Theorem gsumf1ofi 14250

Description: Re-index a finite group sum using a bijection. (Contributed by Mario Carneiro, 15-Dec-2014.) (Revised by Mario Carneiro, 24-Apr-2016.) (Revised by AV, 3-Jun-2019.)

Hypotheses

Ref	Expression
gsumclfi.b	|- B = ( Base ` G )
gsumclfi.z	|- .0. = ( 0g ` G )
gsumclfi.g	|- ( ph -> G e. CMnd )
gsumclfi.a	|- ( ph -> A e. Fin )
gsumclfi.f	|- ( ph -> F : A --> B )
gsumf1o.h	|- ( ph -> H : C -1-1-onto-> A )

Assertion

Ref	Expression
gsumf1ofi	|- ( ph -> ( G gsumg F ) = ( G gsumg ( F o. H ) ) )

Proof of Theorem gsumf1ofi

Dummy variables x f g are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		gsumclfi.a	. . . 4 |- ( ph -> A e. Fin )
2		isfinite4im 11253	. . . 4 |- ( A e. Fin -> ( 1 ... (♯ ` A ) ) ~~ A )
3	1, 2	syl 14	. . 3 |- ( ph -> ( 1 ... (♯ ` A ) ) ~~ A )
4		bren 7034	. . 3 |- ( ( 1 ... (♯ ` A ) ) ~~ A <-> E. f f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A )
5	3, 4	sylib 122	. 2 |- ( ph -> E. f f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A )
6		gsumf1o.h	. . . . . . . . . 10 |- ( ph -> H : C -1-1-onto-> A )
7		f1ocnv 5655	. . . . . . . . . 10 |- ( H : C -1-1-onto-> A -> `' H : A -1-1-onto-> C )
8	6, 7	syl 14	. . . . . . . . 9 |- ( ph -> `' H : A -1-1-onto-> C )
9		f1oeng 7047	. . . . . . . . 9 |- ( ( A e. Fin /\ `' H : A -1-1-onto-> C ) -> A ~~ C )
10	1, 8, 9	syl2anc 415	. . . . . . . 8 |- ( ph -> A ~~ C )
11	10	ensymd 7074	. . . . . . 7 |- ( ph -> C ~~ A )
12		enfii 7180	. . . . . . 7 |- ( ( A e. Fin /\ C ~~ A ) -> C e. Fin )
13	1, 11, 12	syl2anc 415	. . . . . 6 |- ( ph -> C e. Fin )
14		isfinite4im 11253	. . . . . 6 |- ( C e. Fin -> ( 1 ... (♯ ` C ) ) ~~ C )
15	13, 14	syl 14	. . . . 5 |- ( ph -> ( 1 ... (♯ ` C ) ) ~~ C )
16		bren 7034	. . . . 5 |- ( ( 1 ... (♯ ` C ) ) ~~ C <-> E. g g : ( 1 ... (♯ ` C ) ) -1-1-onto-> C )
17	15, 16	sylib 122	. . . 4 |- ( ph -> E. g g : ( 1 ... (♯ ` C ) ) -1-1-onto-> C )
18	17	adantr 276	. . 3 |- ( ( ph /\ f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A ) -> E. g g : ( 1 ... (♯ ` C ) ) -1-1-onto-> C )
19		gsumclfi.b	. . . . . 6 |- B = ( Base ` G )
20		gsumclfi.z	. . . . . 6 |- .0. = ( 0g ` G )
21		gsumclfi.g	. . . . . . 7 |- ( ph -> G e. CMnd )
22	21	ad2antrr 492	. . . . . 6 |- ( ( ( ph /\ f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A ) /\ g : ( 1 ... (♯ ` C ) ) -1-1-onto-> C ) -> G e. CMnd )
23		1zzd 9682	. . . . . 6 |- ( ( ( ph /\ f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A ) /\ g : ( 1 ... (♯ ` C ) ) -1-1-onto-> C ) -> 1 e. ZZ )
24	1	ad2antrr 492	. . . . . . . 8 |- ( ( ( ph /\ f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A ) /\ g : ( 1 ... (♯ ` C ) ) -1-1-onto-> C ) -> A e. Fin )
25		hashcl 11242	. . . . . . . 8 |- ( A e. Fin -> (♯ ` A ) e. NN0 )
26	24, 25	syl 14	. . . . . . 7 |- ( ( ( ph /\ f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A ) /\ g : ( 1 ... (♯ ` C ) ) -1-1-onto-> C ) -> (♯ ` A ) e. NN0 )
27	26	nn0zd 9777	. . . . . 6 |- ( ( ( ph /\ f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A ) /\ g : ( 1 ... (♯ ` C ) ) -1-1-onto-> C ) -> (♯ ` A ) e. ZZ )
28		gsumclfi.f	. . . . . . . 8 |- ( ph -> F : A --> B )
29	28	ad2antrr 492	. . . . . . 7 |- ( ( ( ph /\ f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A ) /\ g : ( 1 ... (♯ ` C ) ) -1-1-onto-> C ) -> F : A --> B )
30		f1of 5642	. . . . . . . 8 |- ( f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A -> f : ( 1 ... (♯ ` A ) ) --> A )
31	30	ad2antlr 493	. . . . . . 7 |- ( ( ( ph /\ f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A ) /\ g : ( 1 ... (♯ ` C ) ) -1-1-onto-> C ) -> f : ( 1 ... (♯ ` A ) ) --> A )
