> ( G gsumg F ) = ( G gsumg ( k e. A |-> .0. ) ) )
34	11, 32, 33	3eqtr4d 2666	. . 3 |- ( ( ph /\ ( F supp .0. ) = (/) ) -> ( G gsumg ( F |` W ) ) = ( G gsumg F ) )
35	34	ex 450	. 2 |- ( ph -> ( ( F supp .0. ) = (/) -> ( G gsumg ( F |` W ) ) = ( G gsumg F ) ) )
36		f1ofo 6144	. . . . . . . . . . . 12 |- ( f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) -> f : ( 1 ... ( # ` ( F supp .0. ) ) ) -onto-> ( F supp .0. ) )
37		forn 6118	. . . . . . . . . . . 12 |- ( f : ( 1 ... ( # ` ( F supp .0. ) ) ) -onto-> ( F supp .0. ) -> ran f = ( F supp .0. ) )
38	36, 37	syl 17	. . . . . . . . . . 11 |- ( f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) -> ran f = ( F supp .0. ) )
39	38	ad2antll 765	. . . . . . . . . 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 481	. . . . . . . . . 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 3639	. . . . . . . . 9 |- ( ( ph /\ ( ( # ` ( F supp .0. ) ) e. NN /\ f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) ) ) -> ran f C_ W )
43		cores 5638	. . . . . . . . 9 |- ( ran f C_ W -> ( ( F |` W ) o. f ) = ( F o. f ) )
44	42, 43	syl 17	. . . . . . . 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 12809	. . . . . . 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 6193	. . . . . 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 2622	. . . . . . 7 |- ( +g ` G ) = ( +g ` G )
49		gsumzcl.z	. . . . . . 7 |- Z = (Cntz ` G )
50	1	adantr 481	. . . . . . 7 |- ( ( ph /\ ( ( # ` ( F supp .0. ) ) e. NN /\ f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) ) ) -> G e. Mnd )
51	4	adantr 481	. . . . . . 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 481	. . . . . . . . 9 |- ( ( ph /\ ( ( # ` ( F supp .0. ) ) e. NN /\ f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) ) ) -> F : A --> B )
53		fssres 6070	. . . . . . . . 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 694	. . . . . . . 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 6030	. . . . . . . . 9 |- ( ph -> ( ( F |` ( A i^i W ) ) : ( A i^i W ) --> B <-> ( F |` W ) : ( A i^i W ) --> B ) )
56	55	biimpa 501	. . . . . . . 8 |- ( ( ph /\ ( F |` ( A i^i W ) ) : ( A i^i W ) --> B ) -> ( F |` W ) : ( A i^i W ) --> B )
57	54, 56	syldan 487	. . . . . . 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 5422	. . . . . . . . . 10 |- ( F |` W ) C_ F
60		rnss 5354	. . . . . . . . . 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 17770	. . . . . . . . 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 694	. . . . . . . 8 |- ( ph -> ran ( F |` W ) C_ ( Z ` ran ( F |` W ) ) )
64	63	adantr 481	. . . . . . 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 794	. . . . . . 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 6136	. . . . . . . . 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 765	. . . . . . . 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 7308	. . . . . . . . . . 11 |- ( F supp .0. ) C_ dom F
69		fdm 6051	. . . . . . . . . . . 12 |- ( F : A --> B -> dom F = A )
70	13, 69	syl 17	. . . . . . . . . . 11 |- ( ph -> dom F = A )
71	68, 70	syl5sseq 3653	. . . . . . . . . 10 |- ( ph -> ( F supp .0. ) C_ A )
72	71, 40	ssind 3837	. . . . . . . . 9 |- ( ph -> ( F supp .0. ) C_ ( A i^i W ) )
73	72	adantr 481	. . . . . . . 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 6106	. . . . . . . 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 693	. . . . . . 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 6490	. . . . . . . . . . . . 13 |- ( ( F : A --> B /\ A e. V ) -> F e. _V )
77	13, 2, 76	syl2anc 693	. . . . . . . . . . . 12 |- ( ph -> F e. _V )
78		ressuppss 7314	. . . . . . . . . . . 12 |- ( ( F e. _V /\ .0. e. _V ) -> ( ( F |` W ) supp .0. ) C_ ( F supp .0. ) )
79	77, 21, 78	sylancl 694	. . . . . . . . . . 11 |- ( ph -> ( ( F |` W ) supp .0. ) C_ ( F supp .0. ) )
80		sseq2 3627	. . . . . . . . . . 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 236	. . . . . . . . . 10 |- ( ran f = ( F supp .0. ) -> ( ph -> ( ( F |` W ) supp .0. ) C_ ran f ) )
82	36, 37, 81	3syl 18	. . . . . . . . 9 |- ( f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) -> ( ph -> ( ( F |` W ) supp .0. ) C_ ran f ) )
83	82	adantl 482	. . . . . . . 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 446	. . . . . . 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 2622	. . . . . . 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 18308	. . . . . 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 481	. . . . . . 7 |- ( ( ph /\ ( ( # ` ( F supp .0. ) ) e. NN /\ f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) ) ) -> A e. V )
88	58	adantr 481	. . . . . . 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 481	. . . . . . . 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 6106	. . . . . . . 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 693	. . . . . . 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 3654	. . . . . . 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 2622	. . . . . . 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 18308	. . . . . 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 2666	. . . . 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 643	. . . 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 1861	. . 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 629	. 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 8281	. . . 4 |- ( F finSupp .0. -> ( Fun F /\ ( F supp .0. ) e. Fin ) )
101	100	simprd 479	. . 3 |- ( F finSupp .0. -> ( F supp .0. ) e. Fin )
102		fz1f1o 14441	. . 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 18	. 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 396	1 |- ( ph -> ( G gsumg ( F |` W ) ) = ( G gsumg F ) )

Syntax hints:   -> wi 4   \/ wo 383   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   _Vcvv 3200   i^i cin 3573   C_ wss 3574   (/)c0 3915   class class class wbr 4653   |-> cmpt 4729   dom cdm 5114   ran crn 5115   |` cres 5116   o. ccom 5118   Fun wfun 5882   Fn wfn 5883   -->wf 5884   -1-1->wf1 5885   -onto->wfo 5886   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   supp csupp 7295   Fincfn 7955   finSupp cfsupp 8275   1c1 9937   NNcn 11020   ...cfz 12326   seqcseq 12801   #chash 13117   Basecbs 15857   +g cplusg 15941   0gc0g 16100   gsum_g cgsu 16101   Mndcmnd 17294  Cntzccntz 17748

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-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-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-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-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-seq 12802  df-hash 13118  df-0g 16102  df-gsum 16103  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-cntz 17750

This theorem is referenced by:  gsumres  18314  gsumzsplit  18327  gsumpt  18361  dmdprdsplitlem  18436  dpjidcl  18457  mplcoe5  19468