Theorem gsumzf1o 16786

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 15874	. . . . . . 7 |- ( ( G e. Mnd /\ A e. V ) -> ( G gsumg ( k e. A |-> .0. ) ) = .0. )
5	1, 2, 4	syl2anc 661	. . . . . 6 |- ( ph -> ( G gsumg ( k e. A |-> .0. ) ) = .0. )
6		gsumzf1o.h	. . . . . . . . 9 |- ( ph -> H : C -1-1-onto-> A )
7		f1of1 5801	. . . . . . . . 9 |- ( H : C -1-1-onto-> A -> H : C -1-1-> A )
8	6, 7	syl 16	. . . . . . . 8 |- ( ph -> H : C -1-1-> A )
9		f1dmex 6751	. . . . . . . 8 |- ( ( H : C -1-1-> A /\ A e. V ) -> C e. _V )
10	8, 2, 9	syl2anc 661	. . . . . . 7 |- ( ph -> C e. _V )
11	3	gsumz 15874	. . . . . . 7 |- ( ( G e. Mnd /\ C e. _V ) -> ( G gsumg ( x e. C |-> .0. ) ) = .0. )
12	1, 10, 11	syl2anc 661	. . . . . 6 |- ( ph -> ( G gsumg ( x e. C |-> .0. ) ) = .0. )
13	5, 12	eqtr4d 2485	. . . . 5 |- ( ph -> ( G gsumg ( k e. A |-> .0. ) ) = ( G gsumg ( x e. C |-> .0. ) ) )
14	13	adantr 465	. . . 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 5862	. . . . . . . 8 |- ( 0g ` G ) e. _V
17	3, 16	eqeltri 2525	. . . . . . 7 |- .0. e. _V
18	17	a1i 11	. . . . . 6 |- ( ph -> .0. e. _V )
19		ssid 3505	. . . . . . 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 16781	. . . . 5 |- ( ( ph /\ ( F supp .0. ) = (/) ) -> F = ( k e. A |-> .0. ) )
22	21	oveq2d 6293	. . . 4 |- ( ( ph /\ ( F supp .0. ) = (/) ) -> ( G gsumg F ) = ( G gsumg ( k e. A |-> .0. ) ) )
23		f1of 5802	. . . . . . . . 9 |- ( H : C -1-1-onto-> A -> H : C --> A )
24	6, 23	syl 16	. . . . . . . 8 |- ( ph -> H : C --> A )
25	24	adantr 465	. . . . . . 7 |- ( ( ph /\ ( F supp .0. ) = (/) ) -> H : C --> A )
26	25	ffvelrnda 6012	. . . . . 6 |- ( ( ( ph /\ ( F supp .0. ) = (/) ) /\ x e. C ) -> ( H ` x ) e. A )
27	25	feqmptd 5907	. . . . . 6 |- ( ( ph /\ ( F supp .0. ) = (/) ) -> H = ( x e. C |-> ( H ` x ) ) )
28		eqidd 2442	. . . . . 6 |- ( k = ( H ` x ) -> .0. = .0. )
29	26, 27, 21, 28	fmptco 6045	. . . . 5 |- ( ( ph /\ ( F supp .0. ) = (/) ) -> ( F o. H ) = ( x e. C |-> .0. ) )
30	29	oveq2d 6293	. . . 4 |- ( ( ph /\ ( F supp .0. ) = (/) ) -> ( G gsumg ( F o. H ) ) = ( G gsumg ( x e. C |-> .0. ) ) )
31	14, 22, 30	3eqtr4d 2492	. . 3 |- ( ( ph /\ ( F supp .0. ) = (/) ) -> ( G gsumg F ) = ( G gsumg ( F o. H ) ) )
32	31	ex 434	. 2 |- ( ph -> ( ( F supp .0. ) = (/) -> ( G gsumg F ) = ( G gsumg ( F o. H ) ) ) )
33		coass 5512	. . . . . . . . . . 11 |- ( ( H o. `' H ) o. f ) = ( H o. ( `' H o. f ) )
34	6	adantr 465	. . . . . . . . . . . . . 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 5828	. . . . . . . . . . . . . 14 |- ( H : C -1-1-onto-> A -> ( H o. `' H ) = ( _I |` A ) )
36	34, 35	syl 16	. . . . . . . . . . . . 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 5150	. . . . . . . . . . . 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 5801	. . . . . . . . . . . . . . 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 728	. . . . . . . . . . . . . 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 6912	. . . . . . . . . . . . . . . 16 |- ( F supp .0. ) C_ dom F
41		fdm 5721	. . . . . . . . . . . . . . . . 17 |- ( F : A --> B -> dom F = A )
42	15, 41	syl 16	. . . . . . . . . . . . . . . 16 |- ( ph -> dom F = A )
43	40, 42	syl5sseq 3534	. . . . . . . . . . . . . . 15 |- ( ph -> ( F supp .0. ) C_ A )
44	43	adantr 465	. . . . . . . . . . . . . 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 5772	. . . . . . . . . . . . . 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 661	. . . . . . . . . . . . 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 5767	. . . . . . . . . . . . 13 |- ( f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-> A -> f : ( 1 ... ( # ` ( F supp .0. ) ) ) --> A )
48		fcoi2 5746	. . . . . . . . . . . . 13 |- ( f : ( 1 ... ( # ` ( F supp .0. ) ) ) --> A -> ( ( _I |` A ) o. f ) = f )