32	29, 31	fcod 5556	. . . . . 6 |- ( ( ( ph /\ f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A ) /\ g : ( 1 ... (♯ ` C ) ) -1-1-onto-> C ) -> ( F o. f ) : ( 1 ... (♯ ` A ) ) --> B )
33		f1ocnv 5655	. . . . . . . 8 |- ( f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A -> `' f : A -1-1-onto-> ( 1 ... (♯ ` A ) ) )
34	33	ad2antlr 493	. . . . . . 7 |- ( ( ( ph /\ f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A ) /\ g : ( 1 ... (♯ ` C ) ) -1-1-onto-> C ) -> `' f : A -1-1-onto-> ( 1 ... (♯ ` A ) ) )
35	6	ad2antrr 492	. . . . . . . 8 |- ( ( ( ph /\ f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A ) /\ g : ( 1 ... (♯ ` C ) ) -1-1-onto-> C ) -> H : C -1-1-onto-> A )
36		simpr 110	. . . . . . . . 9 |- ( ( ( ph /\ f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A ) /\ g : ( 1 ... (♯ ` C ) ) -1-1-onto-> C ) -> g : ( 1 ... (♯ ` C ) ) -1-1-onto-> C )
37	10	ad2antrr 492	. . . . . . . . . . . 12 |- ( ( ( ph /\ f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A ) /\ g : ( 1 ... (♯ ` C ) ) -1-1-onto-> C ) -> A ~~ C )
38	13	ad2antrr 492	. . . . . . . . . . . . 13 |- ( ( ( ph /\ f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A ) /\ g : ( 1 ... (♯ ` C ) ) -1-1-onto-> C ) -> C e. Fin )
39		hashen 11245	. . . . . . . . . . . . 13 |- ( ( A e. Fin /\ C e. Fin ) -> ( (♯ ` A ) = (♯ ` C ) <-> A ~~ C ) )
40	24, 38, 39	syl2anc 415	. . . . . . . . . . . 12 |- ( ( ( ph /\ f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A ) /\ g : ( 1 ... (♯ ` C ) ) -1-1-onto-> C ) -> ( (♯ ` A ) = (♯ ` C ) <-> A ~~ C ) )
41	37, 40	mpbird 167	. . . . . . . . . . 11 |- ( ( ( ph /\ f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A ) /\ g : ( 1 ... (♯ ` C ) ) -1-1-onto-> C ) -> (♯ ` A ) = (♯ ` C ) )
42	41	oveq2d 6104	. . . . . . . . . 10 |- ( ( ( ph /\ f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A ) /\ g : ( 1 ... (♯ ` C ) ) -1-1-onto-> C ) -> ( 1 ... (♯ ` A ) ) = ( 1 ... (♯ ` C ) ) )
43	42	f1oeq2d 5638	. . . . . . . . 9 |- ( ( ( ph /\ f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A ) /\ g : ( 1 ... (♯ ` C ) ) -1-1-onto-> C ) -> ( g : ( 1 ... (♯ ` A ) ) -1-1-onto-> C <-> g : ( 1 ... (♯ ` C ) ) -1-1-onto-> C ) )
44	36, 43	mpbird 167	. . . . . . . 8 |- ( ( ( ph /\ f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A ) /\ g : ( 1 ... (♯ ` C ) ) -1-1-onto-> C ) -> g : ( 1 ... (♯ ` A ) ) -1-1-onto-> C )
45		f1oco 5665	. . . . . . . 8 |- ( ( H : C -1-1-onto-> A /\ g : ( 1 ... (♯ ` A ) ) -1-1-onto-> C ) -> ( H o. g ) : ( 1 ... (♯ ` A ) ) -1-1-onto-> A )
46	35, 44, 45	syl2anc 415	. . . . . . 7 |- ( ( ( ph /\ f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A ) /\ g : ( 1 ... (♯ ` C ) ) -1-1-onto-> C ) -> ( H o. g ) : ( 1 ... (♯ ` A ) ) -1-1-onto-> A )
47		f1oco 5665	. . . . . . 7 |- ( ( `' f : A -1-1-onto-> ( 1 ... (♯ ` A ) ) /\ ( H o. g ) : ( 1 ... (♯ ` A ) ) -1-1-onto-> A ) -> ( `' f o. ( H o. g ) ) : ( 1 ... (♯ ` A ) ) -1-1-onto-> ( 1 ... (♯ ` A ) ) )
48	34, 46, 47	syl2anc 415	. . . . . 6 |- ( ( ( ph /\ f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A ) /\ g : ( 1 ... (♯ ` C ) ) -1-1-onto-> C ) -> ( `' f o. ( H o. g ) ) : ( 1 ... (♯ ` A ) ) -1-1-onto-> ( 1 ... (♯ ` A ) ) )
49	19, 20, 22, 23, 27, 32, 48	gzsumreidx 14231	. . . . 5 |- ( ( ( ph /\ f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A ) /\ g : ( 1 ... (♯ ` C ) ) -1-1-onto-> C ) -> ( G gzsumgz ( F o. f ) ) = ( G gzsumgz ( ( F o. f ) o. ( `' f o. ( H o. g ) ) ) ) )
