Theorem gsumzoppg 18344

Description: The opposite of a group sum is the same as the original. (Contributed by Mario Carneiro, 25-Apr-2016.) (Revised by AV, 6-Jun-2019.)

Hypotheses

Ref	Expression
gsumzoppg.b	|- B = ( Base ` G )
gsumzoppg.0	|- .0. = ( 0g ` G )
gsumzoppg.z	|- Z = (Cntz ` G )
gsumzoppg.o	|- O = (oppg ` G )
gsumzoppg.g	|- ( ph -> G e. Mnd )
gsumzoppg.a	|- ( ph -> A e. V )
gsumzoppg.f	|- ( ph -> F : A --> B )
gsumzoppg.c	|- ( ph -> ran F C_ ( Z ` ran F ) )
gsumzoppg.n	|- ( ph -> F finSupp .0. )

Assertion

Ref	Expression
gsumzoppg	|- ( ph -> ( O gsumg F ) = ( G gsumg F ) )

Proof of Theorem gsumzoppg

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

Step	Hyp	Ref	Expression
1		gsumzoppg.g	. . . . . . . 8 |- ( ph -> G e. Mnd )
2		gsumzoppg.o	. . . . . . . . 9 |- O = (oppg ` G )
3	2	oppgmnd 17784	. . . . . . . 8 |- ( G e. Mnd -> O e. Mnd )
4	1, 3	syl 17	. . . . . . 7 |- ( ph -> O e. Mnd )
5		gsumzoppg.a	. . . . . . 7 |- ( ph -> A e. V )
6		gsumzoppg.0	. . . . . . . . 9 |- .0. = ( 0g ` G )
7	2, 6	oppgid 17786	. . . . . . . 8 |- .0. = ( 0g ` O )
8	7	gsumz 17374	. . . . . . 7 |- ( ( O e. Mnd /\ A e. V ) -> ( O gsumg ( k e. A |-> .0. ) ) = .0. )
9	4, 5, 8	syl2anc 693	. . . . . 6 |- ( ph -> ( O gsumg ( k e. A |-> .0. ) ) = .0. )
10	6	gsumz 17374	. . . . . . 7 |- ( ( G e. Mnd /\ A e. V ) -> ( G gsumg ( k e. A |-> .0. ) ) = .0. )
11	1, 5, 10	syl2anc 693	. . . . . 6 |- ( ph -> ( G gsumg ( k e. A |-> .0. ) ) = .0. )
12	9, 11	eqtr4d 2659	. . . . 5 |- ( ph -> ( O gsumg ( k e. A |-> .0. ) ) = ( G gsumg ( k e. A |-> .0. ) ) )
13	12	adantr 481	. . . 4 |- ( ( ph /\ ( `' F " ( _V \ { .0. } ) ) = (/) ) -> ( O gsumg ( k e. A |-> .0. ) ) = ( G gsumg ( k e. A |-> .0. ) ) )
14		gsumzoppg.f	. . . . . 6 |- ( ph -> F : A --> B )
15		fvex 6201	. . . . . . . 8 |- ( 0g ` G ) e. _V
16	6, 15	eqeltri 2697	. . . . . . 7 |- .0. e. _V
17	16	a1i 11	. . . . . 6 |- ( ph -> .0. e. _V )
18		ssid 3624	. . . . . . 7 |- ( `' F " ( _V \ { .0. } ) ) C_ ( `' F " ( _V \ { .0. } ) )
19		fex 6490	. . . . . . . . . 10 |- ( ( F : A --> B /\ A e. V ) -> F e. _V )
20	14, 5, 19	syl2anc 693	. . . . . . . . 9 |- ( ph -> F e. _V )
21		suppimacnv 7306	. . . . . . . . 9 |- ( ( F e. _V /\ .0. e. _V ) -> ( F supp .0. ) = ( `' F " ( _V \ { .0. } ) ) )
22	20, 16, 21	sylancl 694	. . . . . . . 8 |- ( ph -> ( F supp .0. ) = ( `' F " ( _V \ { .0. } ) ) )
23	22	sseq1d 3632	. . . . . . 7 |- ( ph -> ( ( F supp .0. ) C_ ( `' F " ( _V \ { .0. } ) ) <-> ( `' F " ( _V \ { .0. } ) ) C_ ( `' F " ( _V \ { .0. } ) ) ) )
24	18, 23	mpbiri 248	. . . . . 6 |- ( ph -> ( F supp .0. ) C_ ( `' F " ( _V \ { .0. } ) ) )
