Theorem dmdprd 18397

Description: The domain of definition of the internal direct product, which states that S is a family of subgroups that mutually commute and have trivial intersections. (Contributed by Mario Carneiro, 25-Apr-2016.) (Proof shortened by AV, 11-Jul-2019.)

Hypotheses

Ref	Expression
dmdprd.z	|- Z = (Cntz ` G )
dmdprd.0	|- .0. = ( 0g ` G )
dmdprd.k	|- K = (mrCls ` (SubGrp ` G ) )

Assertion

Ref	Expression
dmdprd	|- ( ( I e. V /\ dom S = I ) -> ( G dom DProd S <-> ( G e. Grp /\ S : I --> (SubGrp ` G ) /\ A. x e. I ( A. y e. ( I \ { x } ) ( S ` x ) C_ ( Z ` ( S ` y ) ) /\ ( ( S ` x ) i^i ( K ` U. ( S " ( I \ { x } ) ) ) ) = { .0. } ) ) ) )

Distinct variable groups:   x, y, G   x, I, y   x, S, y   x, V, y

Allowed substitution hints:   K( x, y)   .0. ( x, y)   Z( x, y)

Proof of Theorem dmdprd

Dummy variables g h f s are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3212	. . . . 5 |- ( S e. { h | ( h : dom h --> (SubGrp ` G ) /\ A. x e. dom h ( A. y e. ( dom h \ { x } ) ( h ` x ) C_ ( Z ` ( h ` y ) ) /\ ( ( h ` x ) i^i ( K ` U. ( h " ( dom h \ { x } ) ) ) ) = { .0. } ) ) } -> S e. _V )
2	1	a1i 11	. . . 4 |- ( ( I e. V /\ dom S = I ) -> ( S e. { h | ( h : dom h --> (SubGrp ` G ) /\ A. x e. dom h ( A. y e. ( dom h \ { x } ) ( h ` x ) C_ ( Z ` ( h ` y ) ) /\ ( ( h ` x ) i^i ( K ` U. ( h " ( dom h \ { x } ) ) ) ) = { .0. } ) ) } -> S e. _V ) )
3		fex 6490	. . . . . . 7 |- ( ( S : I --> (SubGrp ` G ) /\ I e. V ) -> S e. _V )
4	3	expcom 451	. . . . . 6 |- ( I e. V -> ( S : I --> (SubGrp ` G ) -> S e. _V ) )
5	4	adantr 481	. . . . 5 |- ( ( I e. V /\ dom S = I ) -> ( S : I --> (SubGrp ` G ) -> S e. _V ) )
6	5	adantrd 484	. . . 4 |- ( ( I e. V /\ dom S = I ) -> ( ( S : I --> (SubGrp ` G ) /\ A. x e. I ( A. y e. ( I \ { x } ) ( S ` x ) C_ ( Z ` ( S ` y ) ) /\ ( ( S ` x ) i^i ( K ` U. ( S " ( I \ { x } ) ) ) ) = { .0. } ) ) -> S e. _V ) )
7		df-sbc 3436	. . . . . 6 |- ( [. S / h ]. ( h : dom h --> (SubGrp ` G ) /\ A. x e. dom h ( A. y e. ( dom h \ { x } ) ( h ` x ) C_ ( Z ` ( h ` y ) ) /\ ( ( h ` x ) i^i ( K ` U. ( h " ( dom h \ { x } ) ) ) ) = { .0. } ) ) <-> S e. { h | ( h : dom h --> (SubGrp ` G ) /\ A. x e. dom h ( A. y e. ( dom h \ { x } ) ( h ` x ) C_ ( Z ` ( h ` y ) ) /\ ( ( h ` x ) i^i ( K ` U. ( h " ( dom h \ { x } ) ) ) ) = { .0. } ) ) } )
8		simpr 477	. . . . . . 7 |- ( ( ( I e. V /\ dom S = I ) /\ S e. _V ) -> S e. _V )
9		simpr 477	. . . . . . . . . 10 |- ( ( ( I e. V /\ dom S = I ) /\ h = S ) -> h = S )
