Theorem dprd2dlem1 18440

Description: The direct product of a collection of direct products. (Contributed by Mario Carneiro, 26-Apr-2016.)

Hypotheses

Ref	Expression
dprd2d.1	|- ( ph -> Rel A )
dprd2d.2	|- ( ph -> S : A --> (SubGrp ` G ) )
dprd2d.3	|- ( ph -> dom A C_ I )
dprd2d.4	|- ( ( ph /\ i e. I ) -> G dom DProd ( j e. ( A " { i } ) |-> ( i S j ) ) )
dprd2d.5	|- ( ph -> G dom DProd ( i e. I |-> ( G DProd ( j e. ( A " { i } ) |-> ( i S j ) ) ) ) )
dprd2d.k	|- K = (mrCls ` (SubGrp ` G ) )
dprd2d.6	|- ( ph -> C C_ I )

Assertion

Ref	Expression
dprd2dlem1	|- ( ph -> ( K ` U. ( S " ( A |` C ) ) ) = ( G DProd ( i e. C |-> ( G DProd ( j e. ( A " { i } ) |-> ( i S j ) ) ) ) ) )

Distinct variable groups:   i, j, A   C, i   i, G, j   i, I   i, K   ph, i, j   S, i, j

Allowed substitution hints:   C( j)   I( j)   K( j)

