Theorem gsumzf1o 17545

Description: Re-index a finite group sum using a bijection. (Contributed by Mario Carneiro, 24-Apr-2016.) (Revised by AV, 2-Jun-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 ) )
gsumzcl.w	|- ( ph -> F finSupp .0. )
gsumzf1o.h	|- ( ph -> H : C -1-1-onto-> A )

Assertion

Ref	Expression
gsumzf1o	|- ( ph -> ( G gsumg F ) = ( G gsumg ( F o. H ) ) )

Proof of Theorem gsumzf1o

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

Step	Hyp	Ref	Expression
1		gsumzcl.g	. . . . . . 7 |- ( ph -> G e. Mnd )
2		gsumzcl.a	. . . . . . 7 |- ( ph -> A e. V )
3		gsumzcl.0	. . . . . . . 8 |- .0. = ( 0g ` G )
4	3	gsumz 16620	. . . . . . 7 |- ( ( G e. Mnd /\ A e. V ) -> ( G gsumg ( k e. A |-> .0. ) ) = .0. )
5	1, 2, 4	syl2anc 665	. . . . . 6 |- ( ph -> ( G gsumg ( k e. A |-> .0. ) ) = .0. )
6		gsumzf1o.h	. . . . . . . . 9 |- ( ph -> H : C -1-1-onto-> A )
7		f1of1 5830	. . . . . . . . 9 |- ( H : C -1-1-onto-> A -> H : C -1-1-> A )
8	6, 7	syl 17	. . . . . . . 8 |- ( ph -> H : C -1-1-> A )
9		f1dmex 6777	. . . . . . . 8 |- ( ( H : C -1-1-> A /\ A e. V ) -> C e. _V )
10	8, 2, 9	syl2anc 665	. . . . . . 7 |- ( ph -> C e. _V )
11	3	gsumz 16620	. . . . . . 7 |- ( ( G e. Mnd /\ C e. _V ) -> ( G gsumg ( x e. C |-> .0. ) ) = .0. )
12	1, 10, 11	syl2anc 665	. . . . . 6 |- ( ph -> ( G gsumg ( x e. C |-> .0. ) ) = .0. )
13	5, 12	eqtr4d 2466	. . . . 5 |- ( ph -> ( G gsumg ( k e. A |-> .0. ) ) = ( G gsumg ( x e. C |-> .0. ) ) )
14	13	adantr 466	. . . 4 |- ( ( ph /\ ( F supp .0. ) = (/) ) -> ( G gsumg ( k e. A |-> .0. ) ) = ( G gsumg ( x e. C |-> .0. ) ) )
15		gsumzcl.f	. . . . . 6 |- ( ph -> F : A --> B )
16		fvex 5891	. . . . . . . 8 |- ( 0g ` G ) e. _V
17	3, 16	eqeltri 2503	. . . . . . 7 |- .0. e. _V
18	17	a1i 11	. . . . . 6 |- ( ph -> .0. e. _V )
19		ssid 3483	. . . . . . 7 |- ( F supp .0. ) C_ ( F supp .0. )
20	19	a1i 11	. . . . . 6 |- ( ph -> ( F supp .0. ) C_ ( F supp .0. ) )
21	15, 2, 18, 20	gsumcllem 17541	. . . . 5 |- ( ( ph /\ ( F supp .0. ) = (/) ) -> F = ( k e. A |-> .0. ) )
22	21	oveq2d 6321	. . . 4 |- ( ( ph /\ ( F supp .0. ) = (/) ) -> ( G gsumg F ) = ( G gsumg ( k e. A |-> .0. ) ) )
23		f1of 5831	. . . . . . . . 9 |- ( H : C -1-1-onto-> A -> H : C --> A )
24	6, 23	syl 17	. . . . . . . 8 |- ( ph -> H : C --> A )
25	24	adantr 466	. . . . . . 7 |- ( ( ph /\ ( F supp .0. ) = (/) ) -> H : C --> A )
26	25	ffvelrnda 6037	. . . . . 6 |- ( ( ( ph /\ ( F supp .0. ) = (/) ) /\ x e. C ) -> ( H ` x ) e. A )
27	25	feqmptd 5934	. . . . . 6 |- ( ( ph /\ ( F supp .0. ) = (/) ) -> H = ( x e. C |-> ( H ` x ) ) )
28		eqidd 2423	. . . . . 6 |- ( k = ( H ` x ) -> .0. = .0. )
29	26, 27, 21, 28	fmptco 6071	. . . . 5 |- ( ( ph /\ ( F supp .0. ) = (/) ) -> ( F o. H ) = ( x e. C |-> .0. ) )
30	29	oveq2d 6321	. . . 4 |- ( ( ph /\ ( F supp .0. ) = (/) ) -> ( G gsumg ( F o. H ) ) = ( G gsumg ( x e. C |-> .0. ) ) )
31	14, 22, 30	3eqtr4d 2473	. . 3 |- ( ( ph /\ ( F supp .0. ) = (/) ) -> ( G gsumg F ) = ( G gsumg ( F o. H ) ) )