10	9	dmeqd 5326	. . . . . . . . . . 11 |- ( ( ( I e. V /\ dom S = I ) /\ h = S ) -> dom h = dom S )
11		simplr 792	. . . . . . . . . . 11 |- ( ( ( I e. V /\ dom S = I ) /\ h = S ) -> dom S = I )
12	10, 11	eqtrd 2656	. . . . . . . . . 10 |- ( ( ( I e. V /\ dom S = I ) /\ h = S ) -> dom h = I )
13	9, 12	feq12d 6033	. . . . . . . . 9 |- ( ( ( I e. V /\ dom S = I ) /\ h = S ) -> ( h : dom h --> (SubGrp ` G ) <-> S : I --> (SubGrp ` G ) ) )
14	12	difeq1d 3727	. . . . . . . . . . . 12 |- ( ( ( I e. V /\ dom S = I ) /\ h = S ) -> ( dom h \ { x } ) = ( I \ { x } ) )
15	9	fveq1d 6193	. . . . . . . . . . . . 13 |- ( ( ( I e. V /\ dom S = I ) /\ h = S ) -> ( h ` x ) = ( S ` x ) )
16	9	fveq1d 6193	. . . . . . . . . . . . . 14 |- ( ( ( I e. V /\ dom S = I ) /\ h = S ) -> ( h ` y ) = ( S ` y ) )
17	16	fveq2d 6195	. . . . . . . . . . . . 13 |- ( ( ( I e. V /\ dom S = I ) /\ h = S ) -> ( Z ` ( h ` y ) ) = ( Z ` ( S ` y ) ) )
18	15, 17	sseq12d 3634	. . . . . . . . . . . 12 |- ( ( ( I e. V /\ dom S = I ) /\ h = S ) -> ( ( h ` x ) C_ ( Z ` ( h ` y ) ) <-> ( S ` x ) C_ ( Z ` ( S ` y ) ) ) )
19	14, 18	raleqbidv 3152	. . . . . . . . . . 11 |- ( ( ( I e. V /\ dom S = I ) /\ h = S ) -> ( A. y e. ( dom h \ { x } ) ( h ` x ) C_ ( Z ` ( h ` y ) ) <-> A. y e. ( I \ { x } ) ( S ` x ) C_ ( Z ` ( S ` y ) ) ) )
20	9, 14	imaeq12d 5467	. . . . . . . . . . . . . . 15 |- ( ( ( I e. V /\ dom S = I ) /\ h = S ) -> ( h " ( dom h \ { x } ) ) = ( S " ( I \ { x } ) ) )
21	20	unieqd 4446	. . . . . . . . . . . . . 14 |- ( ( ( I e. V /\ dom S = I ) /\ h = S ) -> U. ( h " ( dom h \ { x } ) ) = U. ( S " ( I \ { x } ) ) )
22	21	fveq2d 6195	. . . . . . . . . . . . 13 |- ( ( ( I e. V /\ dom S = I ) /\ h = S ) -> ( K ` U. ( h " ( dom h \ { x } ) ) ) = ( K ` U. ( S " ( I \ { x } ) ) ) )
23	15, 22	ineq12d 3815	. . . . . . . . . . . 12 |- ( ( ( I e. V /\ dom S = I ) /\ h = S ) -> ( ( h ` x ) i^i ( K ` U. ( h " ( dom h \ { x } ) ) ) ) = ( ( S ` x ) i^i ( K ` U. ( S " ( I \ { x } ) ) ) ) )
24	23	eqeq1d 2624	. . . . . . . . . . 11 |- ( ( ( I e. V /\ dom S = I ) /\ h = S ) -> ( ( ( h ` x ) i^i ( K ` U. ( h " ( dom h \ { x } ) ) ) ) = { .0. } <-> ( ( S ` x ) i^i ( K ` U. ( S " ( I \ { x } ) ) ) ) = { .0. } ) )
25	19, 24	anbi12d 747	. . . . . . . . . 10 |- ( ( ( I e. V /\ dom S = I ) /\ h = S ) -> ( ( A. y e. ( dom h \ { x } ) ( h ` x ) C_ ( Z ` ( h ` y ) ) /\ ( ( h ` x ) i^i ( K ` U. ( h " ( dom h \ { x } ) ) ) ) = { .0. } ) <-> ( A. y e. ( I \ { x } ) ( S ` x ) C_ ( Z ` ( S ` y ) ) /\ ( ( S ` x ) i^i ( K ` U. ( S " ( I \ { x } ) ) ) ) = { .0. } ) ) )