49	46, 47, 48	3syl 20	. . . . . . . . . . . 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 2482	. . . . . . . . . . 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 2497	. . . . . . . . . 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 5151	. . . . . . . . 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 5512	. . . . . . . . 9 |- ( ( F o. H ) o. ( `' H o. f ) ) = ( F o. ( H o. ( `' H o. f ) ) )
54	52, 53	syl6eqr 2500	. . . . . . . 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 12089	. . . . . . 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 5854	. . . . . 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 2441	. . . . . . 7 |- ( +g ` G ) = ( +g ` G )
59		gsumzcl.z	. . . . . . 7 |- Z = (Cntz ` G )
60	1	adantr 465	. . . . . . 7 |- ( ( ph /\ ( ( # ` ( F supp .0. ) ) e. NN /\ f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) ) ) -> G e. Mnd )
61	2	adantr 465	. . . . . . 7 |- ( ( ph /\ ( ( # ` ( F supp .0. ) ) e. NN /\ f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) ) ) -> A e. V )
62	15	adantr 465	. . . . . . 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 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 ) )
65		simprl 755	. . . . . . 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 5809	. . . . . . . . . 10 |- ( f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) -> f : ( 1 ... ( # ` ( F supp .0. ) ) ) -onto-> ( F supp .0. ) )
67		forn 5784	. . . . . . . . . 10 |- ( f : ( 1 ... ( # ` ( F supp .0. ) ) ) -onto-> ( F supp .0. ) -> ran f = ( F supp .0. ) )
68	66, 67	syl 16	. . . . . . . . 9 |- ( f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) -> ran f = ( F supp .0. ) )
69	68	ad2antll 728	. . . . . . . 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 3535	. . . . . . 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 2441	. . . . . . 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 16780	. . . . . 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 465	. . . . . . 7 |- ( ( ph /\ ( ( # ` ( F supp .0. ) ) e. NN /\ f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) ) ) -> C e. _V )
74		fco 5727	. . . . . . . . 9 |- ( ( F : A --> B /\ H : C --> A ) -> ( F o. H ) : C --> B )
75	15, 24, 74	syl2anc 661	. . . . . . . 8 |- ( ph -> ( F o. H ) : C --> B )
76	75	adantr 465	. . . . . . 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 5249	. . . . . . . . 9 |- ran ( F o. H ) C_ ran F
78	59	cntzidss 16244	. . . . . . . . 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 662	. . . . . . . 8 |- ( ph -> ran ( F o. H ) C_ ( Z ` ran ( F o. H ) ) )
80	79	adantr 465	. . . . . . 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 5814	. . . . . . . . . 10 |- ( H : C -1-1-onto-> A -> `' H : A -1-1-onto-> C )
82		f1of1 5801	. . . . . . . . . 10 |- ( `' H : A -1-1-onto-> C -> `' H : A -1-1-> C )
83	6, 81, 82	3syl 20	. . . . . . . . 9 |- ( ph -> `' H : A -1-1-> C )
84	83	adantr 465	. . . . . . . 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 5776	. . . . . . . 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 661	. . . . . . 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 6126	. . . . . . . . . . . . . . 15 |- ( ( F : A --> B /\ A e. V ) -> F e. _V )
89	15, 2, 88	syl2anc 661	. . . . . . . . . . . . . 14 |- ( ph -> F e. _V )
90		suppimacnv 6910	. . . . . . . . . . . . . 14 |- ( ( F e. _V /\ .0. e. _V ) -> ( F supp .0. ) = ( `' F " ( _V \ { .0. } ) ) )
91	89, 17, 90	sylancl 662	. . . . . . . . . . . . 13 |- ( ph -> ( F supp .0. ) = ( `' F " ( _V \ { .0. } ) ) )
92	91	eqcomd 2449	. . . . . . . . . . . 12 |- ( ph -> ( `' F " ( _V \ { .0. } ) ) = ( F supp .0. ) )
93	92	adantr 465	. . . . . . . . . . 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 3529	. . . . . . . . . 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 5358	. . . . . . . . . 10 |- ( ( `' F " ( _V \ { .0. } ) ) C_ ran f -> ( `' H " ( `' F " ( _V \ { .0. } ) ) ) C_ ( `' H " ran f ) )