Proof of Theorem dprd2dlem1

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

Step	Hyp	Ref	Expression
1		dprd2d.5	. . . . . 6 |- ( ph -> G dom DProd ( i e. I |-> ( G DProd ( j e. ( A " { i } ) |-> ( i S j ) ) ) ) )
2		dprdgrp 18404	. . . . . 6 |- ( G dom DProd ( i e. I |-> ( G DProd ( j e. ( A " { i } ) |-> ( i S j ) ) ) ) -> G e. Grp )
3	1, 2	syl 17	. . . . 5 |- ( ph -> G e. Grp )
4		eqid 2622	. . . . . 6 |- ( Base ` G ) = ( Base ` G )
5	4	subgacs 17629	. . . . 5 |- ( G e. Grp -> (SubGrp ` G ) e. (ACS ` ( Base ` G ) ) )
6		acsmre 16313	. . . . 5 |- ( (SubGrp ` G ) e. (ACS ` ( Base ` G ) ) -> (SubGrp ` G ) e. (Moore ` ( Base ` G ) ) )
7	3, 5, 6	3syl 18	. . . 4 |- ( ph -> (SubGrp ` G ) e. (Moore ` ( Base ` G ) ) )
8		dprd2d.k	. . . 4 |- K = (mrCls ` (SubGrp ` G ) )
9		dprd2d.2	. . . . . 6 |- ( ph -> S : A --> (SubGrp ` G ) )
10		ffun 6048	. . . . . 6 |- ( S : A --> (SubGrp ` G ) -> Fun S )
11		funiunfv 6506	. . . . . 6 |- ( Fun S -> U_ x e. ( A |` C ) ( S ` x ) = U. ( S " ( A |` C ) ) )
12	9, 10, 11	3syl 18	. . . . 5 |- ( ph -> U_ x e. ( A |` C ) ( S ` x ) = U. ( S " ( A |` C ) ) )
13		resss 5422	. . . . . . . . . 10 |- ( A |` C ) C_ A
14	13	sseli 3599	. . . . . . . . 9 |- ( x e. ( A |` C ) -> x e. A )
15		dprd2d.1	. . . . . . . . . 10 |- ( ph -> Rel A )
16		dprd2d.3	. . . . . . . . . 10 |- ( ph -> dom A C_ I )
17		dprd2d.4	. . . . . . . . . 10 |- ( ( ph /\ i e. I ) -> G dom DProd ( j e. ( A " { i } ) |-> ( i S j ) ) )
18	15, 9, 16, 17, 1, 8	dprd2dlem2 18439	. . . . . . . . 9 |- ( ( ph /\ x e. A ) -> ( S ` x ) C_ ( G DProd ( j e. ( A " { ( 1st ` x ) } ) |-> ( ( 1st ` x ) S j ) ) ) )
19	14, 18	sylan2 491	. . . . . . . 8 |- ( ( ph /\ x e. ( A |` C ) ) -> ( S ` x ) C_ ( G DProd ( j e. ( A " { ( 1st ` x ) } ) |-> ( ( 1st ` x ) S j ) ) ) )
20		1st2nd 7214	. . . . . . . . . . . . 13 |- ( ( Rel A /\ x e. A ) -> x = <. ( 1st ` x ) , ( 2nd ` x ) >. )
21	15, 14, 20	syl2an 494	. . . . . . . . . . . 12 |- ( ( ph /\ x e. ( A |` C ) ) -> x = <. ( 1st ` x ) , ( 2nd ` x ) >. )
22		simpr 477	. . . . . . . . . . . 12 |- ( ( ph /\ x e. ( A |` C ) ) -> x e. ( A |` C ) )
23	21, 22	eqeltrrd 2702	. . . . . . . . . . 11 |- ( ( ph /\ x e. ( A |` C ) ) -> <. ( 1st ` x ) , ( 2nd ` x ) >. e. ( A |` C ) )
24		fvex 6201	. . . . . . . . . . . . 13 |- ( 2nd ` x ) e. _V
25	24	opelres 5401	. . . . . . . . . . . 12 |- ( <. ( 1st ` x ) , ( 2nd ` x ) >. e. ( A |` C ) <-> ( <. ( 1st ` x ) , ( 2nd ` x ) >. e. A /\ ( 1st ` x ) e. C ) )
26	25	simprbi 480	. . . . . . . . . . 11 |- ( <. ( 1st ` x ) , ( 2nd ` x ) >. e. ( A |` C ) -> ( 1st ` x ) e. C )
27	23, 26	syl 17	. . . . . . . . . 10 |- ( ( ph /\ x e. ( A |` C ) ) -> ( 1st ` x ) e. C )
28		ovex 6678	. . . . . . . . . 10 |- ( G DProd ( j e. ( A " { ( 1st ` x ) } ) |-> ( ( 1st ` x ) S j ) ) ) e. _V
29		eqid 2622	. . . . . . . . . . 11 |- ( i e. C |-> ( G DProd ( j e. ( A " { i } ) |-> ( i S j ) ) ) ) = ( i e. C |-> ( G DProd ( j e. ( A " { i } ) |-> ( i S j ) ) ) )
30		sneq 4187	. . . . . . . . . . . . . 14 |- ( i = ( 1st ` x ) -> { i } = { ( 1st ` x ) } )
31	30	imaeq2d 5466	. . . . . . . . . . . . 13 |- ( i = ( 1st ` x ) -> ( A " { i } ) = ( A " { ( 1st ` x ) } ) )
