Theorem gsumzf1o 17053

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 16141	. . . . . . 7 |- ( ( G e. Mnd /\ A e. V ) -> ( G gsumg ( k e. A |-> .0. ) ) = .0. )
5	1, 2, 4	syl2anc 659	. . . . . 6 |- ( ph -> ( G gsumg ( k e. A |-> .0. ) ) = .0. )
6		gsumzf1o.h	. . . . . . . . 9 |- ( ph -> H : C -1-1-onto-> A )
7		f1of1 5736	. . . . . . . . 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 6687	. . . . . . . 8 |- ( ( H : C -1-1-> A /\ A e. V ) -> C e. _V )
10	8, 2, 9	syl2anc 659	. . . . . . 7 |- ( ph -> C e. _V )
11	3	gsumz 16141	. . . . . . 7 |- ( ( G e. Mnd /\ C e. _V ) -> ( G gsumg ( x e. C |-> .0. ) ) = .0. )
12	1, 10, 11	syl2anc 659	. . . . . 6 |- ( ph -> ( G gsumg ( x e. C |-> .0. ) ) = .0. )
13	5, 12	eqtr4d 2436	. . . . 5 |- ( ph -> ( G gsumg ( k e. A |-> .0. ) ) = ( G gsumg ( x e. C |-> .0. ) ) )
14	13	adantr 463	. . . 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 5797	. . . . . . . 8 |- ( 0g ` G ) e. _V
17	3, 16	eqeltri 2476	. . . . . . 7 |- .0. e. _V
18	17	a1i 11	. . . . . 6 |- ( ph -> .0. e. _V )
19		ssid 3449	. . . . . . 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 17048	. . . . 5 |- ( ( ph /\ ( F supp .0. ) = (/) ) -> F = ( k e. A |-> .0. ) )
22	21	oveq2d 6230	. . . 4 |- ( ( ph /\ ( F supp .0. ) = (/) ) -> ( G gsumg F ) = ( G gsumg ( k e. A |-> .0. ) ) )
23		f1of 5737	. . . . . . . . 9 |- ( H : C -1-1-onto-> A -> H : C --> A )
24	6, 23	syl 16	. . . . . . . 8 |- ( ph -> H : C --> A )
25	24	adantr 463	. . . . . . 7 |- ( ( ph /\ ( F supp .0. ) = (/) ) -> H : C --> A )
26	25	ffvelrnda 5946	. . . . . 6 |- ( ( ( ph /\ ( F supp .0. ) = (/) ) /\ x e. C ) -> ( H ` x ) e. A )
27	25	feqmptd 5840	. . . . . 6 |- ( ( ph /\ ( F supp .0. ) = (/) ) -> H = ( x e. C |-> ( H ` x ) ) )
28		eqidd 2393	. . . . . 6 |- ( k = ( H ` x ) -> .0. = .0. )
29	26, 27, 21, 28	fmptco 5979	. . . . 5 |- ( ( ph /\ ( F supp .0. ) = (/) ) -> ( F o. H ) = ( x e. C |-> .0. ) )
30	29	oveq2d 6230	. . . 4 |- ( ( ph /\ ( F supp .0. ) = (/) ) -> ( G gsumg ( F o. H ) ) = ( G gsumg ( x e. C |-> .0. ) ) )
31	14, 22, 30	3eqtr4d 2443	. . 3 |- ( ( ph /\ ( F supp .0. ) = (/) ) -> ( G gsumg F ) = ( G gsumg ( F o. H ) ) )
32	31	ex 432	. 2 |- ( ph -> ( ( F supp .0. ) = (/) -> ( G gsumg F ) = ( G gsumg ( F o. H ) ) ) )
33		coass 5447	. . . . . . . . . . 11 |- ( ( H o. `' H ) o. f ) = ( H o. ( `' H o. f ) )
34	6	adantr 463	. . . . . . . . . . . . . 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 5763	. . . . . . . . . . . . . 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 5090	. . . . . . . . . . . 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 5736	. . . . . . . . . . . . . . 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 726	. . . . . . . . . . . . . 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 6848	. . . . . . . . . . . . . . . 16 |- ( F supp .0. ) C_ dom F
41		fdm 5656	. . . . . . . . . . . . . . . . 17 |- ( F : A --> B -> dom F = A )
42	15, 41	syl 16	. . . . . . . . . . . . . . . 16 |- ( ph -> dom F = A )
43	40, 42	syl5sseq 3478	. . . . . . . . . . . . . . 15 |- ( ph -> ( F supp .0. ) C_ A )
44	43	adantr 463	. . . . . . . . . . . . . 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 5707	. . . . . . . . . . . . . 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 659	. . . . . . . . . . . . 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 5702	. . . . . . . . . . . . 13 |- ( f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-> A -> f : ( 1 ... ( # ` ( F supp .0. ) ) ) --> A )
48		fcoi2 5681	. . . . . . . . . . . . 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 2433	. . . . . . . . . . 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 2448	. . . . . . . . . 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 5091	. . . . . . . . 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 5447	. . . . . . . . 9 |- ( ( F o. H ) o. ( `' H o. f ) ) = ( F o. ( H o. ( `' H o. f ) ) )
54	52, 53	syl6eqr 2451	. . . . . . . 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 12037	. . . . . . 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 5789	. . . . . 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 2392	. . . . . . 7 |- ( +g ` G ) = ( +g ` G )
59		gsumzcl.z	. . . . . . 7 |- Z = (Cntz ` G )
60	1	adantr 463	. . . . . . 7 |- ( ( ph /\ ( ( # ` ( F supp .0. ) ) e. NN /\ f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) ) ) -> G e. Mnd )
61	2	adantr 463	. . . . . . 7 |- ( ( ph /\ ( ( # ` ( F supp .0. ) ) e. NN /\ f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) ) ) -> A e. V )