26	12, 25	raleqbidv 3152	. . . . . . . . 9 |- ( ( ( I e. V /\ dom S = I ) /\ h = S ) -> ( A. x e. dom h ( A. y e. ( dom h \ { x } ) ( h ` x ) C_ ( Z ` ( h ` y ) ) /\ ( ( h ` x ) i^i ( K ` U. ( h " ( dom h \ { x } ) ) ) ) = { .0. } ) <-> A. x e. I ( A. y e. ( I \ { x } ) ( S ` x ) C_ ( Z ` ( S ` y ) ) /\ ( ( S ` x ) i^i ( K ` U. ( S " ( I \ { x } ) ) ) ) = { .0. } ) ) )
27	13, 26	anbi12d 747	. . . . . . . 8 |- ( ( ( I e. V /\ dom S = I ) /\ h = S ) -> ( ( h : dom h --> (SubGrp ` G ) /\ A. x e. dom h ( A. y e. ( dom h \ { x } ) ( h ` x ) C_ ( Z ` ( h ` y ) ) /\ ( ( h ` x ) i^i ( K ` U. ( h " ( dom h \ { x } ) ) ) ) = { .0. } ) ) <-> ( S : I --> (SubGrp ` G ) /\ A. x e. I ( A. y e. ( I \ { x } ) ( S ` x ) C_ ( Z ` ( S ` y ) ) /\ ( ( S ` x ) i^i ( K ` U. ( S " ( I \ { x } ) ) ) ) = { .0. } ) ) ) )
28	27	adantlr 751	. . . . . . 7 |- ( ( ( ( I e. V /\ dom S = I ) /\ S e. _V ) /\ h = S ) -> ( ( h : dom h --> (SubGrp ` G ) /\ A. x e. dom h ( A. y e. ( dom h \ { x } ) ( h ` x ) C_ ( Z ` ( h ` y ) ) /\ ( ( h ` x ) i^i ( K ` U. ( h " ( dom h \ { x } ) ) ) ) = { .0. } ) ) <-> ( S : I --> (SubGrp ` G ) /\ A. x e. I ( A. y e. ( I \ { x } ) ( S ` x ) C_ ( Z ` ( S ` y ) ) /\ ( ( S ` x ) i^i ( K ` U. ( S " ( I \ { x } ) ) ) ) = { .0. } ) ) ) )
29	8, 28	sbcied 3472	. . . . . 6 |- ( ( ( I e. V /\ dom S = I ) /\ S e. _V ) -> ( [. S / h ]. ( h : dom h --> (SubGrp ` G ) /\ A. x e. dom h ( A. y e. ( dom h \ { x } ) ( h ` x ) C_ ( Z ` ( h ` y ) ) /\ ( ( h ` x ) i^i ( K ` U. ( h " ( dom h \ { x } ) ) ) ) = { .0. } ) ) <-> ( S : I --> (SubGrp ` G ) /\ A. x e. I ( A. y e. ( I \ { x } ) ( S ` x ) C_ ( Z ` ( S ` y ) ) /\ ( ( S ` x ) i^i ( K ` U. ( S " ( I \ { x } ) ) ) ) = { .0. } ) ) ) )
30	7, 29	syl5bbr 274	. . . . 5 |- ( ( ( I e. V /\ dom S = I ) /\ S e. _V ) -> ( S e. { h | ( h : dom h --> (SubGrp ` G ) /\ A. x e. dom h ( A. y e. ( dom h \ { x } ) ( h ` x ) C_ ( Z ` ( h ` y ) ) /\ ( ( h ` x ) i^i ( K ` U. ( h " ( dom h \ { x } ) ) ) ) = { .0. } ) ) } <-> ( S : I --> (SubGrp ` G ) /\ A. x e. I ( A. y e. ( I \ { x } ) ( S ` x ) C_ ( Z ` ( S ` y ) ) /\ ( ( S ` x ) i^i ( K ` U. ( S " ( I \ { x } ) ) ) ) = { .0. } ) ) ) )
31	30	ex 450	. . . 4 |- ( ( I e. V /\ dom S = I ) -> ( S e. _V -> ( S e. { h | ( h : dom h --> (SubGrp ` G ) /\ A. x e. dom h ( A. y e. ( dom h \ { x } ) ( h ` x ) C_ ( Z ` ( h ` y ) ) /\ ( ( h ` x ) i^i ( K ` U. ( h " ( dom h \ { x } ) ) ) ) = { .0. } ) ) } <-> ( S : I --> (SubGrp ` G ) /\ A. x e. I ( A. y e. ( I \ { x } ) ( S ` x ) C_ ( Z ` ( S ` y ) ) /\ ( ( S ` x ) i^i ( K ` U. ( S " ( I \ { x } ) ) ) ) = { .0. } ) ) ) ) )
