Theorem dprdfeq0 18421

Description: The zero function is the only function that sums to zero in a direct product. (Contributed by Mario Carneiro, 25-Apr-2016.) (Revised by AV, 14-Jul-2019.)

Hypotheses

Ref	Expression
eldprdi.0	|- .0. = ( 0g ` G )
eldprdi.w	|- W = { h e. X_ i e. I ( S ` i ) | h finSupp .0. }
eldprdi.1	|- ( ph -> G dom DProd S )
eldprdi.2	|- ( ph -> dom S = I )
eldprdi.3	|- ( ph -> F e. W )

Assertion

Ref	Expression
dprdfeq0	|- ( ph -> ( ( G gsumg F ) = .0. <-> F = ( x e. I |-> .0. ) ) )

Distinct variable groups:   x, h, F   h, i, G, x   h, I, i, x   ph, x   .0. , h, x   S, h, i, x

Allowed substitution hints:   ph( h, i)   F( i)   W( x, h, i)   .0. ( i)

Proof of Theorem dprdfeq0

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eldprdi.w	. . . . . . 7 |- W = { h e. X_ i e. I ( S ` i ) | h finSupp .0. }
2		eldprdi.1	. . . . . . 7 |- ( ph -> G dom DProd S )
3		eldprdi.2	. . . . . . 7 |- ( ph -> dom S = I )
4		eldprdi.3	. . . . . . 7 |- ( ph -> F e. W )
5		eqid 2622	. . . . . . 7 |- ( Base ` G ) = ( Base ` G )
6	1, 2, 3, 4, 5	dprdff 18411	. . . . . 6 |- ( ph -> F : I --> ( Base ` G ) )
7	6	feqmptd 6249	. . . . 5 |- ( ph -> F = ( x e. I |-> ( F ` x ) ) )
8	7	adantr 481	. . . 4 |- ( ( ph /\ ( G gsumg F ) = .0. ) -> F = ( x e. I |-> ( F ` x ) ) )
9	1, 2, 3, 4	dprdfcl 18412	. . . . . . . . 9 |- ( ( ph /\ x e. I ) -> ( F ` x ) e. ( S ` x ) )
10	9	adantlr 751	. . . . . . . 8 |- ( ( ( ph /\ ( G gsumg F ) = .0. ) /\ x e. I ) -> ( F ` x ) e. ( S ` x ) )
11		eldprdi.0	. . . . . . . . . . . 12 |- .0. = ( 0g ` G )
12	2	ad2antrr 762	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( G gsumg F ) = .0. ) /\ x e. I ) -> G dom DProd S )
13	3	ad2antrr 762	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( G gsumg F ) = .0. ) /\ x e. I ) -> dom S = I )
14		simpr 477	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ ( G gsumg F ) = .0. ) /\ x e. I ) -> x e. I )
15		eqid 2622	. . . . . . . . . . . . . 14 |- ( y e. I |-> if ( y = x , ( F ` x ) , .0. ) ) = ( y e. I |-> if ( y = x , ( F ` x ) , .0. ) )
16	11, 1, 12, 13, 14, 10, 15	dprdfid 18416	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( G gsumg F ) = .0. ) /\ x e. I ) -> ( ( y e. I |-> if ( y = x , ( F ` x ) , .0. ) ) e. W /\ ( G gsumg ( y e. I |-> if ( y = x , ( F ` x ) , .0. ) ) ) = ( F ` x ) ) )
17	16	simpld 475	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( G gsumg F ) = .0. ) /\ x e. I ) -> ( y e. I |-> if ( y = x , ( F ` x ) , .0. ) ) e. W )
18	4	ad2antrr 762	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( G gsumg F ) = .0. ) /\ x e. I ) -> F e. W )
