Theorem gsumzres 17621

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 4539	. . . . . . . 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 16699	. . . . . . 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 673	. . . . . 6 |- ( ph -> ( G gsumg ( k e. ( A i^i W ) |-> .0. ) ) = .0. )
8	5	gsumz 16699	. . . . . . 7 |- ( ( G e. Mnd /\ A e. V ) -> ( G gsumg ( k e. A |-> .0. ) ) = .0. )
9	1, 2, 8	syl2anc 673	. . . . . 6 |- ( ph -> ( G gsumg ( k e. A |-> .0. ) ) = .0. )
10	7, 9	eqtr4d 2508	. . . . 5 |- ( ph -> ( G gsumg ( k e. ( A i^i W ) |-> .0. ) ) = ( G gsumg ( k e. A |-> .0. ) ) )
11	10	adantr 472	. . . 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 5739	. . . . . . . . . 10 |- ( F : A --> B -> F Fn A )
15		fnresdm 5695	. . . . . . . . . 10 |- ( F Fn A -> ( F |` A ) = F )
16	13, 14, 15	3syl 18	. . . . . . . . 9 |- ( ph -> ( F |` A ) = F )
17	16	reseq1d 5110	. . . . . . . 8 |- ( ph -> ( ( F |` A ) |` W ) = ( F |` W ) )
18	12, 17	syl5eqr 2519	. . . . . . 7 |- ( ph -> ( F |` ( A i^i W ) ) = ( F |` W ) )
19	18	adantr 472	. . . . . 6 |- ( ( ph /\ ( F supp .0. ) = (/) ) -> ( F |` ( A i^i W ) ) = ( F |` W ) )
20		fvex 5889	. . . . . . . . . . 11 |- ( 0g ` G ) e. _V
21	5, 20	eqeltri 2545	. . . . . . . . . 10 |- .0. e. _V
22	21	a1i 11	. . . . . . . . 9 |- ( ph -> .0. e. _V )
23		ssid 3437	. . . . . . . . . 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 17620	. . . . . . . 8 |- ( ( ph /\ ( F supp .0. ) = (/) ) -> F = ( k e. A |-> .0. ) )
26	25	reseq1d 5110	. . . . . . 7 |- ( ( ph /\ ( F supp .0. ) = (/) ) -> ( F |` ( A i^i W ) ) = ( ( k e. A |-> .0. ) |` ( A i^i W ) ) )
27		inss1 3643	. . . . . . . 8 |- ( A i^i W ) C_ A
28		resmpt 5160	. . . . . . . 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 2521	. . . . . 6 |- ( ( ph /\ ( F supp .0. ) = (/) ) -> ( F |` ( A i^i W ) ) = ( k e. ( A i^i W ) |-> .0. ) )
31	19, 30	eqtr3d 2507	. . . . 5 |- ( ( ph /\ ( F supp .0. ) = (/) ) -> ( F |` W ) = ( k e. ( A i^i W ) |-> .0. ) )
32	31	oveq2d 6324	. . . 4 |- ( ( ph /\ ( F supp .0. ) = (/) ) -> ( G gsumg ( F |` W ) ) = ( G gsumg ( k e. ( A i^i W ) |-> .0. ) ) )
33	25	oveq2d 6324	. . . 4 |- ( ( ph /\ ( F supp .0. ) = (/) ) -> ( G gsumg F ) = ( G gsumg ( k e. A |-> .0. ) ) )
34	11, 32, 33	3eqtr4d 2515	. . 3 |- ( ( ph /\ ( F supp .0. ) = (/) ) -> ( G gsumg ( F |` W ) ) = ( G gsumg F ) )
35	34	ex 441	. 2 |- ( ph -> ( ( F supp .0. ) = (/) -> ( G gsumg ( F |` W ) ) = ( G gsumg F ) ) )
36		f1ofo 5835	. . . . . . . . . . . 12 |- ( f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) -> f : ( 1 ... ( # ` ( F supp .0. ) ) ) -onto-> ( F supp .0. ) )
37		forn 5809	. . . . . . . . . . . 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 743	. . . . . . . . . 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 472	. . . . . . . . . 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 3452	. . . . . . . . 9 |- ( ( ph /\ ( ( # ` ( F supp .0. ) ) e. NN /\ f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) ) ) -> ran f C_ W )
43		cores 5345	. . . . . . . . 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 12259	. . . . . . 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 5881	. . . . . 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 2471	. . . . . . 7 |- ( +g ` G ) = ( +g ` G )
49		gsumzcl.z	. . . . . . 7 |- Z = (Cntz ` G )
50	1	adantr 472	. . . . . . 7 |- ( ( ph /\ ( ( # ` ( F supp .0. ) ) e. NN /\ f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) ) ) -> G e. Mnd )
51	4	adantr 472	. . . . . . 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 472	. . . . . . . . 9 |- ( ( ph /\ ( ( # ` ( F supp .0. ) ) e. NN /\ f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) ) ) -> F : A --> B )
53		fssres 5761	. . . . . . . . 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 675	. . . . . . . 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 5724	. . . . . . . . 9 |- ( ph -> ( ( F |` ( A i^i W ) ) : ( A i^i W ) --> B <-> ( F |` W ) : ( A i^i W ) --> B ) )
56	55	biimpa 492	. . . . . . . 8 |- ( ( ph /\ ( F |` ( A i^i W ) ) : ( A i^i W ) --> B ) -> ( F |` W ) : ( A i^i W ) --> B )
57	54, 56	syldan 478	. . . . . . 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 5069	. . . . . . . . . 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 17069	. . . . . . . . 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 675	. . . . . . . 8 |- ( ph -> ran ( F |` W ) C_ ( Z ` ran ( F |` W ) ) )
64	63	adantr 472	. . . . . . 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 772	. . . . . . 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 5827	. . . . . . . . 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 743	. . . . . . . 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 6946	. . . . . . . . . . 11 |- ( F supp .0. ) C_ dom F
69		fdm 5745	. . . . . . . . . . . 12 |- ( F : A --> B -> dom F = A )
70	13, 69	syl 17	. . . . . . . . . . 11 |- ( ph -> dom F = A )
71	68, 70	syl5sseq 3466	. . . . . . . . . 10 |- ( ph -> ( F supp .0. ) C_ A )
72	71, 40	ssind 3647	. . . . . . . . 9 |- ( ph -> ( F supp .0. ) C_ ( A i^i W ) )
73	72	adantr 472	. . . . . . . 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 5797	. . . . . . . 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 673	. . . . . . 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 6155	. . . . . . . . . . . . 13 |- ( ( F : A --> B /\ A e. V ) -> F e. _V )
77	13, 2, 76	syl2anc 673	. . . . . . . . . . . 12 |- ( ph -> F e. _V )
78		ressuppss 6953	. . . . . . . . . . . 12 |- ( ( F e. _V /\ .0. e. _V ) -> ( ( F |` W ) supp .0. ) C_ ( F supp .0. ) )
79	77, 21, 78	sylancl 675	. . . . . . . . . . 11 |- ( ph -> ( ( F |` W ) supp .0. ) C_ ( F supp .0. ) )
80		sseq2 3440	. . . . . . . . . . 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 229	. . . . . . . . . 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 473	. . . . . . . 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 437	. . . . . . 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 2471	. . . . . . 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 17619	. . . . . 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 472	. . . . . . 7 |- ( ( ph /\ ( ( # ` ( F supp .0. ) ) e. NN /\ f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) ) ) -> A e. V )
88	58	adantr 472	. . . . . . 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 472	. . . . . . . 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 5797	. . . . . . . 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 673	. . . . . . 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 3467	. . . . . . 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 2471	. . . . . . 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 17619	. . . . . 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 2515	. . . . 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 626	. . . 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 1787	. . 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 614	. 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 7907	. . . 4 |- ( F finSupp .0. -> ( Fun F /\ ( F supp .0. ) e. Fin ) )
101	100	simprd 470	. . 3 |- ( F finSupp .0. -> ( F supp .0. ) e. Fin )
102		fz1f1o 13853	. . 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 388	1 |- ( ph -> ( G gsumg ( F |` W ) ) = ( G gsumg F ) )