62	15	adantr 463	. . . . . . 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 463	. . . . . . 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 754	. . . . . . 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 5744	. . . . . . . . . 10 |- ( f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) -> f : ( 1 ... ( # ` ( F supp .0. ) ) ) -onto-> ( F supp .0. ) )
67		forn 5719	. . . . . . . . . 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 726	. . . . . . . 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 3479	. . . . . . 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 2392	. . . . . . 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 17047	. . . . . 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 463	. . . . . . 7 |- ( ( ph /\ ( ( # ` ( F supp .0. ) ) e. NN /\ f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) ) ) -> C e. _V )
74		fco 5662	. . . . . . . . 9 |- ( ( F : A --> B /\ H : C --> A ) -> ( F o. H ) : C --> B )
75	15, 24, 74	syl2anc 659	. . . . . . . 8 |- ( ph -> ( F o. H ) : C --> B )
76	75	adantr 463	. . . . . . 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 5189	. . . . . . . . 9 |- ran ( F o. H ) C_ ran F
78	59	cntzidss 16511	. . . . . . . . 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 660	. . . . . . . 8 |- ( ph -> ran ( F o. H ) C_ ( Z ` ran ( F o. H ) ) )
80	79	adantr 463	. . . . . . 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 5749	. . . . . . . . . 10 |- ( H : C -1-1-onto-> A -> `' H : A -1-1-onto-> C )
82		f1of1 5736	. . . . . . . . . 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 463	. . . . . . . 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 5711	. . . . . . . 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 659	. . . . . . 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 6062	. . . . . . . . . . . . . . 15 |- ( ( F : A --> B /\ A e. V ) -> F e. _V )
89	15, 2, 88	syl2anc 659	. . . . . . . . . . . . . 14 |- ( ph -> F e. _V )
90		suppimacnv 6846	. . . . . . . . . . . . . 14 |- ( ( F e. _V /\ .0. e. _V ) -> ( F supp .0. ) = ( `' F " ( _V \ { .0. } ) ) )
91	89, 17, 90	sylancl 660	. . . . . . . . . . . . 13 |- ( ph -> ( F supp .0. ) = ( `' F " ( _V \ { .0. } ) ) )
92	91	eqcomd 2400	. . . . . . . . . . . 12 |- ( ph -> ( `' F " ( _V \ { .0. } ) ) = ( F supp .0. ) )
93	92	adantr 463	. . . . . . . . . . 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 3473	. . . . . . . . . 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 5297	. . . . . . . . . 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 5114	. . . . . . . . . . 11 |- `' ( F o. H ) = ( `' H o. `' F )
98	97	imaeq1i 5259	. . . . . . . . . 10 |- ( `' ( F o. H ) " ( _V \ { .0. } ) ) = ( ( `' H o. `' F ) " ( _V \ { .0. } ) )
99		imaco 5433	. . . . . . . . . 10 |- ( ( `' H o. `' F ) " ( _V \ { .0. } ) ) = ( `' H " ( `' F " ( _V \ { .0. } ) ) )
100	98, 99	eqtri 2421	. . . . . . . . 9 |- ( `' ( F o. H ) " ( _V \ { .0. } ) ) = ( `' H " ( `' F " ( _V \ { .0. } ) ) )
101		rnco2 5435	. . . . . . . . 9 |- ran ( `' H o. f ) = ( `' H " ran f )
102	96, 100, 101	3sstr4g 3471	. . . . . . . 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 6666	. . . . . . . . . . . . 13 |- ( ( H : C -1-1-onto-> A /\ A e. V ) -> H e. _V )
104	6, 2, 103	syl2anc 659	. . . . . . . . . . . 12 |- ( ph -> H e. _V )
105		coexg 6668	. . . . . . . . . . . 12 |- ( ( F e. _V /\ H e. _V ) -> ( F o. H ) e. _V )
106	89, 104, 105	syl2anc 659	. . . . . . . . . . 11 |- ( ph -> ( F o. H ) e. _V )
107		suppimacnv 6846	. . . . . . . . . . 11 |- ( ( ( F o. H ) e. _V /\ .0. e. _V ) -> ( ( F o. H ) supp .0. ) = ( `' ( F o. H ) " ( _V \ { .0. } ) ) )
108	106, 17, 107	sylancl 660	. . . . . . . . . 10 |- ( ph -> ( ( F o. H ) supp .0. ) = ( `' ( F o. H ) " ( _V \ { .0. } ) ) )
109	108	sseq1d 3457	. . . . . . . . 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 463	. . . . . . . 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 2392	. . . . . . 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 17047	. . . . . 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 2443	. . . . 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 613	. . . 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 1739	. . 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 601	. 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 7768	. . . 4 |- ( F finSupp .0. -> ( Fun F /\ ( F supp .0. ) e. Fin ) )
120	119	simprd 461	. . 3 |- ( F finSupp .0. -> ( F supp .0. ) e. Fin )
121		fz1f1o 13553	. . 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 379	1 |- ( ph -> ( G gsumg F ) = ( G gsumg ( F o. H ) ) )