19		eqid 2622	. . . . . . . . . . . 12 |- ( -g ` G ) = ( -g ` G )
20	11, 1, 12, 13, 17, 18, 19	dprdfsub 18420	. . . . . . . . . . 11 |- ( ( ( ph /\ ( G gsumg F ) = .0. ) /\ x e. I ) -> ( ( ( y e. I |-> if ( y = x , ( F ` x ) , .0. ) ) oF ( -g ` G ) F ) e. W /\ ( G gsumg ( ( y e. I |-> if ( y = x , ( F ` x ) , .0. ) ) oF ( -g ` G ) F ) ) = ( ( G gsumg ( y e. I |-> if ( y = x , ( F ` x ) , .0. ) ) ) ( -g ` G ) ( G gsumg F ) ) ) )
21	20	simprd 479	. . . . . . . . . 10 |- ( ( ( ph /\ ( G gsumg F ) = .0. ) /\ x e. I ) -> ( G gsumg ( ( y e. I |-> if ( y = x , ( F ` x ) , .0. ) ) oF ( -g ` G ) F ) ) = ( ( G gsumg ( y e. I |-> if ( y = x , ( F ` x ) , .0. ) ) ) ( -g ` G ) ( G gsumg F ) ) )
22	2, 3	dprddomcld 18400	. . . . . . . . . . . . 13 |- ( ph -> I e. _V )
23	22	ad2antrr 762	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( G gsumg F ) = .0. ) /\ x e. I ) -> I e. _V )
24		fvex 6201	. . . . . . . . . . . . . 14 |- ( F ` x ) e. _V
25		fvex 6201	. . . . . . . . . . . . . . 15 |- ( 0g ` G ) e. _V
26	11, 25	eqeltri 2697	. . . . . . . . . . . . . 14 |- .0. e. _V
27	24, 26	ifex 4156	. . . . . . . . . . . . 13 |- if ( y = x , ( F ` x ) , .0. ) e. _V
28	27	a1i 11	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ ( G gsumg F ) = .0. ) /\ x e. I ) /\ y e. I ) -> if ( y = x , ( F ` x ) , .0. ) e. _V )
29		fvexd 6203	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ ( G gsumg F ) = .0. ) /\ x e. I ) /\ y e. I ) -> ( F ` y ) e. _V )
30		eqidd 2623	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( G gsumg F ) = .0. ) /\ x e. I ) -> ( y e. I |-> if ( y = x , ( F ` x ) , .0. ) ) = ( y e. I |-> if ( y = x , ( F ` x ) , .0. ) ) )
31	6	ad2antrr 762	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( G gsumg F ) = .0. ) /\ x e. I ) -> F : I --> ( Base ` G ) )
32	31	feqmptd 6249	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( G gsumg F ) = .0. ) /\ x e. I ) -> F = ( y e. I |-> ( F ` y ) ) )
33	23, 28, 29, 30, 32	offval2 6914	. . . . . . . . . . 11 |- ( ( ( ph /\ ( G gsumg F ) = .0. ) /\ x e. I ) -> ( ( y e. I |-> if ( y = x , ( F ` x ) , .0. ) ) oF ( -g ` G ) F ) = ( y e. I |-> ( if ( y = x , ( F ` x ) , .0. ) ( -g ` G ) ( F ` y ) ) ) )
34	33	oveq2d 6666	. . . . . . . . . 10 |- ( ( ( ph /\ ( G gsumg F ) = .0. ) /\ x e. I ) -> ( G gsumg ( ( y e. I |-> if ( y = x , ( F ` x ) , .0. ) ) oF ( -g ` G ) F ) ) = ( G gsumg ( y e. I |-> ( if ( y = x , ( F ` x ) , .0. ) ( -g ` G ) ( F ` y ) ) ) ) )
35	16	simprd 479	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( G gsumg F ) = .0. ) /\ x e. I ) -> ( G gsumg ( y e. I |-> if ( y = x , ( F ` x ) , .0. ) ) ) = ( F ` x ) )