25	14, 5, 17, 24	gsumcllem 18309	. . . . 5 |- ( ( ph /\ ( `' F " ( _V \ { .0. } ) ) = (/) ) -> F = ( k e. A |-> .0. ) )
26	25	oveq2d 6666	. . . 4 |- ( ( ph /\ ( `' F " ( _V \ { .0. } ) ) = (/) ) -> ( O gsumg F ) = ( O gsumg ( k e. A |-> .0. ) ) )
27	25	oveq2d 6666	. . . 4 |- ( ( ph /\ ( `' F " ( _V \ { .0. } ) ) = (/) ) -> ( G gsumg F ) = ( G gsumg ( k e. A |-> .0. ) ) )
28	13, 26, 27	3eqtr4d 2666	. . 3 |- ( ( ph /\ ( `' F " ( _V \ { .0. } ) ) = (/) ) -> ( O gsumg F ) = ( G gsumg F ) )
29	28	ex 450	. 2 |- ( ph -> ( ( `' F " ( _V \ { .0. } ) ) = (/) -> ( O gsumg F ) = ( G gsumg F ) ) )
30		simprl 794	. . . . . . . 8 |- ( ( ph /\ ( ( # ` ( `' F " ( _V \ { .0. } ) ) ) e. NN /\ f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { .0. } ) ) ) ) -> ( # ` ( `' F " ( _V \ { .0. } ) ) ) e. NN )
31		nnuz 11723	. . . . . . . 8 |- NN = ( ZZ>= ` 1 )
32	30, 31	syl6eleq 2711	. . . . . . 7 |- ( ( ph /\ ( ( # ` ( `' F " ( _V \ { .0. } ) ) ) e. NN /\ f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { .0. } ) ) ) ) -> ( # ` ( `' F " ( _V \ { .0. } ) ) ) e. ( ZZ>= ` 1 ) )
33	14	adantr 481	. . . . . . . . . . 11 |- ( ( ph /\ ( ( # ` ( `' F " ( _V \ { .0. } ) ) ) e. NN /\ f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { .0. } ) ) ) ) -> F : A --> B )
34		ffn 6045	. . . . . . . . . . . 12 |- ( F : A --> B -> F Fn A )
35		dffn4 6121	. . . . . . . . . . . 12 |- ( F Fn A <-> F : A -onto-> ran F )
36	34, 35	sylib 208	. . . . . . . . . . 11 |- ( F : A --> B -> F : A -onto-> ran F )
37		fof 6115	. . . . . . . . . . 11 |- ( F : A -onto-> ran F -> F : A --> ran F )
38	33, 36, 37	3syl 18	. . . . . . . . . 10 |- ( ( ph /\ ( ( # ` ( `' F " ( _V \ { .0. } ) ) ) e. NN /\ f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { .0. } ) ) ) ) -> F : A --> ran F )
39	1	adantr 481	. . . . . . . . . . . 12 |- ( ( ph /\ ( ( # ` ( `' F " ( _V \ { .0. } ) ) ) e. NN /\ f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { .0. } ) ) ) ) -> G e. Mnd )
40		gsumzoppg.b	. . . . . . . . . . . . 13 |- B = ( Base ` G )
41	40	submacs 17365	. . . . . . . . . . . 12 |- ( G e. Mnd -> (SubMnd ` G ) e. (ACS ` B ) )
42		acsmre 16313	. . . . . . . . . . . 12 |- ( (SubMnd ` G ) e. (ACS ` B ) -> (SubMnd ` G ) e. (Moore ` B ) )
43	39, 41, 42	3syl 18	. . . . . . . . . . 11 |- ( ( ph /\ ( ( # ` ( `' F " ( _V \ { .0. } ) ) ) e. NN /\ f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { .0. } ) ) ) ) -> (SubMnd ` G ) e. (Moore ` B ) )
44		eqid 2622	. . . . . . . . . . 11 |- (mrCls ` (SubMnd ` G ) ) = (mrCls ` (SubMnd ` G ) )
45		frn 6053	. . . . . . . . . . . 12 |- ( F : A --> B -> ran F C_ B )
46	33, 45	syl 17	. . . . . . . . . . 11 |- ( ( ph /\ ( ( # ` ( `' F " ( _V \ { .0. } ) ) ) e. NN /\ f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { .0. } ) ) ) ) -> ran F C_ B )