32	2, 6, 31	pm5.21ndd 369	. . 3 |- ( ( I e. V /\ dom S = I ) -> ( S e. { h | ( h : dom h --> (SubGrp ` G ) /\ A. x e. dom h ( A. y e. ( dom h \ { x } ) ( h ` x ) C_ ( Z ` ( h ` y ) ) /\ ( ( h ` x ) i^i ( K ` U. ( h " ( dom h \ { x } ) ) ) ) = { .0. } ) ) } <-> ( S : I --> (SubGrp ` G ) /\ A. x e. I ( A. y e. ( I \ { x } ) ( S ` x ) C_ ( Z ` ( S ` y ) ) /\ ( ( S ` x ) i^i ( K ` U. ( S " ( I \ { x } ) ) ) ) = { .0. } ) ) ) )
33	32	anbi2d 740	. 2 |- ( ( I e. V /\ dom S = I ) -> ( ( G e. Grp /\ S e. { h | ( h : dom h --> (SubGrp ` G ) /\ A. x e. dom h ( A. y e. ( dom h \ { x } ) ( h ` x ) C_ ( Z ` ( h ` y ) ) /\ ( ( h ` x ) i^i ( K ` U. ( h " ( dom h \ { x } ) ) ) ) = { .0. } ) ) } ) <-> ( G e. Grp /\ ( S : I --> (SubGrp ` G ) /\ A. x e. I ( A. y e. ( I \ { x } ) ( S ` x ) C_ ( Z ` ( S ` y ) ) /\ ( ( S ` x ) i^i ( K ` U. ( S " ( I \ { x } ) ) ) ) = { .0. } ) ) ) ) )
34		df-br 4654	. . 3 |- ( G dom DProd S <-> <. G , S >. e. dom DProd )
35		fvex 6201	. . . . . . . . . . 11 |- ( s ` x ) e. _V
36	35	rgenw 2924	. . . . . . . . . 10 |- A. x e. dom s ( s ` x ) e. _V
37		ixpexg 7932	. . . . . . . . . 10 |- ( A. x e. dom s ( s ` x ) e. _V -> X_ x e. dom s ( s ` x ) e. _V )
38	36, 37	ax-mp 5	. . . . . . . . 9 |- X_ x e. dom s ( s ` x ) e. _V
39	38	mptrabex 6488	. . . . . . . 8 |- ( f e. { h e. X_ x e. dom s ( s ` x ) | h finSupp ( 0g ` g ) } |-> ( g gsumg f ) ) e. _V
40	39	rnex 7100	. . . . . . 7 |- ran ( f e. { h e. X_ x e. dom s ( s ` x ) | h finSupp ( 0g ` g ) } |-> ( g gsumg f ) ) e. _V
41	40	rgen2w 2925	. . . . . 6 |- A. g e. Grp A. s e. { h | ( h : dom h --> (SubGrp ` g ) /\ A. x e. dom h ( A. y e. ( dom h \ { x } ) ( h ` x ) C_ ( (Cntz ` g ) ` ( h ` y ) ) /\ ( ( h ` x ) i^i ( (mrCls ` (SubGrp ` g ) ) ` U. ( h " ( dom h \ { x } ) ) ) ) = { ( 0g ` g ) } ) ) } ran ( f e. { h e. X_ x e. dom s ( s ` x ) | h finSupp ( 0g ` g ) } |-> ( g gsumg f ) ) e. _V
42		df-dprd 18394	. . . . . . 7 |- DProd = ( g e. Grp , s e. { h | ( h : dom h --> (SubGrp ` g ) /\ A. x e. dom h ( A. y e. ( dom h \ { x } ) ( h ` x ) C_ ( (Cntz ` g ) ` ( h ` y ) ) /\ ( ( h ` x ) i^i ( (mrCls ` (SubGrp ` g ) ) ` U. ( h " ( dom h \ { x } ) ) ) ) = { ( 0g ` g ) } ) ) } |-> ran ( f e. { h e. X_ x e. dom s ( s ` x ) | h finSupp ( 0g ` g ) } |-> ( g gsumg f ) ) )