36		simplr 792	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( G gsumg F ) = .0. ) /\ x e. I ) -> ( G gsumg F ) = .0. )
37	35, 36	oveq12d 6668	. . . . . . . . . . 11 |- ( ( ( ph /\ ( G gsumg F ) = .0. ) /\ x e. I ) -> ( ( G gsumg ( y e. I |-> if ( y = x , ( F ` x ) , .0. ) ) ) ( -g ` G ) ( G gsumg F ) ) = ( ( F ` x ) ( -g ` G ) .0. ) )
38		dprdgrp 18404	. . . . . . . . . . . . 13 |- ( G dom DProd S -> G e. Grp )
39	12, 38	syl 17	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( G gsumg F ) = .0. ) /\ x e. I ) -> G e. Grp )
40	31, 14	ffvelrnd 6360	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( G gsumg F ) = .0. ) /\ x e. I ) -> ( F ` x ) e. ( Base ` G ) )
41	5, 11, 19	grpsubid1 17500	. . . . . . . . . . . 12 |- ( ( G e. Grp /\ ( F ` x ) e. ( Base ` G ) ) -> ( ( F ` x ) ( -g ` G ) .0. ) = ( F ` x ) )
42	39, 40, 41	syl2anc 693	. . . . . . . . . . 11 |- ( ( ( ph /\ ( G gsumg F ) = .0. ) /\ x e. I ) -> ( ( F ` x ) ( -g ` G ) .0. ) = ( F ` x ) )
43	37, 42	eqtrd 2656	. . . . . . . . . 10 |- ( ( ( ph /\ ( G gsumg F ) = .0. ) /\ x e. I ) -> ( ( G gsumg ( y e. I |-> if ( y = x , ( F ` x ) , .0. ) ) ) ( -g ` G ) ( G gsumg F ) ) = ( F ` x ) )
44	21, 34, 43	3eqtr3d 2664	. . . . . . . . 9 |- ( ( ( ph /\ ( G gsumg F ) = .0. ) /\ x e. I ) -> ( G gsumg ( y e. I |-> ( if ( y = x , ( F ` x ) , .0. ) ( -g ` G ) ( F ` y ) ) ) ) = ( F ` x ) )
45		eqid 2622	. . . . . . . . . 10 |- (Cntz ` G ) = (Cntz ` G )
46		grpmnd 17429	. . . . . . . . . . . 12 |- ( G e. Grp -> G e. Mnd )
47	2, 38, 46	3syl 18	. . . . . . . . . . 11 |- ( ph -> G e. Mnd )
48	47	ad2antrr 762	. . . . . . . . . 10 |- ( ( ( ph /\ ( G gsumg F ) = .0. ) /\ x e. I ) -> G e. Mnd )
49	5	subgacs 17629	. . . . . . . . . . . . 13 |- ( G e. Grp -> (SubGrp ` G ) e. (ACS ` ( Base ` G ) ) )
50		acsmre 16313	. . . . . . . . . . . . 13 |- ( (SubGrp ` G ) e. (ACS ` ( Base ` G ) ) -> (SubGrp ` G ) e. (Moore ` ( Base ` G ) ) )
51	39, 49, 50	3syl 18	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( G gsumg F ) = .0. ) /\ x e. I ) -> (SubGrp ` G ) e. (Moore ` ( Base ` G ) ) )
52		imassrn 5477	. . . . . . . . . . . . . 14 |- ( S " ( I \ { x } ) ) C_ ran S
53	2, 3	dprdf2 18406	. . . . . . . . . . . . . . . . 17 |- ( ph -> S : I --> (SubGrp ` G ) )
54	53	ad2antrr 762	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ ( G gsumg F ) = .0. ) /\ x e. I ) -> S : I --> (SubGrp ` G ) )
55		frn 6053	. . . . . . . . . . . . . . . 16 |- ( S : I --> (SubGrp ` G ) -> ran S C_ (SubGrp ` G ) )
