Theorem gsumzres 17543

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 4546	. . . . . . . 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 16621	. . . . . . 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 667	. . . . . 6 |- ( ph -> ( G gsumg ( k e. ( A i^i W ) |-> .0. ) ) = .0. )
8	5	gsumz 16621	. . . . . . 7 |- ( ( G e. Mnd /\ A e. V ) -> ( G gsumg ( k e. A |-> .0. ) ) = .0. )
9	1, 2, 8	syl2anc 667	. . . . . 6 |- ( ph -> ( G gsumg ( k e. A |-> .0. ) ) = .0. )
10	7, 9	eqtr4d 2488	. . . . 5 |- ( ph -> ( G gsumg ( k e. ( A i^i W ) |-> .0. ) ) = ( G gsumg ( k e. A |-> .0. ) ) )
11	10	adantr 467	. . . 4 |- ( ( ph /\ ( F supp .0. ) = (/) ) -> ( G gsumg ( k e. ( A i^i W ) |-> .0. ) ) = ( G gsumg ( k e. A |-> .0. ) ) )
12		resres 5117	. . . . . . . 8 |- ( ( F |` A ) |` W ) = ( F |` ( A i^i W ) )
13		gsumzcl.f	. . . . . . . . . 10 |- ( ph -> F : A --> B )
14		ffn 5728	. . . . . . . . . 10 |- ( F : A --> B -> F Fn A )
15		fnresdm 5685	. . . . . . . . . 10 |- ( F Fn A -> ( F |` A ) = F )
16	13, 14, 15	3syl 18	. . . . . . . . 9 |- ( ph -> ( F |` A ) = F )
17	16	reseq1d 5104	. . . . . . . 8 |- ( ph -> ( ( F |` A ) |` W ) = ( F |` W ) )
18	12, 17	syl5eqr 2499	. . . . . . 7 |- ( ph -> ( F |` ( A i^i W ) ) = ( F |` W ) )
19	18	adantr 467	. . . . . 6 |- ( ( ph /\ ( F supp .0. ) = (/) ) -> ( F |` ( A i^i W ) ) = ( F |` W ) )
20		fvex 5875	. . . . . . . . . . 11 |- ( 0g ` G ) e. _V
21	5, 20	eqeltri 2525	. . . . . . . . . 10 |- .0. e. _V
22	21	a1i 11	. . . . . . . . 9 |- ( ph -> .0. e. _V )
23		ssid 3451	. . . . . . . . . 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 17542	. . . . . . . 8 |- ( ( ph /\ ( F supp .0. ) = (/) ) -> F = ( k e. A |-> .0. ) )
26	25	reseq1d 5104	. . . . . . 7 |- ( ( ph /\ ( F supp .0. ) = (/) ) -> ( F |` ( A i^i W ) ) = ( ( k e. A |-> .0. ) |` ( A i^i W ) ) )
27		inss1 3652	. . . . . . . 8 |- ( A i^i W ) C_ A
28		resmpt 5154	. . . . . . . 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 2501	. . . . . 6 |- ( ( ph /\ ( F supp .0. ) = (/) ) -> ( F |` ( A i^i W ) ) = ( k e. ( A i^i W ) |-> .0. ) )
31	19, 30	eqtr3d 2487	. . . . 5 |- ( ( ph /\ ( F supp .0. ) = (/) ) -> ( F |` W ) = ( k e. ( A i^i W ) |-> .0. ) )
32	31	oveq2d 6306	. . . 4 |- ( ( ph /\ ( F supp .0. ) = (/) ) -> ( G gsumg ( F |` W ) ) = ( G gsumg ( k e. ( A i^i W ) |-> .0. ) ) )
33	25	oveq2d 6306	. . . 4 |- ( ( ph /\ ( F supp .0. ) = (/) ) -> ( G gsumg F ) = ( G gsumg ( k e. A |-> .0. ) ) )
34	11, 32, 33	3eqtr4d 2495	. . 3 |- ( ( ph /\ ( F supp .0. ) = (/) ) -> ( G gsumg ( F |` W ) ) = ( G gsumg F ) )
35	34	ex 436	. 2 |- ( ph -> ( ( F supp .0. ) = (/) -> ( G gsumg ( F |` W ) ) = ( G gsumg F ) ) )
36		f1ofo 5821	. . . . . . . . . . . 12 |- ( f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) -> f : ( 1 ... ( # ` ( F supp .0. ) ) ) -onto-> ( F supp .0. ) )
37		forn 5796	. . . . . . . . . . . 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 735	. . . . . . . . . 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 467	. . . . . . . . . 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 3466	. . . . . . . . 9 |- ( ( ph /\ ( ( # ` ( F supp .0. ) ) e. NN /\ f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) ) ) -> ran f C_ W )
43		cores 5338	. . . . . . . . 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 12221	. . . . . . 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 5867	. . . . . 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 2451	. . . . . . 7 |- ( +g ` G ) = ( +g ` G )
49		gsumzcl.z	. . . . . . 7 |- Z = (Cntz ` G )
50	1	adantr 467	. . . . . . 7 |- ( ( ph /\ ( ( # ` ( F supp .0. ) ) e. NN /\ f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) ) ) -> G e. Mnd )
51	4	adantr 467	. . . . . . 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 467	. . . . . . . . 9 |- ( ( ph /\ ( ( # ` ( F supp .0. ) ) e. NN /\ f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) ) ) -> F : A --> B )
53		fssres 5749	. . . . . . . . 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 668	. . . . . . . 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 5714	. . . . . . . . 9 |- ( ph -> ( ( F |` ( A i^i W ) ) : ( A i^i W ) --> B <-> ( F |` W ) : ( A i^i W ) --> B ) )
56	55	biimpa 487	. . . . . . . 8 |- ( ( ph /\ ( F |` ( A i^i W ) ) : ( A i^i W ) --> B ) -> ( F |` W ) : ( A i^i W ) --> B )
57	54, 56	syldan 473	. . . . . . 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 5128	. . . . . . . . . 10 |- ( F |` W ) C_ F
60		rnss 5063	. . . . . . . . . 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 16991	. . . . . . . . 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 668	. . . . . . . 8 |- ( ph -> ran ( F |` W ) C_ ( Z ` ran ( F |` W ) ) )
64	63	adantr 467	. . . . . . 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 764	. . . . . . 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 5813	. . . . . . . . 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 735	. . . . . . . 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 6927	. . . . . . . . . . 11 |- ( F supp .0. ) C_ dom F
69		fdm 5733	. . . . . . . . . . . 12 |- ( F : A --> B -> dom F = A )
70	13, 69	syl 17	. . . . . . . . . . 11 |- ( ph -> dom F = A )
71	68, 70	syl5sseq 3480	. . . . . . . . . 10 |- ( ph -> ( F supp .0. ) C_ A )
72	71, 40	ssind 3656	. . . . . . . . 9 |- ( ph -> ( F supp .0. ) C_ ( A i^i W ) )
73	72	adantr 467	. . . . . . . 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 5784	. . . . . . . 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 667	. . . . . . 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 6138	. . . . . . . . . . . . 13 |- ( ( F : A --> B /\ A e. V ) -> F e. _V )
77	13, 2, 76	syl2anc 667	. . . . . . . . . . . 12 |- ( ph -> F e. _V )
78		ressuppss 6934	. . . . . . . . . . . 12 |- ( ( F e. _V /\ .0. e. _V ) -> ( ( F |` W ) supp .0. ) C_ ( F supp .0. ) )
79	77, 21, 78	sylancl 668	. . . . . . . . . . 11 |- ( ph -> ( ( F |` W ) supp .0. ) C_ ( F supp .0. ) )
80		sseq2 3454	. . . . . . . . . . 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 225	. . . . . . . . . 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 468	. . . . . . . 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 432	. . . . . . 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 2451	. . . . . . 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 17541	. . . . . 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 467	. . . . . . 7 |- ( ( ph /\ ( ( # ` ( F supp .0. ) ) e. NN /\ f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) ) ) -> A e. V )
88	58	adantr 467	. . . . . . 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 467	. . . . . . . 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 5784	. . . . . . . 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 667	. . . . . . 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 3481	. . . . . . 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 2451	. . . . . . 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 17541	. . . . . 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 2495	. . . . 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 620	. . . 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 1779	. . 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 608	. 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 7889	. . . 4 |- ( F finSupp .0. -> ( Fun F /\ ( F supp .0. ) e. Fin ) )
101	100	simprd 465	. . 3 |- ( F finSupp .0. -> ( F supp .0. ) e. Fin )
102		fz1f1o 13776	. . 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 383	1 |- ( ph -> ( G gsumg ( F |` W ) ) = ( G gsumg F ) )