43	42	fmpt2x 7236	. . . . . 6 |- ( A. g e. Grp A. s e. { h | ( h : dom h --> (SubGrp ` g ) /\ A. x e. dom h ( A. y e. ( dom h \ { x } ) ( h ` x ) C_ ( (Cntz ` g ) ` ( h ` y ) ) /\ ( ( h ` x ) i^i ( (mrCls ` (SubGrp ` g ) ) ` U. ( h " ( dom h \ { x } ) ) ) ) = { ( 0g ` g ) } ) ) } ran ( f e. { h e. X_ x e. dom s ( s ` x ) | h finSupp ( 0g ` g ) } |-> ( g gsumg f ) ) e. _V <-> DProd : U_ g e. Grp ( { g } X. { h | ( h : dom h --> (SubGrp ` g ) /\ A. x e. dom h ( A. y e. ( dom h \ { x } ) ( h ` x ) C_ ( (Cntz ` g ) ` ( h ` y ) ) /\ ( ( h ` x ) i^i ( (mrCls ` (SubGrp ` g ) ) ` U. ( h " ( dom h \ { x } ) ) ) ) = { ( 0g ` g ) } ) ) } ) --> _V )
44	41, 43	mpbi 220	. . . . 5 |- DProd : U_ g e. Grp ( { g } X. { h | ( h : dom h --> (SubGrp ` g ) /\ A. x e. dom h ( A. y e. ( dom h \ { x } ) ( h ` x ) C_ ( (Cntz ` g ) ` ( h ` y ) ) /\ ( ( h ` x ) i^i ( (mrCls ` (SubGrp ` g ) ) ` U. ( h " ( dom h \ { x } ) ) ) ) = { ( 0g ` g ) } ) ) } ) --> _V
45	44	fdmi 6052	. . . 4 |- dom DProd = U_ g e. Grp ( { g } X. { h | ( h : dom h --> (SubGrp ` g ) /\ A. x e. dom h ( A. y e. ( dom h \ { x } ) ( h ` x ) C_ ( (Cntz ` g ) ` ( h ` y ) ) /\ ( ( h ` x ) i^i ( (mrCls ` (SubGrp ` g ) ) ` U. ( h " ( dom h \ { x } ) ) ) ) = { ( 0g ` g ) } ) ) } )
46	45	eleq2i 2693	. . 3 |- ( <. G , S >. e. dom DProd <-> <. G , S >. e. U_ g e. Grp ( { g } X. { h | ( h : dom h --> (SubGrp ` g ) /\ A. x e. dom h ( A. y e. ( dom h \ { x } ) ( h ` x ) C_ ( (Cntz ` g ) ` ( h ` y ) ) /\ ( ( h ` x ) i^i ( (mrCls ` (SubGrp ` g ) ) ` U. ( h " ( dom h \ { x } ) ) ) ) = { ( 0g ` g ) } ) ) } ) )
47		fveq2 6191	. . . . . . 7 |- ( g = G -> (SubGrp ` g ) = (SubGrp ` G ) )
48	47	feq3d 6032	. . . . . 6 |- ( g = G -> ( h : dom h --> (SubGrp ` g ) <-> h : dom h --> (SubGrp ` G ) ) )
49		fveq2 6191	. . . . . . . . . . . 12 |- ( g = G -> (Cntz ` g ) = (Cntz ` G ) )
50		dmdprd.z	. . . . . . . . . . . 12 |- Z = (Cntz ` G )
51	49, 50	syl6eqr 2674	. . . . . . . . . . 11 |- ( g = G -> (Cntz ` g ) = Z )
52	51	fveq1d 6193	. . . . . . . . . 10 |- ( g = G -> ( (Cntz ` g ) ` ( h ` y ) ) = ( Z ` ( h ` y ) ) )
53	52	sseq2d 3633	. . . . . . . . 9 |- ( g = G -> ( ( h ` x ) C_ ( (Cntz ` g ) ` ( h ` y ) ) <-> ( h ` x ) C_ ( Z ` ( h ` y ) ) ) )
54	53	ralbidv 2986	. . . . . . . 8 |- ( g = G -> ( A. y e. ( dom h \ { x } ) ( h ` x ) C_ ( (Cntz ` g ) ` ( h ` y ) ) <-> A. y e. ( dom h \ { x } ) ( h ` x ) C_ ( Z ` ( h ` y ) ) ) )
55	47	fveq2d 6195	. . . . . . . . . . . 12 |- ( g = G -> (mrCls ` (SubGrp ` g ) ) = (mrCls ` (SubGrp ` G ) ) )
56		dmdprd.k	. . . . . . . . . . . 12 |- K = (mrCls ` (SubGrp ` G ) )