32	31	ex 435	. 2 |- ( ph -> ( ( F supp .0. ) = (/) -> ( G gsumg F ) = ( G gsumg ( F o. H ) ) ) )
33		coass 5373	. . . . . . . . . . 11 |- ( ( H o. `' H ) o. f ) = ( H o. ( `' H o. f ) )
34	6	adantr 466	. . . . . . . . . . . . . 14 |- ( ( ph /\ ( ( # ` ( F supp .0. ) ) e. NN /\ f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) ) ) -> H : C -1-1-onto-> A )
35		f1ococnv2 5857	. . . . . . . . . . . . . 14 |- ( H : C -1-1-onto-> A -> ( H o. `' H ) = ( _I |` A ) )
36	34, 35	syl 17	. . . . . . . . . . . . 13 |- ( ( ph /\ ( ( # ` ( F supp .0. ) ) e. NN /\ f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) ) ) -> ( H o. `' H ) = ( _I |` A ) )
37	36	coeq1d 5015	. . . . . . . . . . . 12 |- ( ( ph /\ ( ( # ` ( F supp .0. ) ) e. NN /\ f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) ) ) -> ( ( H o. `' H ) o. f ) = ( ( _I |` A ) o. f ) )
38		f1of1 5830	. . . . . . . . . . . . . . 15 |- ( f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) -> f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-> ( F supp .0. ) )
39	38	ad2antll 733	. . . . . . . . . . . . . 14 |- ( ( 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. ) )
40		suppssdm 6938	. . . . . . . . . . . . . . . 16 |- ( F supp .0. ) C_ dom F
41		fdm 5750	. . . . . . . . . . . . . . . . 17 |- ( F : A --> B -> dom F = A )
42	15, 41	syl 17	. . . . . . . . . . . . . . . 16 |- ( ph -> dom F = A )
43	40, 42	syl5sseq 3512	. . . . . . . . . . . . . . 15 |- ( ph -> ( F supp .0. ) C_ A )
44	43	adantr 466	. . . . . . . . . . . . . 14 |- ( ( ph /\ ( ( # ` ( F supp .0. ) ) e. NN /\ f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) ) ) -> ( F supp .0. ) C_ A )
45		f1ss 5801	. . . . . . . . . . . . . 14 |- ( ( f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-> ( F supp .0. ) /\ ( F supp .0. ) C_ A ) -> f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-> A )
46	39, 44, 45	syl2anc 665	. . . . . . . . . . . . 13 |- ( ( 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 )
47		f1f 5796	. . . . . . . . . . . . 13 |- ( f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-> A -> f : ( 1 ... ( # ` ( F supp .0. ) ) ) --> A )
48		fcoi2 5775	. . . . . . . . . . . . 13 |- ( f : ( 1 ... ( # ` ( F supp .0. ) ) ) --> A -> ( ( _I |` A ) o. f ) = f )
49	46, 47, 48	3syl 18	. . . . . . . . . . . 12 |- ( ( ph /\ ( ( # ` ( F supp .0. ) ) e. NN /\ f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) ) ) -> ( ( _I |` A ) o. f ) = f )
50	37, 49	eqtrd 2463	. . . . . . . . . . 11 |- ( ( ph /\ ( ( # ` ( F supp .0. ) ) e. NN /\ f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) ) ) -> ( ( H o. `' H ) o. f ) = f )
51	33, 50	syl5reqr 2478	. . . . . . . . . 10 |- ( ( ph /\ ( ( # ` ( F supp .0. ) ) e. NN /\ f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) ) ) -> f = ( H o. ( `' H o. f ) ) )
52	51	coeq2d 5016	. . . . . . . . 9 |- ( ( ph /\ ( ( # ` ( F supp .0. ) ) e. NN /\ f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) ) ) -> ( F o. f ) = ( F o. ( H o. ( `' H o. f ) ) ) )