96	94, 95	syl 16	. . . . . . . . 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 5174	. . . . . . . . . . 11 |- `' ( F o. H ) = ( `' H o. `' F )
98	97	imaeq1i 5320	. . . . . . . . . 10 |- ( `' ( F o. H ) " ( _V \ { .0. } ) ) = ( ( `' H o. `' F ) " ( _V \ { .0. } ) )
99		imaco 5498	. . . . . . . . . 10 |- ( ( `' H o. `' F ) " ( _V \ { .0. } ) ) = ( `' H " ( `' F " ( _V \ { .0. } ) ) )
100	98, 99	eqtri 2470	. . . . . . . . 9 |- ( `' ( F o. H ) " ( _V \ { .0. } ) ) = ( `' H " ( `' F " ( _V \ { .0. } ) ) )
101		rnco2 5500	. . . . . . . . 9 |- ran ( `' H o. f ) = ( `' H " ran f )
102	96, 100, 101	3sstr4g 3527	. . . . . . . 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 6730	. . . . . . . . . . . . 13 |- ( ( H : C -1-1-onto-> A /\ A e. V ) -> H e. _V )
104	6, 2, 103	syl2anc 661	. . . . . . . . . . . 12 |- ( ph -> H e. _V )
105		coexg 6732	. . . . . . . . . . . 12 |- ( ( F e. _V /\ H e. _V ) -> ( F o. H ) e. _V )
106	89, 104, 105	syl2anc 661	. . . . . . . . . . 11 |- ( ph -> ( F o. H ) e. _V )
107		suppimacnv 6910	. . . . . . . . . . 11 |- ( ( ( F o. H ) e. _V /\ .0. e. _V ) -> ( ( F o. H ) supp .0. ) = ( `' ( F o. H ) " ( _V \ { .0. } ) ) )
108	106, 17, 107	sylancl 662	. . . . . . . . . 10 |- ( ph -> ( ( F o. H ) supp .0. ) = ( `' ( F o. H ) " ( _V \ { .0. } ) ) )
109	108	sseq1d 3513	. . . . . . . . 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 465	. . . . . . . 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 232	. . . . . . 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 2441	. . . . . . 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 16780	. . . . . 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 2492	. . . . 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 615	. . . 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 1709	. . 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 603	. 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 7833	. . . 4 |- ( F finSupp .0. -> ( Fun F /\ ( F supp .0. ) e. Fin ) )
120	119	simprd 463	. . 3 |- ( F finSupp .0. -> ( F supp .0. ) e. Fin )
121		fz1f1o 13506	. . 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 20	. 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 381	1 |- ( ph -> ( G gsumg F ) = ( G gsumg ( F o. H ) ) )

Syntax hints:   -> wi 4   <-> wb 184   \/ wo 368   /\ wa 369   = wceq 1381   E.wex 1597   e. wcel 1802   _Vcvv 3093   \ cdif 3455   C_ wss 3458   (/)c0 3767   {csn 4010   class class class wbr 4433   |-> cmpt 4491   _I cid 4776   `'ccnv 4984   dom cdm 4985   ran crn 4986   |` cres 4987   "cima 4988   o. ccom 4989   Fun wfun 5568   -->wf 5570   -1-1->wf1 5571   -onto->wfo 5572   -1-1-onto->wf1o 5573   ` cfv 5574  (class class class)co 6277   supp csupp 6899   Fincfn 7514   finSupp cfsupp 7827   1c1 9491   NNcn 10537   ...cfz 11676   seqcseq 12081   #chash 12379   Basecbs 14504   +g cplusg 14569   0gc0g 14709   gsum_g cgsu 14710   Mndcmnd 15788  Cntzccntz 16222

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-int 4268  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-se 4825  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-isom 5583  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6682  df-1st 6781  df-2nd 6782  df-supp 6900  df-recs 7040  df-rdg 7074  df-1o 7128  df-oadd 7132  df-er 7309  df-en 7515  df-dom 7516  df-sdom 7517  df-fin 7518  df-fsupp 7828  df-oi 7933  df-card 8318  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9807  df-neg 9808  df-nn 10538  df-n0 10797  df-z 10866  df-uz 11086  df-fz 11677  df-fzo 11799  df-seq 12082  df-hash 12380  df-0g 14711  df-gsum 14712  df-mgm 15741  df-sgrp 15780  df-mnd 15790  df-cntz 16224

This theorem is referenced by:  gsumf1o  16793  smadiadetlem3  19037