Theorem gsumzf1o 17624

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 16699	. . . . . . 7 |- ( ( G e. Mnd /\ A e. V ) -> ( G gsumg ( k e. A |-> .0. ) ) = .0. )
5	1, 2, 4	syl2anc 673	. . . . . 6 |- ( ph -> ( G gsumg ( k e. A |-> .0. ) ) = .0. )
6		gsumzf1o.h	. . . . . . . . 9 |- ( ph -> H : C -1-1-onto-> A )
7		f1of1 5827	. . . . . . . . 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 6782	. . . . . . . 8 |- ( ( H : C -1-1-> A /\ A e. V ) -> C e. _V )
10	8, 2, 9	syl2anc 673	. . . . . . 7 |- ( ph -> C e. _V )
11	3	gsumz 16699	. . . . . . 7 |- ( ( G e. Mnd /\ C e. _V ) -> ( G gsumg ( x e. C |-> .0. ) ) = .0. )
12	1, 10, 11	syl2anc 673	. . . . . 6 |- ( ph -> ( G gsumg ( x e. C |-> .0. ) ) = .0. )
13	5, 12	eqtr4d 2508	. . . . 5 |- ( ph -> ( G gsumg ( k e. A |-> .0. ) ) = ( G gsumg ( x e. C |-> .0. ) ) )
14	13	adantr 472	. . . 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 5889	. . . . . . . 8 |- ( 0g ` G ) e. _V
17	3, 16	eqeltri 2545	. . . . . . 7 |- .0. e. _V
18	17	a1i 11	. . . . . 6 |- ( ph -> .0. e. _V )
19		ssid 3437	. . . . . . 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 17620	. . . . 5 |- ( ( ph /\ ( F supp .0. ) = (/) ) -> F = ( k e. A |-> .0. ) )
22	21	oveq2d 6324	. . . 4 |- ( ( ph /\ ( F supp .0. ) = (/) ) -> ( G gsumg F ) = ( G gsumg ( k e. A |-> .0. ) ) )
23		f1of 5828	. . . . . . . . 9 |- ( H : C -1-1-onto-> A -> H : C --> A )
24	6, 23	syl 17	. . . . . . . 8 |- ( ph -> H : C --> A )
25	24	adantr 472	. . . . . . 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 5932	. . . . . 6 |- ( ( ph /\ ( F supp .0. ) = (/) ) -> H = ( x e. C |-> ( H ` x ) ) )
28		eqidd 2472	. . . . . 6 |- ( k = ( H ` x ) -> .0. = .0. )
29	26, 27, 21, 28	fmptco 6072	. . . . 5 |- ( ( ph /\ ( F supp .0. ) = (/) ) -> ( F o. H ) = ( x e. C |-> .0. ) )
30	29	oveq2d 6324	. . . 4 |- ( ( ph /\ ( F supp .0. ) = (/) ) -> ( G gsumg ( F o. H ) ) = ( G gsumg ( x e. C |-> .0. ) ) )
31	14, 22, 30	3eqtr4d 2515	. . 3 |- ( ( ph /\ ( F supp .0. ) = (/) ) -> ( G gsumg F ) = ( G gsumg ( F o. H ) ) )
32	31	ex 441	. 2 |- ( ph -> ( ( F supp .0. ) = (/) -> ( G gsumg F ) = ( G gsumg ( F o. H ) ) ) )
33		coass 5361	. . . . . . . . . . 11 |- ( ( H o. `' H ) o. f ) = ( H o. ( `' H o. f ) )
34	6	adantr 472	. . . . . . . . . . . . . 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 5854	. . . . . . . . . . . . . 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 5001	. . . . . . . . . . . 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 5827	. . . . . . . . . . . . . . 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 743	. . . . . . . . . . . . . 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 6946	. . . . . . . . . . . . . . . 16 |- ( F supp .0. ) C_ dom F
41		fdm 5745	. . . . . . . . . . . . . . . . 17 |- ( F : A --> B -> dom F = A )
42	15, 41	syl 17	. . . . . . . . . . . . . . . 16 |- ( ph -> dom F = A )
43	40, 42	syl5sseq 3466	. . . . . . . . . . . . . . 15 |- ( ph -> ( F supp .0. ) C_ A )
44	43	adantr 472	. . . . . . . . . . . . . 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 5797	. . . . . . . . . . . . . 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 673	. . . . . . . . . . . . 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 5792	. . . . . . . . . . . . 13 |- ( f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-> A -> f : ( 1 ... ( # ` ( F supp .0. ) ) ) --> A )
48		fcoi2 5770	. . . . . . . . . . . . 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 2505	. . . . . . . . . . 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 2520	. . . . . . . . . 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 5002	. . . . . . . . 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 5361	. . . . . . . . 9 |- ( ( F o. H ) o. ( `' H o. f ) ) = ( F o. ( H o. ( `' H o. f ) ) )
54	52, 53	syl6eqr 2523	. . . . . . . 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 12259	. . . . . . 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 5881	. . . . . 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 2471	. . . . . . 7 |- ( +g ` G ) = ( +g ` G )
59		gsumzcl.z	. . . . . . 7 |- Z = (Cntz ` G )
60	1	adantr 472	. . . . . . 7 |- ( ( ph /\ ( ( # ` ( F supp .0. ) ) e. NN /\ f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) ) ) -> G e. Mnd )
61	2	adantr 472	. . . . . . 7 |- ( ( ph /\ ( ( # ` ( F supp .0. ) ) e. NN /\ f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) ) ) -> A e. V )
62	15	adantr 472	. . . . . . 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 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 ) )
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		f1ofo 5835	. . . . . . . . . 10 |- ( f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) -> f : ( 1 ... ( # ` ( F supp .0. ) ) ) -onto-> ( F supp .0. ) )