Syntax hints:   -> wi 4   \/ wo 370   /\ wa 371   = wceq 1444   E.wex 1663   e. wcel 1887   _Vcvv 3045   i^i cin 3403   C_ wss 3404   (/)c0 3731   class class class wbr 4402   |-> cmpt 4461   dom cdm 4834   ran crn 4835   |` cres 4836   o. ccom 4838   Fun wfun 5576   Fn wfn 5577   -->wf 5578   -1-1->wf1 5579   -onto->wfo 5580   -1-1-onto->wf1o 5581   ` cfv 5582  (class class class)co 6290   supp csupp 6914   Fincfn 7569   finSupp cfsupp 7883   1c1 9540   NNcn 10609   ...cfz 11784   seqcseq 12213   #chash 12515   Basecbs 15121   +g cplusg 15190   0gc0g 15338   gsum_g cgsu 15339   Mndcmnd 16535  Cntzccntz 16969

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-se 4794  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-isom 5591  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-supp 6915  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-oadd 7186  df-er 7363  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-fsupp 7884  df-oi 8025  df-card 8373  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-nn 10610  df-n0 10870  df-z 10938  df-uz 11160  df-fz 11785  df-fzo 11916  df-seq 12214  df-hash 12516  df-0g 15340  df-gsum 15341  df-mgm 16488  df-sgrp 16527  df-mnd 16537  df-cntz 16971

This theorem is referenced by:  gsumres  17547  gsumzsplit  17560  gsumpt  17594  dmdprdsplitlem  17670  dpjidcl  17691  mplcoe5  18692