Theorem gsumzres 17041

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 4599	. . . . . . . 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 16132	. . . . . . 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 16132	. . . . . . 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 2501	. . . . 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 5296	. . . . . . . 8 |- ( ( F |` A ) |` W ) = ( F |` ( A i^i W ) )
13		gsumzcl.f	. . . . . . . . . 10 |- ( ph -> F : A --> B )
14		ffn 5737	. . . . . . . . . 10 |- ( F : A --> B -> F Fn A )
15		fnresdm 5696	. . . . . . . . . 10 |- ( F Fn A -> ( F |` A ) = F )
16	13, 14, 15	3syl 20	. . . . . . . . 9 |- ( ph -> ( F |` A ) = F )
17	16	reseq1d 5282	. . . . . . . 8 |- ( ph -> ( ( F |` A ) |` W ) = ( F |` W ) )
18	12, 17	syl5eqr 2512	. . . . . . 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 5882	. . . . . . . . . . 11 |- ( 0g ` G ) e. _V
21	5, 20	eqeltri 2541	. . . . . . . . . 10 |- .0. e. _V
22	21	a1i 11	. . . . . . . . 9 |- ( ph -> .0. e. _V )
23		ssid 3518	. . . . . . . . . 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 17039	. . . . . . . 8 |- ( ( ph /\ ( F supp .0. ) = (/) ) -> F = ( k e. A |-> .0. ) )
26	25	reseq1d 5282	. . . . . . 7 |- ( ( ph /\ ( F supp .0. ) = (/) ) -> ( F |` ( A i^i W ) ) = ( ( k e. A |-> .0. ) |` ( A i^i W ) ) )
27		inss1 3714	. . . . . . . 8 |- ( A i^i W ) C_ A
28		resmpt 5333	. . . . . . . 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 2514	. . . . . 6 |- ( ( ph /\ ( F supp .0. ) = (/) ) -> ( F |` ( A i^i W ) ) = ( k e. ( A i^i W ) |-> .0. ) )
31	19, 30	eqtr3d 2500	. . . . 5 |- ( ( ph /\ ( F supp .0. ) = (/) ) -> ( F |` W ) = ( k e. ( A i^i W ) |-> .0. ) )
32	31	oveq2d 6312	. . . 4 |- ( ( ph /\ ( F supp .0. ) = (/) ) -> ( G gsumg ( F |` W ) ) = ( G gsumg ( k e. ( A i^i W ) |-> .0. ) ) )
33	25	oveq2d 6312	. . . 4 |- ( ( ph /\ ( F supp .0. ) = (/) ) -> ( G gsumg F ) = ( G gsumg ( k e. A |-> .0. ) ) )
34	11, 32, 33	3eqtr4d 2508	. . 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 5829	. . . . . . . . . . . 12 |- ( f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) -> f : ( 1 ... ( # ` ( F supp .0. ) ) ) -onto-> ( F supp .0. ) )
37		forn 5804	. . . . . . . . . . . 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 3533	. . . . . . . . 9 |- ( ( ph /\ ( ( # ` ( F supp .0. ) ) e. NN /\ f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) ) ) -> ran f C_ W )
43		cores 5516	. . . . . . . . 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 12118	. . . . . . 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 5874	. . . . . 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 2457	. . . . . . 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 5757	. . . . . . . . 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 5723	. . . . . . . . 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 5307	. . . . . . . . . 10 |- ( F |` W ) C_ F
60		rnss 5241	. . . . . . . . . 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 16502	. . . . . . . . 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 756	. . . . . . 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 5821	. . . . . . . . 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 6930	. . . . . . . . . . 11 |- ( F supp .0. ) C_ dom F
69		fdm 5741	. . . . . . . . . . . 12 |- ( F : A --> B -> dom F = A )
70	13, 69	syl 16	. . . . . . . . . . 11 |- ( ph -> dom F = A )
71	68, 70	syl5sseq 3547	. . . . . . . . . 10 |- ( ph -> ( F supp .0. ) C_ A )
72	71, 40	ssind 3718	. . . . . . . . 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 5792	. . . . . . . 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 6146	. . . . . . . . . . . . 13 |- ( ( F : A --> B /\ A e. V ) -> F e. _V )
77	13, 2, 76	syl2anc 661	. . . . . . . . . . . 12 |- ( ph -> F e. _V )
78		ressuppss 6937	. . . . . . . . . . . 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 3521	. . . . . . . . . . 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 2457	. . . . . . 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 17038	. . . . . 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 5792	. . . . . . . 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 3548	. . . . . . 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 2457	. . . . . . 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 17038	. . . . . 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 2508	. . . . 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 1725	. . 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 7853	. . . 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 13544	. . 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 1395   E.wex 1613   e. wcel 1819   _Vcvv 3109   i^i cin 3470   C_ wss 3471   (/)c0 3793   class class class wbr 4456   |-> cmpt 4515   dom cdm 5008   ran crn 5009   |` cres 5010   o. ccom 5012   Fun wfun 5588   Fn wfn 5589   -->wf 5590   -1-1->wf1 5591   -onto->wfo 5592   -1-1-onto->wf1o 5593   ` cfv 5594  (class class class)co 6296   supp csupp 6917   Fincfn 7535   finSupp cfsupp 7847   1c1 9510   NNcn 10556   ...cfz 11697   seqcseq 12110   #chash 12408   Basecbs 14644   +g cplusg 14712   0gc0g 14857   gsum_g cgsu 14858   Mndcmnd 16046  Cntzccntz 16480

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-supp 6918  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-fsupp 7848  df-oi 7953  df-card 8337  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-n0 10817  df-z 10886  df-uz 11107  df-fz 11698  df-fzo 11822  df-seq 12111  df-hash 12409  df-0g 14859  df-gsum 14860  df-mgm 15999  df-sgrp 16038  df-mnd 16048  df-cntz 16482

This theorem is referenced by:  gsumres  17048  gsumzsplit  17071  gsumpt  17115  dmdprdsplitlem  17211  dpjidcl  17234  mplcoe5  18258