67		forn 5809	. . . . . . . . . 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 743	. . . . . . . 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 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 )
71		eqid 2471	. . . . . . 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 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. ) ) ) )
73	10	adantr 472	. . . . . . 7 |- ( ( ph /\ ( ( # ` ( F supp .0. ) ) e. NN /\ f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) ) ) -> C e. _V )
74		fco 5751	. . . . . . . . 9 |- ( ( F : A --> B /\ H : C --> A ) -> ( F o. H ) : C --> B )
75	15, 24, 74	syl2anc 673	. . . . . . . 8 |- ( ph -> ( F o. H ) : C --> B )
76	75	adantr 472	. . . . . . 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 5101	. . . . . . . . 9 |- ran ( F o. H ) C_ ran F
78	59	cntzidss 17069	. . . . . . . . 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 675	. . . . . . . 8 |- ( ph -> ran ( F o. H ) C_ ( Z ` ran ( F o. H ) ) )
80	79	adantr 472	. . . . . . 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 5840	. . . . . . . . . 10 |- ( H : C -1-1-onto-> A -> `' H : A -1-1-onto-> C )
82		f1of1 5827	. . . . . . . . . 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 472	. . . . . . . 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 5801	. . . . . . . 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 673	. . . . . . 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 6155	. . . . . . . . . . . . . . 15 |- ( ( F : A --> B /\ A e. V ) -> F e. _V )
89	15, 2, 88	syl2anc 673	. . . . . . . . . . . . . 14 |- ( ph -> F e. _V )
90		suppimacnv 6944	. . . . . . . . . . . . . 14 |- ( ( F e. _V /\ .0. e. _V ) -> ( F supp .0. ) = ( `' F " ( _V \ { .0. } ) ) )
91	89, 17, 90	sylancl 675	. . . . . . . . . . . . 13 |- ( ph -> ( F supp .0. ) = ( `' F " ( _V \ { .0. } ) ) )
92	91	eqcomd 2477	. . . . . . . . . . . 12 |- ( ph -> ( `' F " ( _V \ { .0. } ) ) = ( F supp .0. ) )
93	92	adantr 472	. . . . . . . . . . 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 3461	. . . . . . . . . 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 5210	. . . . . . . . . 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 5025	. . . . . . . . . . 11 |- `' ( F o. H ) = ( `' H o. `' F )
98	97	imaeq1i 5171	. . . . . . . . . 10 |- ( `' ( F o. H ) " ( _V \ { .0. } ) ) = ( ( `' H o. `' F ) " ( _V \ { .0. } ) )
99		imaco 5347	. . . . . . . . . 10 |- ( ( `' H o. `' F ) " ( _V \ { .0. } ) ) = ( `' H " ( `' F " ( _V \ { .0. } ) ) )
100	98, 99	eqtri 2493	. . . . . . . . 9 |- ( `' ( F o. H ) " ( _V \ { .0. } ) ) = ( `' H " ( `' F " ( _V \ { .0. } ) ) )
101		rnco2 5349	. . . . . . . . 9 |- ran ( `' H o. f ) = ( `' H " ran f )
102	96, 100, 101	3sstr4g 3459	. . . . . . . 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 6761	. . . . . . . . . . . . 13 |- ( ( H : C -1-1-onto-> A /\ A e. V ) -> H e. _V )
104	6, 2, 103	syl2anc 673	. . . . . . . . . . . 12 |- ( ph -> H e. _V )
105		coexg 6763	. . . . . . . . . . . 12 |- ( ( F e. _V /\ H e. _V ) -> ( F o. H ) e. _V )
106	89, 104, 105	syl2anc 673	. . . . . . . . . . 11 |- ( ph -> ( F o. H ) e. _V )
107		suppimacnv 6944	. . . . . . . . . . 11 |- ( ( ( F o. H ) e. _V /\ .0. e. _V ) -> ( ( F o. H ) supp .0. ) = ( `' ( F o. H ) " ( _V \ { .0. } ) ) )
108	106, 17, 107	sylancl 675	. . . . . . . . . 10 |- ( ph -> ( ( F o. H ) supp .0. ) = ( `' ( F o. H ) " ( _V \ { .0. } ) ) )
109	108	sseq1d 3445	. . . . . . . . 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 472	. . . . . . . 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 240	. . . . . . 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 2471	. . . . . . 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 17619	. . . . . 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 2515	. . . . 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 626	. . . 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 1787	. . 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 614	. 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 7907	. . . 4 |- ( F finSupp .0. -> ( Fun F /\ ( F supp .0. ) e. Fin ) )
120	119	simprd 470	. . 3 |- ( F finSupp .0. -> ( F supp .0. ) e. Fin )
121		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. ) ) ) )
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 388	1 |- ( ph -> ( G gsumg F ) = ( G gsumg ( F o. H ) ) )

Syntax hints:   -> wi 4   <-> wb 189   \/ wo 375   /\ wa 376   = wceq 1452   E.wex 1671   e. wcel 1904   _Vcvv 3031   \ cdif 3387   C_ wss 3390   (/)c0 3722   {csn 3959   class class class wbr 4395   |-> cmpt 4454   _I cid 4749   `'ccnv 4838   dom cdm 4839   ran crn 4840   |` cres 4841   "cima 4842   o. ccom 4843   Fun wfun 5583   -->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:  gsumf1o  17628  smadiadetlem3  19770