Theorem gsumzf1o 18313

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 17374	. . . . . . 7 |- ( ( G e. Mnd /\ A e. V ) -> ( G gsumg ( k e. A |-> .0. ) ) = .0. )
5	1, 2, 4	syl2anc 693	. . . . . 6 |- ( ph -> ( G gsumg ( k e. A |-> .0. ) ) = .0. )
6		gsumzf1o.h	. . . . . . . . 9 |- ( ph -> H : C -1-1-onto-> A )
7		f1of1 6136	. . . . . . . . 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 7136	. . . . . . . 8 |- ( ( H : C -1-1-> A /\ A e. V ) -> C e. _V )
10	8, 2, 9	syl2anc 693	. . . . . . 7 |- ( ph -> C e. _V )
11	3	gsumz 17374	. . . . . . 7 |- ( ( G e. Mnd /\ C e. _V ) -> ( G gsumg ( x e. C |-> .0. ) ) = .0. )
12	1, 10, 11	syl2anc 693	. . . . . 6 |- ( ph -> ( G gsumg ( x e. C |-> .0. ) ) = .0. )
13	5, 12	eqtr4d 2659	. . . . 5 |- ( ph -> ( G gsumg ( k e. A |-> .0. ) ) = ( G gsumg ( x e. C |-> .0. ) ) )
14	13	adantr 481	. . . 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 6201	. . . . . . . 8 |- ( 0g ` G ) e. _V
17	3, 16	eqeltri 2697	. . . . . . 7 |- .0. e. _V
18	17	a1i 11	. . . . . 6 |- ( ph -> .0. e. _V )
19		ssid 3624	. . . . . . 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 18309	. . . . 5 |- ( ( ph /\ ( F supp .0. ) = (/) ) -> F = ( k e. A |-> .0. ) )
22	21	oveq2d 6666	. . . 4 |- ( ( ph /\ ( F supp .0. ) = (/) ) -> ( G gsumg F ) = ( G gsumg ( k e. A |-> .0. ) ) )
23		f1of 6137	. . . . . . . . 9 |- ( H : C -1-1-onto-> A -> H : C --> A )
24	6, 23	syl 17	. . . . . . . 8 |- ( ph -> H : C --> A )
25	24	adantr 481	. . . . . . 7 |- ( ( ph /\ ( F supp .0. ) = (/) ) -> H : C --> A )
26	25	ffvelrnda 6359	. . . . . 6 |- ( ( ( ph /\ ( F supp .0. ) = (/) ) /\ x e. C ) -> ( H ` x ) e. A )
27	25	feqmptd 6249	. . . . . 6 |- ( ( ph /\ ( F supp .0. ) = (/) ) -> H = ( x e. C |-> ( H ` x ) ) )
28		eqidd 2623	. . . . . 6 |- ( k = ( H ` x ) -> .0. = .0. )
29	26, 27, 21, 28	fmptco 6396	. . . . 5 |- ( ( ph /\ ( F supp .0. ) = (/) ) -> ( F o. H ) = ( x e. C |-> .0. ) )
30	29	oveq2d 6666	. . . 4 |- ( ( ph /\ ( F supp .0. ) = (/) ) -> ( G gsumg ( F o. H ) ) = ( G gsumg ( x e. C |-> .0. ) ) )
31	14, 22, 30	3eqtr4d 2666	. . 3 |- ( ( ph /\ ( F supp .0. ) = (/) ) -> ( G gsumg F ) = ( G gsumg ( F o. H ) ) )
32	31	ex 450	. 2 |- ( ph -> ( ( F supp .0. ) = (/) -> ( G gsumg F ) = ( G gsumg ( F o. H ) ) ) )
33		coass 5654	. . . . . . . . . . 11 |- ( ( H o. `' H ) o. f ) = ( H o. ( `' H o. f ) )
34	6	adantr 481	. . . . . . . . . . . . . 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 6163	. . . . . . . . . . . . . 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 5283	. . . . . . . . . . . 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 6136	. . . . . . . . . . . . . . 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 765	. . . . . . . . . . . . . 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 7308	. . . . . . . . . . . . . . . 16 |- ( F supp .0. ) C_ dom F
41		fdm 6051	. . . . . . . . . . . . . . . . 17 |- ( F : A --> B -> dom F = A )
42	15, 41	syl 17	. . . . . . . . . . . . . . . 16 |- ( ph -> dom F = A )
43	40, 42	syl5sseq 3653	. . . . . . . . . . . . . . 15 |- ( ph -> ( F supp .0. ) C_ A )
44	43	adantr 481	. . . . . . . . . . . . . 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 6106	. . . . . . . . . . . . . 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 693	. . . . . . . . . . . . 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 6101	. . . . . . . . . . . . 13 |- ( f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-> A -> f : ( 1 ... ( # ` ( F supp .0. ) ) ) --> A )