56	54, 55	syl 17	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ ( G gsumg F ) = .0. ) /\ x e. I ) -> ran S C_ (SubGrp ` G ) )
57		mresspw 16252	. . . . . . . . . . . . . . . 16 |- ( (SubGrp ` G ) e. (Moore ` ( Base ` G ) ) -> (SubGrp ` G ) C_ ~P ( Base ` G ) )
58	51, 57	syl 17	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ ( G gsumg F ) = .0. ) /\ x e. I ) -> (SubGrp ` G ) C_ ~P ( Base ` G ) )
59	56, 58	sstrd 3613	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ ( G gsumg F ) = .0. ) /\ x e. I ) -> ran S C_ ~P ( Base ` G ) )
60	52, 59	syl5ss 3614	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( G gsumg F ) = .0. ) /\ x e. I ) -> ( S " ( I \ { x } ) ) C_ ~P ( Base ` G ) )
61		sspwuni 4611	. . . . . . . . . . . . 13 |- ( ( S " ( I \ { x } ) ) C_ ~P ( Base ` G ) <-> U. ( S " ( I \ { x } ) ) C_ ( Base ` G ) )
62	60, 61	sylib 208	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( G gsumg F ) = .0. ) /\ x e. I ) -> U. ( S " ( I \ { x } ) ) C_ ( Base ` G ) )
63		eqid 2622	. . . . . . . . . . . . 13 |- (mrCls ` (SubGrp ` G ) ) = (mrCls ` (SubGrp ` G ) )
64	63	mrccl 16271	. . . . . . . . . . . 12 |- ( ( (SubGrp ` G ) e. (Moore ` ( Base ` G ) ) /\ U. ( S " ( I \ { x } ) ) C_ ( Base ` G ) ) -> ( (mrCls ` (SubGrp ` G ) ) ` U. ( S " ( I \ { x } ) ) ) e. (SubGrp ` G ) )
65	51, 62, 64	syl2anc 693	. . . . . . . . . . 11 |- ( ( ( ph /\ ( G gsumg F ) = .0. ) /\ x e. I ) -> ( (mrCls ` (SubGrp ` G ) ) ` U. ( S " ( I \ { x } ) ) ) e. (SubGrp ` G ) )
66		subgsubm 17616	. . . . . . . . . . 11 |- ( ( (mrCls ` (SubGrp ` G ) ) ` U. ( S " ( I \ { x } ) ) ) e. (SubGrp ` G ) -> ( (mrCls ` (SubGrp ` G ) ) ` U. ( S " ( I \ { x } ) ) ) e. (SubMnd ` G ) )
67	65, 66	syl 17	. . . . . . . . . 10 |- ( ( ( ph /\ ( G gsumg F ) = .0. ) /\ x e. I ) -> ( (mrCls ` (SubGrp ` G ) ) ` U. ( S " ( I \ { x } ) ) ) e. (SubMnd ` G ) )
68		oveq1 6657	. . . . . . . . . . . . 13 |- ( ( F ` x ) = if ( y = x , ( F ` x ) , .0. ) -> ( ( F ` x ) ( -g ` G ) ( F ` y ) ) = ( if ( y = x , ( F ` x ) , .0. ) ( -g ` G ) ( F ` y ) ) )
69	68	eleq1d 2686	. . . . . . . . . . . 12 |- ( ( F ` x ) = if ( y = x , ( F ` x ) , .0. ) -> ( ( ( F ` x ) ( -g ` G ) ( F ` y ) ) e. ( (mrCls ` (SubGrp ` G ) ) ` U. ( S " ( I \ { x } ) ) ) <-> ( if ( y = x , ( F ` x ) , .0. ) ( -g ` G ) ( F ` y ) ) e. ( (mrCls ` (SubGrp ` G ) ) ` U. ( S " ( I \ { x } ) ) ) ) )
70		oveq1 6657	. . . . . . . . . . . . 13 |- ( .0. = if ( y = x , ( F ` x ) , .0. ) -> ( .0. ( -g ` G ) ( F ` y ) ) = ( if ( y = x , ( F ` x ) , .0. ) ( -g ` G ) ( F ` y ) ) )