53		coass 5373	. . . . . . . . 9 |- ( ( F o. H ) o. ( `' H o. f ) ) = ( F o. ( H o. ( `' H o. f ) ) )
54	52, 53	syl6eqr 2481	. . . . . . . 8 |- ( ( ph /\ ( ( # ` ( F supp .0. ) ) e. NN /\ f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) ) ) -> ( F o. f ) = ( ( F o. H ) o. ( `' H o. f ) ) )
55	54	seqeq3d 12227	. . . . . . 7 |- ( ( ph /\ ( ( # ` ( F supp .0. ) ) e. NN /\ f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) ) ) -> seq 1 ( ( +g ` G ) , ( F o. f ) ) = seq 1 ( ( +g ` G ) , ( ( F o. H ) o. ( `' H o. f ) ) ) )
56	55	fveq1d 5883	. . . . . 6 |- ( ( ph /\ ( ( # ` ( F supp .0. ) ) e. NN /\ f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) ) ) -> ( seq 1 ( ( +g ` G ) , ( F o. f ) ) ` ( # ` ( F supp .0. ) ) ) = ( seq 1 ( ( +g ` G ) , ( ( F o. H ) o. ( `' H o. f ) ) ) ` ( # ` ( F supp .0. ) ) ) )
57		gsumzcl.b	. . . . . . 7 |- B = ( Base ` G )
58		eqid 2422	. . . . . . 7 |- ( +g ` G ) = ( +g ` G )
59		gsumzcl.z	. . . . . . 7 |- Z = (Cntz ` G )
60	1	adantr 466	. . . . . . 7 |- ( ( ph /\ ( ( # ` ( F supp .0. ) ) e. NN /\ f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) ) ) -> G e. Mnd )
61	2	adantr 466	. . . . . . 7 |- ( ( ph /\ ( ( # ` ( F supp .0. ) ) e. NN /\ f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) ) ) -> A e. V )
62	15	adantr 466	. . . . . . 7 |- ( ( ph /\ ( ( # ` ( F supp .0. ) ) e. NN /\ f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) ) ) -> F : A --> B )
63		gsumzcl.c	. . . . . . . 8 |- ( ph -> ran F C_ ( Z ` ran F ) )
64	63	adantr 466	. . . . . . 7 |- ( ( ph /\ ( ( # ` ( F supp .0. ) ) e. NN /\ f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) ) ) -> ran F C_ ( Z ` ran F ) )
65		simprl 762	. . . . . . 7 |- ( ( ph /\ ( ( # ` ( F supp .0. ) ) e. NN /\ f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) ) ) -> ( # ` ( F supp .0. ) ) e. NN )
66		f1ofo 5838	. . . . . . . . . 10 |- ( f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) -> f : ( 1 ... ( # ` ( F supp .0. ) ) ) -onto-> ( F supp .0. ) )
67		forn 5813	. . . . . . . . . 10 |- ( f : ( 1 ... ( # ` ( F supp .0. ) ) ) -onto-> ( F supp .0. ) -> ran f = ( F supp .0. ) )
68	66, 67	syl 17	. . . . . . . . 9 |- ( f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) -> ran f = ( F supp .0. ) )
69	68	ad2antll 733	. . . . . . . 8 |- ( ( ph /\ ( ( # ` ( F supp .0. ) ) e. NN /\ f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) ) ) -> ran f = ( F supp .0. ) )
70	19, 69	syl5sseqr 3513	. . . . . . 7 |- ( ( ph /\ ( ( # ` ( F supp .0. ) ) e. NN /\ f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) ) ) -> ( F supp .0. ) C_ ran f )
71		eqid 2422	. . . . . . 7 |- ( ( F o. f ) supp .0. ) = ( ( F o. f ) supp .0. )
72	57, 3, 58, 59, 60, 61, 62, 64, 65, 46, 70, 71	gsumval3 17540	. . . . . 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. ) ) ) )
73	10	adantr 466	. . . . . . 7 |- ( ( ph /\ ( ( # ` ( F supp .0. ) ) e. NN /\ f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) ) ) -> C e. _V )