Syntax hints:   -> wi 4   <-> wb 184   \/ wo 366   /\ wa 367   = wceq 1399   E.wex 1627   e. wcel 1836   _Vcvv 3047   \ cdif 3399   C_ wss 3402   (/)c0 3724   {csn 3957   class class class wbr 4380   |-> cmpt 4438   _I cid 4717   `'ccnv 4925   dom cdm 4926   ran crn 4927   |` cres 4928   "cima 4929   o. ccom 4930   Fun wfun 5503   -->wf 5505   -1-1->wf1 5506   -onto->wfo 5507   -1-1-onto->wf1o 5508   ` cfv 5509  (class class class)co 6214   supp csupp 6835   Fincfn 7453   finSupp cfsupp 7762   1c1 9422   NNcn 10470   ...cfz 11611   seqcseq 12029   #chash 12326   Basecbs 14653   +g cplusg 14721   0gc0g 14866   gsum_g cgsu 14867   Mndcmnd 16055  Cntzccntz 16489

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-8 1838  ax-9 1840  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2016  ax-ext 2370  ax-rep 4491  ax-sep 4501  ax-nul 4509  ax-pow 4556  ax-pr 4614  ax-un 6509  ax-cnex 9477  ax-resscn 9478  ax-1cn 9479  ax-icn 9480  ax-addcl 9481  ax-addrcl 9482  ax-mulcl 9483  ax-mulrcl 9484  ax-mulcom 9485  ax-addass 9486  ax-mulass 9487  ax-distr 9488  ax-i2m1 9489  ax-1ne0 9490  ax-1rid 9491  ax-rnegex 9492  ax-rrecex 9493  ax-cnre 9494  ax-pre-lttri 9495  ax-pre-lttrn 9496  ax-pre-ltadd 9497  ax-pre-mulgt0 9498

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1628  df-nf 1632  df-sb 1758  df-eu 2232  df-mo 2233  df-clab 2378  df-cleq 2384  df-clel 2387  df-nfc 2542  df-ne 2589  df-nel 2590  df-ral 2747  df-rex 2748  df-reu 2749  df-rmo 2750  df-rab 2751  df-v 3049  df-sbc 3266  df-csb 3362  df-dif 3405  df-un 3407  df-in 3409  df-ss 3416  df-pss 3418  df-nul 3725  df-if 3871  df-pw 3942  df-sn 3958  df-pr 3960  df-tp 3962  df-op 3964  df-uni 4177  df-int 4213  df-iun 4258  df-br 4381  df-opab 4439  df-mpt 4440  df-tr 4474  df-eprel 4718  df-id 4722  df-po 4727  df-so 4728  df-fr 4765  df-se 4766  df-we 4767  df-ord 4808  df-on 4809  df-lim 4810  df-suc 4811  df-xp 4932  df-rel 4933  df-cnv 4934  df-co 4935  df-dm 4936  df-rn 4937  df-res 4938  df-ima 4939  df-iota 5473  df-fun 5511  df-fn 5512  df-f 5513  df-f1 5514  df-fo 5515  df-f1o 5516  df-fv 5517  df-isom 5518  df-riota 6176  df-ov 6217  df-oprab 6218  df-mpt2 6219  df-om 6618  df-1st 6717  df-2nd 6718  df-supp 6836  df-recs 6978  df-rdg 7012  df-1o 7066  df-oadd 7070  df-er 7247  df-en 7454  df-dom 7455  df-sdom 7456  df-fin 7457  df-fsupp 7763  df-oi 7868  df-card 8251  df-pnf 9559  df-mnf 9560  df-xr 9561  df-ltxr 9562  df-le 9563  df-sub 9738  df-neg 9739  df-nn 10471  df-n0 10731  df-z 10800  df-uz 11020  df-fz 11612  df-fzo 11736  df-seq 12030  df-hash 12327  df-0g 14868  df-gsum 14869  df-mgm 16008  df-sgrp 16047  df-mnd 16057  df-cntz 16491

This theorem is referenced by:  gsumf1o  17060  smadiadetlem3  19274