71	70	eleq1d 2686	. . . . . . . . . . . 12 |- ( .0. = if ( y = x , ( F ` x ) , .0. ) -> ( ( .0. ( -g ` G ) ( F ` y ) ) e. ( (mrCls ` (SubGrp ` G ) ) ` U. ( S " ( I \ { x } ) ) ) <-> ( if ( y = x , ( F ` x ) , .0. ) ( -g ` G ) ( F ` y ) ) e. ( (mrCls ` (SubGrp ` G ) ) ` U. ( S " ( I \ { x } ) ) ) ) )
72		simpr 477	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ph /\ ( G gsumg F ) = .0. ) /\ x e. I ) /\ y e. I ) /\ y = x ) -> y = x )
73	72	fveq2d 6195	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ph /\ ( G gsumg F ) = .0. ) /\ x e. I ) /\ y e. I ) /\ y = x ) -> ( F ` y ) = ( F ` x ) )
74	73	oveq2d 6666	. . . . . . . . . . . . 13 |- ( ( ( ( ( ph /\ ( G gsumg F ) = .0. ) /\ x e. I ) /\ y e. I ) /\ y = x ) -> ( ( F ` x ) ( -g ` G ) ( F ` y ) ) = ( ( F ` x ) ( -g ` G ) ( F ` x ) ) )
75	5, 11, 19	grpsubid 17499	. . . . . . . . . . . . . . . 16 |- ( ( G e. Grp /\ ( F ` x ) e. ( Base ` G ) ) -> ( ( F ` x ) ( -g ` G ) ( F ` x ) ) = .0. )
76	39, 40, 75	syl2anc 693	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ ( G gsumg F ) = .0. ) /\ x e. I ) -> ( ( F ` x ) ( -g ` G ) ( F ` x ) ) = .0. )
77	11	subg0cl 17602	. . . . . . . . . . . . . . . 16 |- ( ( (mrCls ` (SubGrp ` G ) ) ` U. ( S " ( I \ { x } ) ) ) e. (SubGrp ` G ) -> .0. e. ( (mrCls ` (SubGrp ` G ) ) ` U. ( S " ( I \ { x } ) ) ) )
78	65, 77	syl 17	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ ( G gsumg F ) = .0. ) /\ x e. I ) -> .0. e. ( (mrCls ` (SubGrp ` G ) ) ` U. ( S " ( I \ { x } ) ) ) )
79	76, 78	eqeltrd 2701	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ ( G gsumg F ) = .0. ) /\ x e. I ) -> ( ( F ` x ) ( -g ` G ) ( F ` x ) ) e. ( (mrCls ` (SubGrp ` G ) ) ` U. ( S " ( I \ { x } ) ) ) )
80	79	ad2antrr 762	. . . . . . . . . . . . 13 |- ( ( ( ( ( ph /\ ( G gsumg F ) = .0. ) /\ x e. I ) /\ y e. I ) /\ y = x ) -> ( ( F ` x ) ( -g ` G ) ( F ` x ) ) e. ( (mrCls ` (SubGrp ` G ) ) ` U. ( S " ( I \ { x } ) ) ) )
81	74, 80	eqeltrd 2701	. . . . . . . . . . . 12 |- ( ( ( ( ( ph /\ ( G gsumg F ) = .0. ) /\ x e. I ) /\ y e. I ) /\ y = x ) -> ( ( F ` x ) ( -g ` G ) ( F ` y ) ) e. ( (mrCls ` (SubGrp ` G ) ) ` U. ( S " ( I \ { x } ) ) ) )
82	65	ad2antrr 762	. . . . . . . . . . . . 13 |- ( ( ( ( ( ph /\ ( G gsumg F ) = .0. ) /\ x e. I ) /\ y e. I ) /\ -. y = x ) -> ( (mrCls ` (SubGrp ` G ) ) ` U. ( S " ( I \ { x } ) ) ) e. (SubGrp ` G ) )
83	82, 77	syl 17	. . . . . . . . . . . . 13 |- ( ( ( ( ( ph /\ ( G gsumg F ) = .0. ) /\ x e. I ) /\ y e. I ) /\ -. y = x ) -> .0. e. ( (mrCls ` (SubGrp ` G ) ) ` U. ( S " ( I \ { x } ) ) ) )
84	51, 63, 62	mrcssidd 16285	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ ( G gsumg F ) = .0. ) /\ x e. I ) -> U. ( S " ( I \ { x } ) ) C_ ( (mrCls ` (SubGrp ` G ) ) ` U. ( S " ( I \ { x } ) ) ) )