48		fcoi2 6079	. . . . . . . . . . . . 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 2656	. . . . . . . . . . 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 2671	. . . . . . . . . 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 5284	. . . . . . . . 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 5654	. . . . . . . . 9 |- ( ( F o. H ) o. ( `' H o. f ) ) = ( F o. ( H o. ( `' H o. f ) ) )
54	52, 53	syl6eqr 2674	. . . . . . . 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 12809	. . . . . . 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 6193	. . . . . 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 2622	. . . . . . 7 |- ( +g ` G ) = ( +g ` G )
59		gsumzcl.z	. . . . . . 7 |- Z = (Cntz ` G )
60	1	adantr 481	. . . . . . 7 |- ( ( ph /\ ( ( # ` ( F supp .0. ) ) e. NN /\ f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) ) ) -> G e. Mnd )
61	2	adantr 481	. . . . . . 7 |- ( ( ph /\ ( ( # ` ( F supp .0. ) ) e. NN /\ f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) ) ) -> A e. V )
62	15	adantr 481	. . . . . . 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 481	. . . . . . 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 794	. . . . . . 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 6144	. . . . . . . . . 10 |- ( f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) -> f : ( 1 ... ( # ` ( F supp .0. ) ) ) -onto-> ( F supp .0. ) )
67		forn 6118	. . . . . . . . . 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 765	. . . . . . . 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 3654	. . . . . . 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 2622	. . . . . . 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 18308	. . . . . 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 481	. . . . . . 7 |- ( ( ph /\ ( ( # ` ( F supp .0. ) ) e. NN /\ f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) ) ) -> C e. _V )
74		fco 6058	. . . . . . . . 9 |- ( ( F : A --> B /\ H : C --> A ) -> ( F o. H ) : C --> B )
75	15, 24, 74	syl2anc 693	. . . . . . . 8 |- ( ph -> ( F o. H ) : C --> B )
76	75	adantr 481	. . . . . . 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 5386	. . . . . . . . 9 |- ran ( F o. H ) C_ ran F
78	59	cntzidss 17770	. . . . . . . . 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 694	. . . . . . . 8 |- ( ph -> ran ( F o. H ) C_ ( Z ` ran ( F o. H ) ) )
80	79	adantr 481	. . . . . . 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 6149	. . . . . . . . . 10 |- ( H : C -1-1-onto-> A -> `' H : A -1-1-onto-> C )
82		f1of1 6136	. . . . . . . . . 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 481	. . . . . . . 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 6110	. . . . . . . 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 693	. . . . . . 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 6490	. . . . . . . . . . . . . . 15 |- ( ( F : A --> B /\ A e. V ) -> F e. _V )
89	15, 2, 88	syl2anc 693	. . . . . . . . . . . . . 14 |- ( ph -> F e. _V )
90		suppimacnv 7306	. . . . . . . . . . . . . 14 |- ( ( F e. _V /\ .0. e. _V ) -> ( F supp .0. ) = ( `' F " ( _V \ { .0. } ) ) )
91	89, 17, 90	sylancl 694	. . . . . . . . . . . . 13 |- ( ph -> ( F supp .0. ) = ( `' F " ( _V \ { .0. } ) ) )