Syntax hints:   -> wi 4   \/ wo 375   /\ wa 376   = wceq 1452   E.wex 1671   e. wcel 1904   _Vcvv 3031   i^i cin 3389   C_ wss 3390   (/)c0 3722   class class class wbr 4395   |-> cmpt 4454   dom cdm 4839   ran crn 4840   |` cres 4841   o. ccom 4843   Fun wfun 5583   Fn wfn 5584   -->wf 5585   -1-1->wf1 5586   -onto->wfo 5587   -1-1-onto->wf1o 5588   ` cfv 5589  (class class class)co 6308   supp csupp 6933   Fincfn 7587   finSupp cfsupp 7901   1c1 9558   NNcn 10631   ...cfz 11810   seqcseq 12251   #chash 12553   Basecbs 15199   +g cplusg 15268   0gc0g 15416   gsum_g cgsu 15417   Mndcmnd 16613  Cntzccntz 17047

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-supp 6934  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-fsupp 7902  df-oi 8043  df-card 8391  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-n0 10894  df-z 10962  df-uz 11183  df-fz 11811  df-fzo 11943  df-seq 12252  df-hash 12554  df-0g 15418  df-gsum 15419  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-cntz 17049

This theorem is referenced by:  gsumres  17625  gsumzsplit  17638  gsumpt  17672  dmdprdsplitlem  17748  dpjidcl  17769  mplcoe5  18769