47	43, 44, 46	mrcssidd 16285	. . . . . . . . . 10 |- ( ( ph /\ ( ( # ` ( `' F " ( _V \ { .0. } ) ) ) e. NN /\ f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { .0. } ) ) ) ) -> ran F C_ ( (mrCls ` (SubMnd ` G ) ) ` ran F ) )
48	38, 47	fssd 6057	. . . . . . . . 9 |- ( ( ph /\ ( ( # ` ( `' F " ( _V \ { .0. } ) ) ) e. NN /\ f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { .0. } ) ) ) ) -> F : A --> ( (mrCls ` (SubMnd ` G ) ) ` ran F ) )
49		f1of1 6136	. . . . . . . . . . . 12 |- ( f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { .0. } ) ) -> f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-> ( `' F " ( _V \ { .0. } ) ) )
50	49	ad2antll 765	. . . . . . . . . . 11 |- ( ( ph /\ ( ( # ` ( `' F " ( _V \ { .0. } ) ) ) e. NN /\ f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { .0. } ) ) ) ) -> f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-> ( `' F " ( _V \ { .0. } ) ) )
51		cnvimass 5485	. . . . . . . . . . . 12 |- ( `' F " ( _V \ { .0. } ) ) C_ dom F
52		fdm 6051	. . . . . . . . . . . . 13 |- ( F : A --> B -> dom F = A )
53	33, 52	syl 17	. . . . . . . . . . . 12 |- ( ( ph /\ ( ( # ` ( `' F " ( _V \ { .0. } ) ) ) e. NN /\ f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { .0. } ) ) ) ) -> dom F = A )
54	51, 53	syl5sseq 3653	. . . . . . . . . . 11 |- ( ( ph /\ ( ( # ` ( `' F " ( _V \ { .0. } ) ) ) e. NN /\ f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { .0. } ) ) ) ) -> ( `' F " ( _V \ { .0. } ) ) C_ A )
55		f1ss 6106	. . . . . . . . . . 11 |- ( ( f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-> ( `' F " ( _V \ { .0. } ) ) /\ ( `' F " ( _V \ { .0. } ) ) C_ A ) -> f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-> A )
56	50, 54, 55	syl2anc 693	. . . . . . . . . 10 |- ( ( ph /\ ( ( # ` ( `' F " ( _V \ { .0. } ) ) ) e. NN /\ f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { .0. } ) ) ) ) -> f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-> A )
57		f1f 6101	. . . . . . . . . 10 |- ( f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-> A -> f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) --> A )
58	56, 57	syl 17	. . . . . . . . 9 |- ( ( ph /\ ( ( # ` ( `' F " ( _V \ { .0. } ) ) ) e. NN /\ f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { .0. } ) ) ) ) -> f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) --> A )
59		fco 6058	. . . . . . . . 9 |- ( ( F : A --> ( (mrCls ` (SubMnd ` G ) ) ` ran F ) /\ f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) --> A ) -> ( F o. f ) : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) --> ( (mrCls ` (SubMnd ` G ) ) ` ran F ) )