74		fco 5756	. . . . . . . . 9 |- ( ( F : A --> B /\ H : C --> A ) -> ( F o. H ) : C --> B )
75	15, 24, 74	syl2anc 665	. . . . . . . 8 |- ( ph -> ( F o. H ) : C --> B )
76	75	adantr 466	. . . . . . 7 |- ( ( ph /\ ( ( # ` ( F supp .0. ) ) e. NN /\ f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) ) ) -> ( F o. H ) : C --> B )
77		rncoss 5114	. . . . . . . . 9 |- ran ( F o. H ) C_ ran F
78	59	cntzidss 16990	. . . . . . . . 9 |- ( ( ran F C_ ( Z ` ran F ) /\ ran ( F o. H ) C_ ran F ) -> ran ( F o. H ) C_ ( Z ` ran ( F o. H ) ) )
79	63, 77, 78	sylancl 666	. . . . . . . 8 |- ( ph -> ran ( F o. H ) C_ ( Z ` ran ( F o. H ) ) )
80	79	adantr 466	. . . . . . 7 |- ( ( ph /\ ( ( # ` ( F supp .0. ) ) e. NN /\ f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) ) ) -> ran ( F o. H ) C_ ( Z ` ran ( F o. H ) ) )
81		f1ocnv 5843	. . . . . . . . . 10 |- ( H : C -1-1-onto-> A -> `' H : A -1-1-onto-> C )
82		f1of1 5830	. . . . . . . . . 10 |- ( `' H : A -1-1-onto-> C -> `' H : A -1-1-> C )
83	6, 81, 82	3syl 18	. . . . . . . . 9 |- ( ph -> `' H : A -1-1-> C )
84	83	adantr 466	. . . . . . . 8 |- ( ( ph /\ ( ( # ` ( F supp .0. ) ) e. NN /\ f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) ) ) -> `' H : A -1-1-> C )
85		f1co 5805	. . . . . . . 8 |- ( ( `' H : A -1-1-> C /\ f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-> A ) -> ( `' H o. f ) : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-> C )
86	84, 46, 85	syl2anc 665	. . . . . . 7 |- ( ( ph /\ ( ( # ` ( F supp .0. ) ) e. NN /\ f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) ) ) -> ( `' H o. f ) : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-> C )
87	19	a1i 11	. . . . . . . . . . 11 |- ( ( ph /\ ( ( # ` ( F supp .0. ) ) e. NN /\ f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) ) ) -> ( F supp .0. ) C_ ( F supp .0. ) )
88		fex 6153	. . . . . . . . . . . . . . 15 |- ( ( F : A --> B /\ A e. V ) -> F e. _V )
89	15, 2, 88	syl2anc 665	. . . . . . . . . . . . . 14 |- ( ph -> F e. _V )
90		suppimacnv 6936	. . . . . . . . . . . . . 14 |- ( ( F e. _V /\ .0. e. _V ) -> ( F supp .0. ) = ( `' F " ( _V \ { .0. } ) ) )
91	89, 17, 90	sylancl 666	. . . . . . . . . . . . 13 |- ( ph -> ( F supp .0. ) = ( `' F " ( _V \ { .0. } ) ) )
92	91	eqcomd 2430	. . . . . . . . . . . 12 |- ( ph -> ( `' F " ( _V \ { .0. } ) ) = ( F supp .0. ) )
93	92	adantr 466	. . . . . . . . . . 11 |- ( ( ph /\ ( ( # ` ( F supp .0. ) ) e. NN /\ f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) ) ) -> ( `' F " ( _V \ { .0. } ) ) = ( F supp .0. ) )
94	87, 93, 69	3sstr4d 3507	. . . . . . . . . 10 |- ( ( ph /\ ( ( # ` ( F supp .0. ) ) e. NN /\ f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) ) ) -> ( `' F " ( _V \ { .0. } ) ) C_ ran f )
95		imass2 5223	. . . . . . . . . 10 |- ( ( `' F " ( _V \ { .0. } ) ) C_ ran f -> ( `' H " ( `' F " ( _V \ { .0. } ) ) ) C_ ( `' H " ran f ) )