50		coass 5309	. . . . . . . 8 |- ( ( ( F o. f ) o. `' f ) o. ( H o. g ) ) = ( ( F o. f ) o. ( `' f o. ( H o. g ) ) )
51		f1of 5642	. . . . . . . . . . . . 13 |- ( `' f : A -1-1-onto-> ( 1 ... (♯ ` A ) ) -> `' f : A --> ( 1 ... (♯ ` A ) ) )
52	34, 51	syl 14	. . . . . . . . . . . 12 |- ( ( ( ph /\ f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A ) /\ g : ( 1 ... (♯ ` C ) ) -1-1-onto-> C ) -> `' f : A --> ( 1 ... (♯ ` A ) ) )
53	32, 52	fcod 5556	. . . . . . . . . . 11 |- ( ( ( ph /\ f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A ) /\ g : ( 1 ... (♯ ` C ) ) -1-1-onto-> C ) -> ( ( F o. f ) o. `' f ) : A --> B )
54	53	ffnd 5537	. . . . . . . . . 10 |- ( ( ( ph /\ f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A ) /\ g : ( 1 ... (♯ ` C ) ) -1-1-onto-> C ) -> ( ( F o. f ) o. `' f ) Fn A )
55	29	ffnd 5537	. . . . . . . . . 10 |- ( ( ( ph /\ f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A ) /\ g : ( 1 ... (♯ ` C ) ) -1-1-onto-> C ) -> F Fn A )
56		simpr 110	. . . . . . . . . . . . . 14 |- ( ( ph /\ f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A ) -> f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A )
57	56	ad2antrr 492	. . . . . . . . . . . . 13 |- ( ( ( ( ph /\ f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A ) /\ g : ( 1 ... (♯ ` C ) ) -1-1-onto-> C ) /\ x e. A ) -> f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A )
58	57, 33, 51	3syl 17	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A ) /\ g : ( 1 ... (♯ ` C ) ) -1-1-onto-> C ) /\ x e. A ) -> `' f : A --> ( 1 ... (♯ ` A ) ) )
59		fvco3 5779	. . . . . . . . . . . 12 |- ( ( `' f : A --> ( 1 ... (♯ ` A ) ) /\ x e. A ) -> ( ( ( F o. f ) o. `' f ) ` x ) = ( ( F o. f ) ` ( `' f ` x ) ) )
60	58, 59	sylancom 424	. . . . . . . . . . 11 |- ( ( ( ( ph /\ f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A ) /\ g : ( 1 ... (♯ ` C ) ) -1-1-onto-> C ) /\ x e. A ) -> ( ( ( F o. f ) o. `' f ) ` x ) = ( ( F o. f ) ` ( `' f ` x ) ) )
61	57, 30	syl 14	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A ) /\ g : ( 1 ... (♯ ` C ) ) -1-1-onto-> C ) /\ x e. A ) -> f : ( 1 ... (♯ ` A ) ) --> A )
62	52	ffvelcdmda 5846	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A ) /\ g : ( 1 ... (♯ ` C ) ) -1-1-onto-> C ) /\ x e. A ) -> ( `' f ` x ) e. ( 1 ... (♯ ` A ) ) )
63		fvco3 5779	. . . . . . . . . . . 12 |- ( ( f : ( 1 ... (♯ ` A ) ) --> A /\ ( `' f ` x ) e. ( 1 ... (♯ ` A ) ) ) -> ( ( F o. f ) ` ( `' f ` x ) ) = ( F ` ( f ` ( `' f ` x ) ) ) )
64	61, 62, 63	syl2anc 415	. . . . . . . . . . 11 |- ( ( ( ( ph /\ f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A ) /\ g : ( 1 ... (♯ ` C ) ) -1-1-onto-> C ) /\ x e. A ) -> ( ( F o. f ) ` ( `' f ` x ) ) = ( F ` ( f ` ( `' f ` x ) ) ) )
65		f1ocnvfv2 5987	. . . . . . . . . . . . 13 |- ( ( f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A /\ x e. A ) -> ( f ` ( `' f ` x ) ) = x )
66	57, 65	sylancom 424	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A ) /\ g : ( 1 ... (♯ ` C ) ) -1-1-onto-> C ) /\ x e. A ) -> ( f ` ( `' f ` x ) ) = x )
67	66	fveq2d 5702	. . . . . . . . . . 11 |- ( ( ( ( ph /\ f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A ) /\ g : ( 1 ... (♯ ` C ) ) -1-1-onto-> C ) /\ x e. A ) -> ( F ` ( f ` ( `' f ` x ) ) ) = ( F ` x ) )
68	60, 64, 67	3eqtrd 2275	. . . . . . . . . 10 |- ( ( ( ( ph /\ f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A ) /\ g : ( 1 ... (♯ ` C ) ) -1-1-onto-> C ) /\ x e. A ) -> ( ( ( F o. f ) o. `' f ) ` x ) = ( F ` x ) )