60	48, 58, 59	syl2anc 693	. . . . . . . 8 |- ( ( ph /\ ( ( # ` ( `' F " ( _V \ { .0. } ) ) ) e. NN /\ f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { .0. } ) ) ) ) -> ( F o. f ) : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) --> ( (mrCls ` (SubMnd ` G ) ) ` ran F ) )
61	60	ffvelrnda 6359	. . . . . . 7 |- ( ( ( ph /\ ( ( # ` ( `' F " ( _V \ { .0. } ) ) ) e. NN /\ f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { .0. } ) ) ) ) /\ x e. ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) ) -> ( ( F o. f ) ` x ) e. ( (mrCls ` (SubMnd ` G ) ) ` ran F ) )
62	44	mrccl 16271	. . . . . . . . . 10 |- ( ( (SubMnd ` G ) e. (Moore ` B ) /\ ran F C_ B ) -> ( (mrCls ` (SubMnd ` G ) ) ` ran F ) e. (SubMnd ` G ) )
63	43, 46, 62	syl2anc 693	. . . . . . . . 9 |- ( ( ph /\ ( ( # ` ( `' F " ( _V \ { .0. } ) ) ) e. NN /\ f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { .0. } ) ) ) ) -> ( (mrCls ` (SubMnd ` G ) ) ` ran F ) e. (SubMnd ` G ) )
64	2	oppgsubm 17792	. . . . . . . . 9 |- (SubMnd ` G ) = (SubMnd ` O )
65	63, 64	syl6eleq 2711	. . . . . . . 8 |- ( ( ph /\ ( ( # ` ( `' F " ( _V \ { .0. } ) ) ) e. NN /\ f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { .0. } ) ) ) ) -> ( (mrCls ` (SubMnd ` G ) ) ` ran F ) e. (SubMnd ` O ) )
66		eqid 2622	. . . . . . . . . 10 |- ( +g ` O ) = ( +g ` O )
67	66	submcl 17353	. . . . . . . . 9 |- ( ( ( (mrCls ` (SubMnd ` G ) ) ` ran F ) e. (SubMnd ` O ) /\ x e. ( (mrCls ` (SubMnd ` G ) ) ` ran F ) /\ y e. ( (mrCls ` (SubMnd ` G ) ) ` ran F ) ) -> ( x ( +g ` O ) y ) e. ( (mrCls ` (SubMnd ` G ) ) ` ran F ) )
68	67	3expb 1266	. . . . . . . 8 |- ( ( ( (mrCls ` (SubMnd ` G ) ) ` ran F ) e. (SubMnd ` O ) /\ ( x e. ( (mrCls ` (SubMnd ` G ) ) ` ran F ) /\ y e. ( (mrCls ` (SubMnd ` G ) ) ` ran F ) ) ) -> ( x ( +g ` O ) y ) e. ( (mrCls ` (SubMnd ` G ) ) ` ran F ) )
69	65, 68	sylan 488	. . . . . . 7 |- ( ( ( ph /\ ( ( # ` ( `' F " ( _V \ { .0. } ) ) ) e. NN /\ f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { .0. } ) ) ) ) /\ ( x e. ( (mrCls ` (SubMnd ` G ) ) ` ran F ) /\ y e. ( (mrCls ` (SubMnd ` G ) ) ` ran F ) ) ) -> ( x ( +g ` O ) y ) e. ( (mrCls ` (SubMnd ` G ) ) ` ran F ) )
70		gsumzoppg.c	. . . . . . . . . . . . . 14 |- ( ph -> ran F C_ ( Z ` ran F ) )
71	70	adantr 481	. . . . . . . . . . . . 13 |- ( ( ph /\ ( ( # ` ( `' F " ( _V \ { .0. } ) ) ) e. NN /\ f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { .0. } ) ) ) ) -> ran F C_ ( Z ` ran F ) )
72		gsumzoppg.z	. . . . . . . . . . . . . 14 |- Z = (Cntz ` G )
73		eqid 2622	. . . . . . . . . . . . . 14 |- ( G ↾s ( (mrCls ` (SubMnd ` G ) ) ` ran F ) ) = ( G ↾s ( (mrCls ` (SubMnd ` G ) ) ` ran F ) )