92	91	eqcomd 2628	. . . . . . . . . . . 12 |- ( ph -> ( `' F " ( _V \ { .0. } ) ) = ( F supp .0. ) )
93	92	adantr 481	. . . . . . . . . . 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 3648	. . . . . . . . . 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 5501	. . . . . . . . . 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 5308	. . . . . . . . . . 11 |- `' ( F o. H ) = ( `' H o. `' F )
98	97	imaeq1i 5463	. . . . . . . . . 10 |- ( `' ( F o. H ) " ( _V \ { .0. } ) ) = ( ( `' H o. `' F ) " ( _V \ { .0. } ) )
99		imaco 5640	. . . . . . . . . 10 |- ( ( `' H o. `' F ) " ( _V \ { .0. } ) ) = ( `' H " ( `' F " ( _V \ { .0. } ) ) )
100	98, 99	eqtri 2644	. . . . . . . . 9 |- ( `' ( F o. H ) " ( _V \ { .0. } ) ) = ( `' H " ( `' F " ( _V \ { .0. } ) ) )
101		rnco2 5642	. . . . . . . . 9 |- ran ( `' H o. f ) = ( `' H " ran f )
102	96, 100, 101	3sstr4g 3646	. . . . . . . 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 7115	. . . . . . . . . . . . 13 |- ( ( H : C -1-1-onto-> A /\ A e. V ) -> H e. _V )
104	6, 2, 103	syl2anc 693	. . . . . . . . . . . 12 |- ( ph -> H e. _V )
105		coexg 7117	. . . . . . . . . . . 12 |- ( ( F e. _V /\ H e. _V ) -> ( F o. H ) e. _V )
106	89, 104, 105	syl2anc 693	. . . . . . . . . . 11 |- ( ph -> ( F o. H ) e. _V )
107		suppimacnv 7306	. . . . . . . . . . 11 |- ( ( ( F o. H ) e. _V /\ .0. e. _V ) -> ( ( F o. H ) supp .0. ) = ( `' ( F o. H ) " ( _V \ { .0. } ) ) )
108	106, 17, 107	sylancl 694	. . . . . . . . . 10 |- ( ph -> ( ( F o. H ) supp .0. ) = ( `' ( F o. H ) " ( _V \ { .0. } ) ) )
109	108	sseq1d 3632	. . . . . . . . 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 481	. . . . . . . 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 247	. . . . . . 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 2622	. . . . . . 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 18308	. . . . . 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 2666	. . . . 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 643	. . . 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 1861	. . 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 629	. 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 8281	. . . 4 |- ( F finSupp .0. -> ( Fun F /\ ( F supp .0. ) e. Fin ) )
120	119	simprd 479	. . 3 |- ( F finSupp .0. -> ( F supp .0. ) e. Fin )
121		fz1f1o 14441	. . 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 396	1 |- ( ph -> ( G gsumg F ) = ( G gsumg ( F o. H ) ) )

Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   _Vcvv 3200   \ cdif 3571   C_ wss 3574   (/)c0 3915   {csn 4177   class class class wbr 4653   |-> cmpt 4729   _I cid 5023   `'ccnv 5113   dom cdm 5114   ran crn 5115   |` cres 5116   "cima 5117   o. ccom 5118   Fun wfun 5882   -->wf 5884   -1-1->wf1 5885   -onto->wfo 5886   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   supp csupp 7295   Fincfn 7955   finSupp cfsupp 8275   1c1 9937   NNcn 11020   ...cfz 12326   seqcseq 12801   #chash 13117   Basecbs 15857   +g cplusg 15941   0gc0g 16100   gsum_g cgsu 16101   Mndcmnd 17294  Cntzccntz 17748

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-seq 12802  df-hash 13118  df-0g 16102  df-gsum 16103  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-cntz 17750

This theorem is referenced by:  gsumf1o  18317  smadiadetlem3  20474