96	94, 95	syl 17	. . . . . . . . 9 |- ( ( ph /\ ( ( # ` ( F supp .0. ) ) e. NN /\ f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) ) ) -> ( `' H " ( `' F " ( _V \ { .0. } ) ) ) C_ ( `' H " ran f ) )
97		cnvco 5039	. . . . . . . . . . 11 |- `' ( F o. H ) = ( `' H o. `' F )
98	97	imaeq1i 5184	. . . . . . . . . 10 |- ( `' ( F o. H ) " ( _V \ { .0. } ) ) = ( ( `' H o. `' F ) " ( _V \ { .0. } ) )
99		imaco 5359	. . . . . . . . . 10 |- ( ( `' H o. `' F ) " ( _V \ { .0. } ) ) = ( `' H " ( `' F " ( _V \ { .0. } ) ) )
100	98, 99	eqtri 2451	. . . . . . . . 9 |- ( `' ( F o. H ) " ( _V \ { .0. } ) ) = ( `' H " ( `' F " ( _V \ { .0. } ) ) )
101		rnco2 5361	. . . . . . . . 9 |- ran ( `' H o. f ) = ( `' H " ran f )
102	96, 100, 101	3sstr4g 3505	. . . . . . . 8 |- ( ( ph /\ ( ( # ` ( F supp .0. ) ) e. NN /\ f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) ) ) -> ( `' ( F o. H ) " ( _V \ { .0. } ) ) C_ ran ( `' H o. f ) )
103		f1oexrnex 6756	. . . . . . . . . . . . 13 |- ( ( H : C -1-1-onto-> A /\ A e. V ) -> H e. _V )
104	6, 2, 103	syl2anc 665	. . . . . . . . . . . 12 |- ( ph -> H e. _V )
105		coexg 6758	. . . . . . . . . . . 12 |- ( ( F e. _V /\ H e. _V ) -> ( F o. H ) e. _V )
106	89, 104, 105	syl2anc 665	. . . . . . . . . . 11 |- ( ph -> ( F o. H ) e. _V )
107		suppimacnv 6936	. . . . . . . . . . 11 |- ( ( ( F o. H ) e. _V /\ .0. e. _V ) -> ( ( F o. H ) supp .0. ) = ( `' ( F o. H ) " ( _V \ { .0. } ) ) )
108	106, 17, 107	sylancl 666	. . . . . . . . . 10 |- ( ph -> ( ( F o. H ) supp .0. ) = ( `' ( F o. H ) " ( _V \ { .0. } ) ) )
109	108	sseq1d 3491	. . . . . . . . 9 |- ( ph -> ( ( ( F o. H ) supp .0. ) C_ ran ( `' H o. f ) <-> ( `' ( F o. H ) " ( _V \ { .0. } ) ) C_ ran ( `' H o. f ) ) )
110	109	adantr 466	. . . . . . . 8 |- ( ( ph /\ ( ( # ` ( F supp .0. ) ) e. NN /\ f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) ) ) -> ( ( ( F o. H ) supp .0. ) C_ ran ( `' H o. f ) <-> ( `' ( F o. H ) " ( _V \ { .0. } ) ) C_ ran ( `' H o. f ) ) )
111	102, 110	mpbird 235	. . . . . . 7 |- ( ( ph /\ ( ( # ` ( F supp .0. ) ) e. NN /\ f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) ) ) -> ( ( F o. H ) supp .0. ) C_ ran ( `' H o. f ) )
112		eqid 2422	. . . . . . 7 |- ( ( ( F o. H ) o. ( `' H o. f ) ) supp .0. ) = ( ( ( F o. H ) o. ( `' H o. f ) ) supp .0. )
113	57, 3, 58, 59, 60, 73, 76, 80, 65, 86, 111, 112	gsumval3 17540	. . . . . 6 |- ( ( ph /\ ( ( # ` ( F supp .0. ) ) e. NN /\ f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) ) ) -> ( G gsumg ( F o. H ) ) = ( seq 1 ( ( +g ` G ) , ( ( F o. H ) o. ( `' H o. f ) ) ) ` ( # ` ( F supp .0. ) ) ) )
114	56, 72, 113	3eqtr4d 2473	. . . . 5 |- ( ( ph /\ ( ( # ` ( F supp .0. ) ) e. NN /\ f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) ) ) -> ( G gsumg F ) = ( G gsumg ( F o. H ) ) )
115	114	expr 618	. . . 4 |- ( ( ph /\ ( # ` ( F supp .0. ) ) e. NN ) -> ( f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) -> ( G gsumg F ) = ( G gsumg ( F o. H ) ) ) )