74	72, 44, 73	cntzspan 18247	. . . . . . . . . . . . 13 |- ( ( G e. Mnd /\ ran F C_ ( Z ` ran F ) ) -> ( G ↾s ( (mrCls ` (SubMnd ` G ) ) ` ran F ) ) e. CMnd )
75	39, 71, 74	syl2anc 693	. . . . . . . . . . . 12 |- ( ( ph /\ ( ( # ` ( `' F " ( _V \ { .0. } ) ) ) e. NN /\ f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { .0. } ) ) ) ) -> ( G ↾s ( (mrCls ` (SubMnd ` G ) ) ` ran F ) ) e. CMnd )
76	73, 72	submcmn2 18244	. . . . . . . . . . . . 13 |- ( ( (mrCls ` (SubMnd ` G ) ) ` ran F ) e. (SubMnd ` G ) -> ( ( G ↾s ( (mrCls ` (SubMnd ` G ) ) ` ran F ) ) e. CMnd <-> ( (mrCls ` (SubMnd ` G ) ) ` ran F ) C_ ( Z ` ( (mrCls ` (SubMnd ` G ) ) ` ran F ) ) ) )
77	63, 76	syl 17	. . . . . . . . . . . 12 |- ( ( ph /\ ( ( # ` ( `' F " ( _V \ { .0. } ) ) ) e. NN /\ f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { .0. } ) ) ) ) -> ( ( G ↾s ( (mrCls ` (SubMnd ` G ) ) ` ran F ) ) e. CMnd <-> ( (mrCls ` (SubMnd ` G ) ) ` ran F ) C_ ( Z ` ( (mrCls ` (SubMnd ` G ) ) ` ran F ) ) ) )
78	75, 77	mpbid 222	. . . . . . . . . . 11 |- ( ( ph /\ ( ( # ` ( `' F " ( _V \ { .0. } ) ) ) e. NN /\ f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { .0. } ) ) ) ) -> ( (mrCls ` (SubMnd ` G ) ) ` ran F ) C_ ( Z ` ( (mrCls ` (SubMnd ` G ) ) ` ran F ) ) )
79	78	sselda 3603	. . . . . . . . . 10 |- ( ( ( ph /\ ( ( # ` ( `' F " ( _V \ { .0. } ) ) ) e. NN /\ f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { .0. } ) ) ) ) /\ x e. ( (mrCls ` (SubMnd ` G ) ) ` ran F ) ) -> x e. ( Z ` ( (mrCls ` (SubMnd ` G ) ) ` ran F ) ) )
80		eqid 2622	. . . . . . . . . . 11 |- ( +g ` G ) = ( +g ` G )
81	80, 72	cntzi 17762	. . . . . . . . . 10 |- ( ( x e. ( Z ` ( (mrCls ` (SubMnd ` G ) ) ` ran F ) ) /\ y e. ( (mrCls ` (SubMnd ` G ) ) ` ran F ) ) -> ( x ( +g ` G ) y ) = ( y ( +g ` G ) x ) )
82	79, 81	sylan 488	. . . . . . . . 9 |- ( ( ( ( ph /\ ( ( # ` ( `' F " ( _V \ { .0. } ) ) ) e. NN /\ f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { .0. } ) ) ) ) /\ x e. ( (mrCls ` (SubMnd ` G ) ) ` ran F ) ) /\ y e. ( (mrCls ` (SubMnd ` G ) ) ` ran F ) ) -> ( x ( +g ` G ) y ) = ( y ( +g ` G ) x ) )
83	80, 2, 66	oppgplus 17779	. . . . . . . . 9 |- ( x ( +g ` O ) y ) = ( y ( +g ` G ) x )
84	82, 83	syl6reqr 2675	. . . . . . . 8 |- ( ( ( ( ph /\ ( ( # ` ( `' F " ( _V \ { .0. } ) ) ) e. NN /\ f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { .0. } ) ) ) ) /\ x e. ( (mrCls ` (SubMnd ` G ) ) ` ran F ) ) /\ y e. ( (mrCls ` (SubMnd ` G ) ) ` ran F ) ) -> ( x ( +g ` O ) y ) = ( x ( +g ` G ) y ) )