57	55, 56	syl6eqr 2674	. . . . . . . . . . 11 |- ( g = G -> (mrCls ` (SubGrp ` g ) ) = K )
58	57	fveq1d 6193	. . . . . . . . . 10 |- ( g = G -> ( (mrCls ` (SubGrp ` g ) ) ` U. ( h " ( dom h \ { x } ) ) ) = ( K ` U. ( h " ( dom h \ { x } ) ) ) )
59	58	ineq2d 3814	. . . . . . . . 9 |- ( g = G -> ( ( h ` x ) i^i ( (mrCls ` (SubGrp ` g ) ) ` U. ( h " ( dom h \ { x } ) ) ) ) = ( ( h ` x ) i^i ( K ` U. ( h " ( dom h \ { x } ) ) ) ) )
60		fveq2 6191	. . . . . . . . . . 11 |- ( g = G -> ( 0g ` g ) = ( 0g ` G ) )
61		dmdprd.0	. . . . . . . . . . 11 |- .0. = ( 0g ` G )
62	60, 61	syl6eqr 2674	. . . . . . . . . 10 |- ( g = G -> ( 0g ` g ) = .0. )
63	62	sneqd 4189	. . . . . . . . 9 |- ( g = G -> { ( 0g ` g ) } = { .0. } )
64	59, 63	eqeq12d 2637	. . . . . . . 8 |- ( g = G -> ( ( ( h ` x ) i^i ( (mrCls ` (SubGrp ` g ) ) ` U. ( h " ( dom h \ { x } ) ) ) ) = { ( 0g ` g ) } <-> ( ( h ` x ) i^i ( K ` U. ( h " ( dom h \ { x } ) ) ) ) = { .0. } ) )
65	54, 64	anbi12d 747	. . . . . . 7 |- ( g = G -> ( ( A. y e. ( dom h \ { x } ) ( h ` x ) C_ ( (Cntz ` g ) ` ( h ` y ) ) /\ ( ( h ` x ) i^i ( (mrCls ` (SubGrp ` g ) ) ` U. ( h " ( dom h \ { x } ) ) ) ) = { ( 0g ` g ) } ) <-> ( A. y e. ( dom h \ { x } ) ( h ` x ) C_ ( Z ` ( h ` y ) ) /\ ( ( h ` x ) i^i ( K ` U. ( h " ( dom h \ { x } ) ) ) ) = { .0. } ) ) )
66	65	ralbidv 2986	. . . . . 6 |- ( g = G -> ( A. x e. dom h ( A. y e. ( dom h \ { x } ) ( h ` x ) C_ ( (Cntz ` g ) ` ( h ` y ) ) /\ ( ( h ` x ) i^i ( (mrCls ` (SubGrp ` g ) ) ` U. ( h " ( dom h \ { x } ) ) ) ) = { ( 0g ` g ) } ) <-> A. x e. dom h ( A. y e. ( dom h \ { x } ) ( h ` x ) C_ ( Z ` ( h ` y ) ) /\ ( ( h ` x ) i^i ( K ` U. ( h " ( dom h \ { x } ) ) ) ) = { .0. } ) ) )
67	48, 66	anbi12d 747	. . . . 5 |- ( g = G -> ( ( h : dom h --> (SubGrp ` g ) /\ A. x e. dom h ( A. y e. ( dom h \ { x } ) ( h ` x ) C_ ( (Cntz ` g ) ` ( h ` y ) ) /\ ( ( h ` x ) i^i ( (mrCls ` (SubGrp ` g ) ) ` U. ( h " ( dom h \ { x } ) ) ) ) = { ( 0g ` g ) } ) ) <-> ( h : dom h --> (SubGrp ` G ) /\ A. x e. dom h ( A. y e. ( dom h \ { x } ) ( h ` x ) C_ ( Z ` ( h ` y ) ) /\ ( ( h ` x ) i^i ( K ` U. ( h " ( dom h \ { x } ) ) ) ) = { .0. } ) ) ) )
68	67	abbidv 2741	. . . 4 |- ( g = G -> { h | ( h : dom h --> (SubGrp ` g ) /\ A. x e. dom h ( A. y e. ( dom h \ { x } ) ( h ` x ) C_ ( (Cntz ` g ) ` ( h ` y ) ) /\ ( ( h ` x ) i^i ( (mrCls ` (SubGrp ` g ) ) ` U. ( h " ( dom h \ { x } ) ) ) ) = { ( 0g ` g ) } ) ) } = { h | ( h : dom h --> (SubGrp ` G ) /\ A. x e. dom h ( A. y e. ( dom h \ { x } ) ( h ` x ) C_ ( Z ` ( h ` y ) ) /\ ( ( h ` x ) i^i ( K ` U. ( h " ( dom h \ { x } ) ) ) ) = { .0. } ) ) } )