116	115	exlimdv 1772	. . 3 |- ( ( ph /\ ( # ` ( F supp .0. ) ) e. NN ) -> ( E. f f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) -> ( G gsumg F ) = ( G gsumg ( F o. H ) ) ) )
117	116	expimpd 606	. 2 |- ( ph -> ( ( ( # ` ( F supp .0. ) ) e. NN /\ E. f f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) ) -> ( G gsumg F ) = ( G gsumg ( F o. H ) ) ) )
118		gsumzcl.w	. . 3 |- ( ph -> F finSupp .0. )
119		fsuppimp 7898	. . . 4 |- ( F finSupp .0. -> ( Fun F /\ ( F supp .0. ) e. Fin ) )
120	119	simprd 464	. . 3 |- ( F finSupp .0. -> ( F supp .0. ) e. Fin )
121		fz1f1o 13775	. . 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. ) ) ) )
122	118, 120, 121	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. ) ) ) )
123	32, 117, 122	mpjaod 382	1 |- ( ph -> ( G gsumg F ) = ( G gsumg ( F o. H ) ) )

Syntax hints:   -> wi 4   <-> wb 187   \/ wo 369   /\ wa 370   = wceq 1437   E.wex 1657   e. wcel 1872   _Vcvv 3080   \ cdif 3433   C_ wss 3436   (/)c0 3761   {csn 3998   class class class wbr 4423   |-> cmpt 4482   _I cid 4763   `'ccnv 4852   dom cdm 4853   ran crn 4854   |` cres 4855   "cima 4856   o. ccom 4857   Fun wfun 5595   -->wf 5597   -1-1->wf1 5598   -onto->wfo 5599   -1-1-onto->wf1o 5600   ` cfv 5601  (class class class)co 6305   supp csupp 6925   Fincfn 7580   finSupp cfsupp 7892   1c1 9547   NNcn 10616   ...cfz 11791   seqcseq 12219   #chash 12521   Basecbs 15120   +g cplusg 15189   0gc0g 15337   gsum_g cgsu 15338   Mndcmnd 16534  Cntzccntz 16968

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-rep 4536  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597  ax-cnex 9602  ax-resscn 9603  ax-1cn 9604  ax-icn 9605  ax-addcl 9606  ax-addrcl 9607  ax-mulcl 9608  ax-mulrcl 9609  ax-mulcom 9610  ax-addass 9611  ax-mulass 9612  ax-distr 9613  ax-i2m1 9614  ax-1ne0 9615  ax-1rid 9616  ax-rnegex 9617  ax-rrecex 9618  ax-cnre 9619  ax-pre-lttri 9620  ax-pre-lttrn 9621  ax-pre-ltadd 9622  ax-pre-mulgt0 9623

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-nel 2617  df-ral 2776  df-rex 2777  df-reu 2778  df-rmo 2779  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-tp 4003  df-op 4005  df-uni 4220  df-int 4256  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-tr 4519  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-se 4813  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-isom 5610  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-supp 6926  df-wrecs 7039  df-recs 7101  df-rdg 7139  df-1o 7193  df-oadd 7197  df-er 7374  df-en 7581  df-dom 7582  df-sdom 7583  df-fin 7584  df-fsupp 7893  df-oi 8034  df-card 8381  df-pnf 9684  df-mnf 9685  df-xr 9686  df-ltxr 9687  df-le 9688  df-sub 9869  df-neg 9870  df-nn 10617  df-n0 10877  df-z 10945  df-uz 11167  df-fz 11792  df-fzo 11923  df-seq 12220  df-hash 12522  df-0g 15339  df-gsum 15340  df-mgm 16487  df-sgrp 16526  df-mnd 16536  df-cntz 16970

This theorem is referenced by:  gsumf1o  17549  smadiadetlem3  19691