85	84	anasss 679	. . . . . . 7 |- ( ( ( ph /\ ( ( # ` ( `' F " ( _V \ { .0. } ) ) ) e. NN /\ f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { .0. } ) ) ) ) /\ ( x e. ( (mrCls ` (SubMnd ` G ) ) ` ran F ) /\ y e. ( (mrCls ` (SubMnd ` G ) ) ` ran F ) ) ) -> ( x ( +g ` O ) y ) = ( x ( +g ` G ) y ) )
86	32, 61, 69, 85	seqfeq4 12850	. . . . . 6 |- ( ( ph /\ ( ( # ` ( `' F " ( _V \ { .0. } ) ) ) e. NN /\ f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { .0. } ) ) ) ) -> ( seq 1 ( ( +g ` O ) , ( F o. f ) ) ` ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) = ( seq 1 ( ( +g ` G ) , ( F o. f ) ) ` ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) )
87	2, 40	oppgbas 17781	. . . . . . 7 |- B = ( Base ` O )
88		eqid 2622	. . . . . . 7 |- (Cntz ` O ) = (Cntz ` O )
89	39, 3	syl 17	. . . . . . 7 |- ( ( ph /\ ( ( # ` ( `' F " ( _V \ { .0. } ) ) ) e. NN /\ f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { .0. } ) ) ) ) -> O e. Mnd )
90	5	adantr 481	. . . . . . 7 |- ( ( ph /\ ( ( # ` ( `' F " ( _V \ { .0. } ) ) ) e. NN /\ f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { .0. } ) ) ) ) -> A e. V )
91	2, 72	oppgcntz 17794	. . . . . . . 8 |- ( Z ` ran F ) = ( (Cntz ` O ) ` ran F )
92	71, 91	syl6sseq 3651	. . . . . . 7 |- ( ( ph /\ ( ( # ` ( `' F " ( _V \ { .0. } ) ) ) e. NN /\ f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { .0. } ) ) ) ) -> ran F C_ ( (Cntz ` O ) ` ran F ) )
93		suppssdm 7308	. . . . . . . . . . . . . . 15 |- ( F supp .0. ) C_ dom F
94	22, 93	syl6eqssr 3656	. . . . . . . . . . . . . 14 |- ( ph -> ( `' F " ( _V \ { .0. } ) ) C_ dom F )
95	94	adantl 482	. . . . . . . . . . . . 13 |- ( ( dom F = A /\ ph ) -> ( `' F " ( _V \ { .0. } ) ) C_ dom F )
96		eqcom 2629	. . . . . . . . . . . . . . 15 |- ( dom F = A <-> A = dom F )
97	96	biimpi 206	. . . . . . . . . . . . . 14 |- ( dom F = A -> A = dom F )
98	97	adantr 481	. . . . . . . . . . . . 13 |- ( ( dom F = A /\ ph ) -> A = dom F )
99	95, 98	sseqtr4d 3642	. . . . . . . . . . . 12 |- ( ( dom F = A /\ ph ) -> ( `' F " ( _V \ { .0. } ) ) C_ A )
100	99	ex 450	. . . . . . . . . . 11 |- ( dom F = A -> ( ph -> ( `' F " ( _V \ { .0. } ) ) C_ A ) )
101	52, 100	syl 17	. . . . . . . . . 10 |- ( F : A --> B -> ( ph -> ( `' F " ( _V \ { .0. } ) ) C_ A ) )
102	14, 101	mpcom 38	. . . . . . . . 9 |- ( ph -> ( `' F " ( _V \ { .0. } ) ) C_ A )
103	102	adantr 481	. . . . . . . 8 |- ( ( ph /\ ( ( # ` ( `' F " ( _V \ { .0. } ) ) ) e. NN /\ f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { .0. } ) ) ) ) -> ( `' F " ( _V \ { .0. } ) ) C_ A )
104	50, 103, 55	syl2anc 693	. . . . . . 7 |- ( ( ph /\ ( ( # ` ( `' F " ( _V \ { .0. } ) ) ) e. NN /\ f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { .0. } ) ) ) ) -> f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-> A )