69	68	opeliunxp2 5260	. . 3 |- ( <. G , S >. e. U_ g e. Grp ( { g } X. { h | ( h : dom h --> (SubGrp ` g ) /\ A. x e. dom h ( A. y e. ( dom h \ { x } ) ( h ` x ) C_ ( (Cntz ` g ) ` ( h ` y ) ) /\ ( ( h ` x ) i^i ( (mrCls ` (SubGrp ` g ) ) ` U. ( h " ( dom h \ { x } ) ) ) ) = { ( 0g ` g ) } ) ) } ) <-> ( G e. Grp /\ S e. { h | ( h : dom h --> (SubGrp ` G ) /\ A. x e. dom h ( A. y e. ( dom h \ { x } ) ( h ` x ) C_ ( Z ` ( h ` y ) ) /\ ( ( h ` x ) i^i ( K ` U. ( h " ( dom h \ { x } ) ) ) ) = { .0. } ) ) } ) )
70	34, 46, 69	3bitri 286	. 2 |- ( G dom DProd S <-> ( G e. Grp /\ S e. { h | ( h : dom h --> (SubGrp ` G ) /\ A. x e. dom h ( A. y e. ( dom h \ { x } ) ( h ` x ) C_ ( Z ` ( h ` y ) ) /\ ( ( h ` x ) i^i ( K ` U. ( h " ( dom h \ { x } ) ) ) ) = { .0. } ) ) } ) )
71		3anass 1042	. 2 |- ( ( G e. Grp /\ S : I --> (SubGrp ` G ) /\ A. x e. I ( A. y e. ( I \ { x } ) ( S ` x ) C_ ( Z ` ( S ` y ) ) /\ ( ( S ` x ) i^i ( K ` U. ( S " ( I \ { x } ) ) ) ) = { .0. } ) ) <-> ( G e. Grp /\ ( S : I --> (SubGrp ` G ) /\ A. x e. I ( A. y e. ( I \ { x } ) ( S ` x ) C_ ( Z ` ( S ` y ) ) /\ ( ( S ` x ) i^i ( K ` U. ( S " ( I \ { x } ) ) ) ) = { .0. } ) ) ) )
72	33, 70, 71	3bitr4g 303	1 |- ( ( I e. V /\ dom S = I ) -> ( G dom DProd S <-> ( G e. Grp /\ S : I --> (SubGrp ` G ) /\ A. x e. I ( A. y e. ( I \ { x } ) ( S ` x ) C_ ( Z ` ( S ` y ) ) /\ ( ( S ` x ) i^i ( K ` U. ( S " ( I \ { x } ) ) ) ) = { .0. } ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   {cab 2608   A.wral 2912   {crab 2916   _Vcvv 3200   [.wsbc 3435   \ cdif 3571   i^i cin 3573   C_ wss 3574   {csn 4177   <.cop 4183   U.cuni 4436   U_ciun 4520   class class class wbr 4653   |-> cmpt 4729   X. cxp 5112   dom cdm 5114   ran crn 5115   "cima 5117   -->wf 5884   ` cfv 5888  (class class class)co 6650   X_cixp 7908   finSupp cfsupp 8275   0gc0g 16100   gsum_g cgsu 16101  mrClscmrc 16243   Grpcgrp 17422  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

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-ral 2917  df-rex 2918  df-reu 2919  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-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  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-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-ixp 7909  df-dprd 18394

This theorem is referenced by:  dmdprdd  18398  dprdgrp  18404  dprdf  18405  dprdcntz  18407  dprddisj  18408  dprdres  18427  subgdmdprd  18433