69	54, 55, 68	eqfnfvd 5812	. . . . . . . . 9 |- ( ( ( ph /\ f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A ) /\ g : ( 1 ... (♯ ` C ) ) -1-1-onto-> C ) -> ( ( F o. f ) o. `' f ) = F )
70	69	coeq1d 4943	. . . . . . . 8 |- ( ( ( ph /\ f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A ) /\ g : ( 1 ... (♯ ` C ) ) -1-1-onto-> C ) -> ( ( ( F o. f ) o. `' f ) o. ( H o. g ) ) = ( F o. ( H o. g ) ) )
71	50, 70	eqtr3id 2285	. . . . . . 7 |- ( ( ( ph /\ f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A ) /\ g : ( 1 ... (♯ ` C ) ) -1-1-onto-> C ) -> ( ( F o. f ) o. ( `' f o. ( H o. g ) ) ) = ( F o. ( H o. g ) ) )
72		coass 5309	. . . . . . 7 |- ( ( F o. H ) o. g ) = ( F o. ( H o. g ) )
73	71, 72	eqtr4di 2289	. . . . . 6 |- ( ( ( ph /\ f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A ) /\ g : ( 1 ... (♯ ` C ) ) -1-1-onto-> C ) -> ( ( F o. f ) o. ( `' f o. ( H o. g ) ) ) = ( ( F o. H ) o. g ) )
74	73	oveq2d 6104	. . . . 5 |- ( ( ( ph /\ f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A ) /\ g : ( 1 ... (♯ ` C ) ) -1-1-onto-> C ) -> ( G gzsumgz ( ( F o. f ) o. ( `' f o. ( H o. g ) ) ) ) = ( G gzsumgz ( ( F o. H ) o. g ) ) )
75	49, 74	eqtrd 2271	. . . 4 |- ( ( ( ph /\ f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A ) /\ g : ( 1 ... (♯ ` C ) ) -1-1-onto-> C ) -> ( G gzsumgz ( F o. f ) ) = ( G gzsumgz ( ( F o. H ) o. g ) ) )
76	21	adantr 276	. . . . . 6 |- ( ( ph /\ f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A ) -> G e. CMnd )
77	28	adantr 276	. . . . . 6 |- ( ( ph /\ f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A ) -> F : A --> B )
78	1	adantr 276	. . . . . 6 |- ( ( ph /\ f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A ) -> A e. Fin )
79	19, 76, 77, 78, 56	gsumvalfi 14242	. . . . 5 |- ( ( ph /\ f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A ) -> ( G gsumg F ) = ( G gzsumgz ( F o. f ) ) )
80	79	adantr 276	. . . 4 |- ( ( ( ph /\ f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A ) /\ g : ( 1 ... (♯ ` C ) ) -1-1-onto-> C ) -> ( G gsumg F ) = ( G gzsumgz ( F o. f ) ) )
81		f1of 5642	. . . . . . . 8 |- ( H : C -1-1-onto-> A -> H : C --> A )
82	6, 81	syl 14	. . . . . . 7 |- ( ph -> H : C --> A )
83	82	ad2antrr 492	. . . . . 6 |- ( ( ( ph /\ f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A ) /\ g : ( 1 ... (♯ ` C ) ) -1-1-onto-> C ) -> H : C --> A )
84	29, 83	fcod 5556	. . . . 5 |- ( ( ( ph /\ f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A ) /\ g : ( 1 ... (♯ ` C ) ) -1-1-onto-> C ) -> ( F o. H ) : C --> B )
85	19, 22, 84, 38, 36	gsumvalfi 14242	. . . 4 |- ( ( ( ph /\ f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A ) /\ g : ( 1 ... (♯ ` C ) ) -1-1-onto-> C ) -> ( G gsumg ( F o. H ) ) = ( G gzsumgz ( ( F o. H ) o. g ) ) )
86	75, 80, 85	3eqtr4d 2281	. . 3 |- ( ( ( ph /\ f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A ) /\ g : ( 1 ... (♯ ` C ) ) -1-1-onto-> C ) -> ( G gsumg F ) = ( G gsumg ( F o. H ) ) )
87	18, 86	exlimddv 1954	. 2 |- ( ( ph /\ f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A ) -> ( G gsumg F ) = ( G gsumg ( F o. H ) ) )
88	5, 87	exlimddv 1954	1 |- ( ph -> ( G gsumg F ) = ( G gsumg ( F o. H ) ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   E.wex 1545   e. wcel 2209   class class class wbr 4131   `'ccnv 4775   o. ccom 4780   -->wf 5376   -1-1-onto->wf1o 5379   ` cfv 5380  (class class class)co 6088   ~~ cen 7024   Fincfn 7026   1c1 8186   NN0cn0 9574   ...cfz 10428  ♯chash 11236   Basecbs 13410   0gc0g 13669   gzsum_gz cgzsu 13670  CMndccmn 14177   gsum_g cgsu 14240