105	23	adantr 481	. . . . . . . . 9 |- ( ( ph /\ ( ( # ` ( `' F " ( _V \ { .0. } ) ) ) e. NN /\ f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { .0. } ) ) ) ) -> ( ( F supp .0. ) C_ ( `' F " ( _V \ { .0. } ) ) <-> ( `' F " ( _V \ { .0. } ) ) C_ ( `' F " ( _V \ { .0. } ) ) ) )
106	18, 105	mpbiri 248	. . . . . . . 8 |- ( ( ph /\ ( ( # ` ( `' F " ( _V \ { .0. } ) ) ) e. NN /\ f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { .0. } ) ) ) ) -> ( F supp .0. ) C_ ( `' F " ( _V \ { .0. } ) ) )
107		f1ofo 6144	. . . . . . . . . . 11 |- ( f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { .0. } ) ) -> f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -onto-> ( `' F " ( _V \ { .0. } ) ) )
108		forn 6118	. . . . . . . . . . 11 |- ( f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -onto-> ( `' F " ( _V \ { .0. } ) ) -> ran f = ( `' F " ( _V \ { .0. } ) ) )
109	107, 108	syl 17	. . . . . . . . . 10 |- ( f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { .0. } ) ) -> ran f = ( `' F " ( _V \ { .0. } ) ) )
110	109	sseq2d 3633	. . . . . . . . 9 |- ( f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { .0. } ) ) -> ( ( F supp .0. ) C_ ran f <-> ( F supp .0. ) C_ ( `' F " ( _V \ { .0. } ) ) ) )
111	110	ad2antll 765	. . . . . . . 8 |- ( ( ph /\ ( ( # ` ( `' F " ( _V \ { .0. } ) ) ) e. NN /\ f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { .0. } ) ) ) ) -> ( ( F supp .0. ) C_ ran f <-> ( F supp .0. ) C_ ( `' F " ( _V \ { .0. } ) ) ) )
112	106, 111	mpbird 247	. . . . . . 7 |- ( ( ph /\ ( ( # ` ( `' F " ( _V \ { .0. } ) ) ) e. NN /\ f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { .0. } ) ) ) ) -> ( F supp .0. ) C_ ran f )
113		eqid 2622	. . . . . . 7 |- ( ( F o. f ) supp .0. ) = ( ( F o. f ) supp .0. )
114	87, 7, 66, 88, 89, 90, 33, 92, 30, 104, 112, 113	gsumval3 18308	. . . . . 6 |- ( ( ph /\ ( ( # ` ( `' F " ( _V \ { .0. } ) ) ) e. NN /\ f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { .0. } ) ) ) ) -> ( O gsumg F ) = ( seq 1 ( ( +g ` O ) , ( F o. f ) ) ` ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) )
115	24	adantr 481	. . . . . . . 8 |- ( ( ph /\ ( ( # ` ( `' F " ( _V \ { .0. } ) ) ) e. NN /\ f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { .0. } ) ) ) ) -> ( F supp .0. ) C_ ( `' F " ( _V \ { .0. } ) ) )
116	115, 111	mpbird 247	. . . . . . 7 |- ( ( ph /\ ( ( # ` ( `' F " ( _V \ { .0. } ) ) ) e. NN /\ f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { .0. } ) ) ) ) -> ( F supp .0. ) C_ ran f )