85	84	ad2antrr 762	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ph /\ ( G gsumg F ) = .0. ) /\ x e. I ) /\ y e. I ) /\ -. y = x ) -> U. ( S " ( I \ { x } ) ) C_ ( (mrCls ` (SubGrp ` G ) ) ` U. ( S " ( I \ { x } ) ) ) )
86	1, 12, 13, 18	dprdfcl 18412	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ph /\ ( G gsumg F ) = .0. ) /\ x e. I ) /\ y e. I ) -> ( F ` y ) e. ( S ` y ) )
87	86	adantr 481	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ph /\ ( G gsumg F ) = .0. ) /\ x e. I ) /\ y e. I ) /\ -. y = x ) -> ( F ` y ) e. ( S ` y ) )
88		ffn 6045	. . . . . . . . . . . . . . . . . 18 |- ( S : I --> (SubGrp ` G ) -> S Fn I )
89	54, 88	syl 17	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ ( G gsumg F ) = .0. ) /\ x e. I ) -> S Fn I )
90	89	ad2antrr 762	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ph /\ ( G gsumg F ) = .0. ) /\ x e. I ) /\ y e. I ) /\ -. y = x ) -> S Fn I )
91		difssd 3738	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ph /\ ( G gsumg F ) = .0. ) /\ x e. I ) /\ y e. I ) /\ -. y = x ) -> ( I \ { x } ) C_ I )
92		df-ne 2795	. . . . . . . . . . . . . . . . . 18 |- ( y =/= x <-> -. y = x )
93		eldifsn 4317	. . . . . . . . . . . . . . . . . . 19 |- ( y e. ( I \ { x } ) <-> ( y e. I /\ y =/= x ) )
94	93	biimpri 218	. . . . . . . . . . . . . . . . . 18 |- ( ( y e. I /\ y =/= x ) -> y e. ( I \ { x } ) )
95	92, 94	sylan2br 493	. . . . . . . . . . . . . . . . 17 |- ( ( y e. I /\ -. y = x ) -> y e. ( I \ { x } ) )
96	95	adantll 750	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ph /\ ( G gsumg F ) = .0. ) /\ x e. I ) /\ y e. I ) /\ -. y = x ) -> y e. ( I \ { x } ) )
97		fnfvima 6496	. . . . . . . . . . . . . . . 16 |- ( ( S Fn I /\ ( I \ { x } ) C_ I /\ y e. ( I \ { x } ) ) -> ( S ` y ) e. ( S " ( I \ { x } ) ) )
98	90, 91, 96, 97	syl3anc 1326	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ph /\ ( G gsumg F ) = .0. ) /\ x e. I ) /\ y e. I ) /\ -. y = x ) -> ( S ` y ) e. ( S " ( I \ { x } ) ) )
99		elunii 4441	. . . . . . . . . . . . . . 15 |- ( ( ( F ` y ) e. ( S ` y ) /\ ( S ` y ) e. ( S " ( I \ { x } ) ) ) -> ( F ` y ) e. U. ( S " ( I \ { x } ) ) )
100	87, 98, 99	syl2anc 693	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ph /\ ( G gsumg F ) = .0. ) /\ x e. I ) /\ y e. I ) /\ -. y = x ) -> ( F ` y ) e. U. ( S " ( I \ { x } ) ) )
101	85, 100	sseldd 3604	. . . . . . . . . . . . 13 |- ( ( ( ( ( ph /\ ( G gsumg F ) = .0. ) /\ x e. I ) /\ y e. I ) /\ -. y = x ) -> ( F ` y ) e. ( (mrCls ` (SubGrp ` G ) ) ` U. ( S " ( I \ { x } ) ) ) )