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737  ax-cnex 8276  ax-resscn 8277  ax-1cn 8278  ax-1re 8279  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-mulrcl 8284  ax-addcom 8285  ax-mulcom 8286  ax-addass 8287  ax-mulass 8288  ax-distr 8289  ax-i2m1 8290  ax-0lt1 8291  ax-1rid 8292  ax-0id 8293  ax-rnegex 8294  ax-precex 8295  ax-cnre 8296  ax-pre-ltirr 8297  ax-pre-ltwlin 8298  ax-pre-lttrn 8299  ax-pre-apti 8300  ax-pre-ltadd 8301  ax-pre-mulgt0 8302

This proof depends on definitions:  df-bi 117  df-stab 843  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-id 4440  df-iord 4513  df-on 4515  df-ilim 4516  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-recs 6580  df-frec 6666  df-1o 6691  df-er 6811  df-en 7027  df-dom 7028  df-fin 7029  df-pnf 8368  df-mnf 8369  df-xr 8370  df-ltxr 8371  df-le 8372  df-sub 8506  df-neg 8507  df-reap 8911  df-ap 8918  df-inn 9314  df-2 9372  df-n0 9575  df-z 9656  df-uz 9938  df-fz 10429  df-fzo 10567  df-seqfrec 10906  df-ihash 11237  df-ndx 13413  df-slot 13414  df-base 13416  df-plusg 13503  df-0g 13671  df-gzsum 13672  df-mgm 13735  df-sgrp 13776  df-mnd 13789  df-cmn 14179  df-gsumfi 14241

This theorem is used by:  gsum2dfilem2  14272  gsumcom2fi  14275  psrass1filem  15177  lgseisenlem3  16396