117	40, 6, 80, 72, 39, 90, 33, 71, 30, 104, 116, 113	gsumval3 18308	. . . . . 6 |- ( ( ph /\ ( ( # ` ( `' F " ( _V \ { .0. } ) ) ) e. NN /\ f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { .0. } ) ) ) ) -> ( G gsumg F ) = ( seq 1 ( ( +g ` G ) , ( F o. f ) ) ` ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) )
118	86, 114, 117	3eqtr4d 2666	. . . . 5 |- ( ( ph /\ ( ( # ` ( `' F " ( _V \ { .0. } ) ) ) e. NN /\ f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { .0. } ) ) ) ) -> ( O gsumg F ) = ( G gsumg F ) )
119	118	expr 643	. . . 4 |- ( ( ph /\ ( # ` ( `' F " ( _V \ { .0. } ) ) ) e. NN ) -> ( f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { .0. } ) ) -> ( O gsumg F ) = ( G gsumg F ) ) )
120	119	exlimdv 1861	. . 3 |- ( ( ph /\ ( # ` ( `' F " ( _V \ { .0. } ) ) ) e. NN ) -> ( E. f f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { .0. } ) ) -> ( O gsumg F ) = ( G gsumg F ) ) )
121	120	expimpd 629	. 2 |- ( ph -> ( ( ( # ` ( `' F " ( _V \ { .0. } ) ) ) e. NN /\ E. f f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { .0. } ) ) ) -> ( O gsumg F ) = ( G gsumg F ) ) )
122		gsumzoppg.n	. . . . 5 |- ( ph -> F finSupp .0. )
123	122	fsuppimpd 8282	. . . 4 |- ( ph -> ( F supp .0. ) e. Fin )
124	22, 123	eqeltrrd 2702	. . 3 |- ( ph -> ( `' F " ( _V \ { .0. } ) ) e. Fin )
125		fz1f1o 14441	. . 3 |- ( ( `' F " ( _V \ { .0. } ) ) e. Fin -> ( ( `' F " ( _V \ { .0. } ) ) = (/) \/ ( ( # ` ( `' F " ( _V \ { .0. } ) ) ) e. NN /\ E. f f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { .0. } ) ) ) ) )
126	124, 125	syl 17	. 2 |- ( ph -> ( ( `' F " ( _V \ { .0. } ) ) = (/) \/ ( ( # ` ( `' F " ( _V \ { .0. } ) ) ) e. NN /\ E. f f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { .0. } ) ) ) ) )
127	29, 121, 126	mpjaod 396	1 |- ( ph -> ( O gsumg F ) = ( G gsumg F ) )

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   `'ccnv 5113   dom cdm 5114   ran crn 5115   "cima 5117   o. ccom 5118   Fn wfn 5883   -->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   ZZ>=cuz 11687   ...cfz 12326   seqcseq 12801   #chash 13117   Basecbs 15857   ↾_s cress 15858   +g cplusg 15941   0gc0g 16100   gsum_g cgsu 16101  Moorecmre 16242  mrClscmrc 16243  ACScacs 16245   Mndcmnd 17294  SubMndcsubmnd 17334  Cntzccntz 17748  opp_gcoppg 17775  CMndccmn 18193

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-iin 4523  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-tpos 7352  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-2 11079  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-seq 12802  df-hash 13118  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-0g 16102  df-gsum 16103  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-cntz 17750  df-oppg 17776  df-cmn 18195

This theorem is referenced by:  gsumzinv  18345