102	19	subgsubcl 17605	. . . . . . . . . . . . 13 |- ( ( ( (mrCls ` (SubGrp ` G ) ) ` U. ( S " ( I \ { x } ) ) ) e. (SubGrp ` G ) /\ .0. e. ( (mrCls ` (SubGrp ` G ) ) ` U. ( S " ( I \ { x } ) ) ) /\ ( F ` y ) e. ( (mrCls ` (SubGrp ` G ) ) ` U. ( S " ( I \ { x } ) ) ) ) -> ( .0. ( -g ` G ) ( F ` y ) ) e. ( (mrCls ` (SubGrp ` G ) ) ` U. ( S " ( I \ { x } ) ) ) )
103	82, 83, 101, 102	syl3anc 1326	. . . . . . . . . . . 12 |- ( ( ( ( ( ph /\ ( G gsumg F ) = .0. ) /\ x e. I ) /\ y e. I ) /\ -. y = x ) -> ( .0. ( -g ` G ) ( F ` y ) ) e. ( (mrCls ` (SubGrp ` G ) ) ` U. ( S " ( I \ { x } ) ) ) )
104	69, 71, 81, 103	ifbothda 4123	. . . . . . . . . . 11 |- ( ( ( ( ph /\ ( G gsumg F ) = .0. ) /\ x e. I ) /\ y e. I ) -> ( if ( y = x , ( F ` x ) , .0. ) ( -g ` G ) ( F ` y ) ) e. ( (mrCls ` (SubGrp ` G ) ) ` U. ( S " ( I \ { x } ) ) ) )
105		eqid 2622	. . . . . . . . . . 11 |- ( y e. I |-> ( if ( y = x , ( F ` x ) , .0. ) ( -g ` G ) ( F ` y ) ) ) = ( y e. I |-> ( if ( y = x , ( F ` x ) , .0. ) ( -g ` G ) ( F ` y ) ) )
106	104, 105	fmptd 6385	. . . . . . . . . 10 |- ( ( ( ph /\ ( G gsumg F ) = .0. ) /\ x e. I ) -> ( y e. I |-> ( if ( y = x , ( F ` x ) , .0. ) ( -g ` G ) ( F ` y ) ) ) : I --> ( (mrCls ` (SubGrp ` G ) ) ` U. ( S " ( I \ { x } ) ) ) )
107	20	simpld 475	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( G gsumg F ) = .0. ) /\ x e. I ) -> ( ( y e. I |-> if ( y = x , ( F ` x ) , .0. ) ) oF ( -g ` G ) F ) e. W )
108	33, 107	eqeltrrd 2702	. . . . . . . . . . 11 |- ( ( ( ph /\ ( G gsumg F ) = .0. ) /\ x e. I ) -> ( y e. I |-> ( if ( y = x , ( F ` x ) , .0. ) ( -g ` G ) ( F ` y ) ) ) e. W )
109	1, 12, 13, 108, 45	dprdfcntz 18414	. . . . . . . . . 10 |- ( ( ( ph /\ ( G gsumg F ) = .0. ) /\ x e. I ) -> ran ( y e. I |-> ( if ( y = x , ( F ` x ) , .0. ) ( -g ` G ) ( F ` y ) ) ) C_ ( (Cntz ` G ) ` ran ( y e. I |-> ( if ( y = x , ( F ` x ) , .0. ) ( -g ` G ) ( F ` y ) ) ) ) )
110	1, 12, 13, 108	dprdffsupp 18413	. . . . . . . . . 10 |- ( ( ( ph /\ ( G gsumg F ) = .0. ) /\ x e. I ) -> ( y e. I |-> ( if ( y = x , ( F ` x ) , .0. ) ( -g ` G ) ( F ` y ) ) ) finSupp .0. )
111	11, 45, 48, 23, 67, 106, 109, 110	gsumzsubmcl 18318	. . . . . . . . 9 |- ( ( ( ph /\ ( G gsumg F ) = .0. ) /\ x e. I ) -> ( G gsumg ( y e. I |-> ( if ( y = x , ( F ` x ) , .0. ) ( -g ` G ) ( F ` y ) ) ) ) e. ( (mrCls ` (SubGrp ` G ) ) ` U. ( S " ( I \ { x } ) ) ) )
112	44, 111	eqeltrrd 2702	. . . . . . . 8 |- ( ( ( ph /\ ( G gsumg F ) = .0. ) /\ x e. I ) -> ( F ` x ) e. ( (mrCls ` (SubGrp ` G ) ) ` U. ( S " ( I \ { x } ) ) ) )
113	10, 112	elind 3798	. . . . . . 7 |- ( ( ( ph /\ ( G gsumg F ) = .0. ) /\ x e. I ) -> ( F ` x ) e. ( ( S ` x ) i^i ( (mrCls ` (SubGrp ` G ) ) ` U. ( S " ( I \ { x } ) ) ) ) )
114	12, 13, 14, 11, 63	dprddisj 18408	. . . . . . 7 |- ( ( ( ph /\ ( G gsumg F ) = .0. ) /\ x e. I ) -> ( ( S ` x ) i^i ( (mrCls ` (SubGrp ` G ) ) ` U. ( S " ( I \ { x } ) ) ) ) = { .0. } )
115	113, 114	eleqtrd 2703	. . . . . 6 |- ( ( ( ph /\ ( G gsumg F ) = .0. ) /\ x e. I ) -> ( F ` x ) e. { .0. } )
116		elsni 4194	. . . . . 6 |- ( ( F ` x ) e. { .0. } -> ( F ` x ) = .0. )
117	115, 116	syl 17	. . . . 5 |- ( ( ( ph /\ ( G gsumg F ) = .0. ) /\ x e. I ) -> ( F ` x ) = .0. )
118	117	mpteq2dva 4744	. . . 4 |- ( ( ph /\ ( G gsumg F ) = .0. ) -> ( x e. I |-> ( F ` x ) ) = ( x e. I |-> .0. ) )
119	8, 118	eqtrd 2656	. . 3 |- ( ( ph /\ ( G gsumg F ) = .0. ) -> F = ( x e. I |-> .0. ) )
120	119	ex 450	. 2 |- ( ph -> ( ( G gsumg F ) = .0. -> F = ( x e. I |-> .0. ) ) )
121	11	gsumz 17374	. . . 4 |- ( ( G e. Mnd /\ I e. _V ) -> ( G gsumg ( x e. I |-> .0. ) ) = .0. )
122	47, 22, 121	syl2anc 693	. . 3 |- ( ph -> ( G gsumg ( x e. I |-> .0. ) ) = .0. )
123		oveq2 6658	. . . 4 |- ( F = ( x e. I |-> .0. ) -> ( G gsumg F ) = ( G gsumg ( x e. I |-> .0. ) ) )
124	123	eqeq1d 2624	. . 3 |- ( F = ( x e. I |-> .0. ) -> ( ( G gsumg F ) = .0. <-> ( G gsumg ( x e. I |-> .0. ) ) = .0. ) )
125	122, 124	syl5ibrcom 237	. 2 |- ( ph -> ( F = ( x e. I |-> .0. ) -> ( G gsumg F ) = .0. ) )
126	120, 125	impbid 202	1 |- ( ph -> ( ( G gsumg F ) = .0. <-> F = ( x e. I |-> .0. ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   {crab 2916   _Vcvv 3200   \ cdif 3571   i^i cin 3573   C_ wss 3574   ifcif 4086   ~Pcpw 4158   {csn 4177   U.cuni 4436   class class class wbr 4653   |-> cmpt 4729   dom cdm 5114   ran crn 5115   "cima 5117   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   oFcof 6895   X_cixp 7908   finSupp cfsupp 8275   Basecbs 15857   0gc0g 16100   gsum_g cgsu 16101  Moorecmre 16242  mrClscmrc 16243  ACScacs 16245   Mndcmnd 17294  SubMndcsubmnd 17334   Grpcgrp 17422   -gcsg 17424  SubGrpcsubg 17588  Cntzccntz 17748   DProd cdprd 18392

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-inf2 8538  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-of 6897  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-map 7859  df-ixp 7909  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-mhm 17335  df-submnd 17336  df-grp 17425  df-minusg 17426  df-sbg 17427  df-mulg 17541  df-subg 17591  df-ghm 17658  df-gim 17701  df-cntz 17750  df-oppg 17776  df-cmn 18195  df-dprd 18394

This theorem is referenced by:  dprdf11  18422