32		oveq1 6657	. . . . . . . . . . . . 13 |- ( i = ( 1st ` x ) -> ( i S j ) = ( ( 1st ` x ) S j ) )
33	31, 32	mpteq12dv 4733	. . . . . . . . . . . 12 |- ( i = ( 1st ` x ) -> ( j e. ( A " { i } ) |-> ( i S j ) ) = ( j e. ( A " { ( 1st ` x ) } ) |-> ( ( 1st ` x ) S j ) ) )
34	33	oveq2d 6666	. . . . . . . . . . 11 |- ( i = ( 1st ` x ) -> ( G DProd ( j e. ( A " { i } ) |-> ( i S j ) ) ) = ( G DProd ( j e. ( A " { ( 1st ` x ) } ) |-> ( ( 1st ` x ) S j ) ) ) )
35	29, 34	elrnmpt1s 5373	. . . . . . . . . 10 |- ( ( ( 1st ` x ) e. C /\ ( G DProd ( j e. ( A " { ( 1st ` x ) } ) |-> ( ( 1st ` x ) S j ) ) ) e. _V ) -> ( G DProd ( j e. ( A " { ( 1st ` x ) } ) |-> ( ( 1st ` x ) S j ) ) ) e. ran ( i e. C |-> ( G DProd ( j e. ( A " { i } ) |-> ( i S j ) ) ) ) )
36	27, 28, 35	sylancl 694	. . . . . . . . 9 |- ( ( ph /\ x e. ( A |` C ) ) -> ( G DProd ( j e. ( A " { ( 1st ` x ) } ) |-> ( ( 1st ` x ) S j ) ) ) e. ran ( i e. C |-> ( G DProd ( j e. ( A " { i } ) |-> ( i S j ) ) ) ) )
37		elssuni 4467	. . . . . . . . 9 |- ( ( G DProd ( j e. ( A " { ( 1st ` x ) } ) |-> ( ( 1st ` x ) S j ) ) ) e. ran ( i e. C |-> ( G DProd ( j e. ( A " { i } ) |-> ( i S j ) ) ) ) -> ( G DProd ( j e. ( A " { ( 1st ` x ) } ) |-> ( ( 1st ` x ) S j ) ) ) C_ U. ran ( i e. C |-> ( G DProd ( j e. ( A " { i } ) |-> ( i S j ) ) ) ) )
38	36, 37	syl 17	. . . . . . . 8 |- ( ( ph /\ x e. ( A |` C ) ) -> ( G DProd ( j e. ( A " { ( 1st ` x ) } ) |-> ( ( 1st ` x ) S j ) ) ) C_ U. ran ( i e. C |-> ( G DProd ( j e. ( A " { i } ) |-> ( i S j ) ) ) ) )
39	19, 38	sstrd 3613	. . . . . . 7 |- ( ( ph /\ x e. ( A |` C ) ) -> ( S ` x ) C_ U. ran ( i e. C |-> ( G DProd ( j e. ( A " { i } ) |-> ( i S j ) ) ) ) )
40	39	ralrimiva 2966	. . . . . 6 |- ( ph -> A. x e. ( A |` C ) ( S ` x ) C_ U. ran ( i e. C |-> ( G DProd ( j e. ( A " { i } ) |-> ( i S j ) ) ) ) )
41		iunss 4561	. . . . . 6 |- ( U_ x e. ( A |` C ) ( S ` x ) C_ U. ran ( i e. C |-> ( G DProd ( j e. ( A " { i } ) |-> ( i S j ) ) ) ) <-> A. x e. ( A |` C ) ( S ` x ) C_ U. ran ( i e. C |-> ( G DProd ( j e. ( A " { i } ) |-> ( i S j ) ) ) ) )
42	40, 41	sylibr 224	. . . . 5 |- ( ph -> U_ x e. ( A |` C ) ( S ` x ) C_ U. ran ( i e. C |-> ( G DProd ( j e. ( A " { i } ) |-> ( i S j ) ) ) ) )
43	12, 42	eqsstr3d 3640	. . . 4 |- ( ph -> U. ( S " ( A |` C ) ) C_ U. ran ( i e. C |-> ( G DProd ( j e. ( A " { i } ) |-> ( i S j ) ) ) ) )
44		dprd2d.6	. . . . . . . . . . . 12 |- ( ph -> C C_ I )
45	44	sselda 3603	. . . . . . . . . . 11 |- ( ( ph /\ i e. C ) -> i e. I )
46	45, 17	syldan 487	. . . . . . . . . 10 |- ( ( ph /\ i e. C ) -> G dom DProd ( j e. ( A " { i } ) |-> ( i S j ) ) )
47		ovex 6678	. . . . . . . . . . . 12 |- ( i S j ) e. _V
48		eqid 2622	. . . . . . . . . . . 12 |- ( j e. ( A " { i } ) |-> ( i S j ) ) = ( j e. ( A " { i } ) |-> ( i S j ) )
49	47, 48	dmmpti 6023	. . . . . . . . . . 11 |- dom ( j e. ( A " { i } ) |-> ( i S j ) ) = ( A " { i } )
50	49	a1i 11	. . . . . . . . . 10 |- ( ( ph /\ i e. C ) -> dom ( j e. ( A " { i } ) |-> ( i S j ) ) = ( A " { i } ) )
51		imassrn 5477	. . . . . . . . . . . . . 14 |- ( S " ( A |` C ) ) C_ ran S
52		frn 6053	. . . . . . . . . . . . . . . 16 |- ( S : A --> (SubGrp ` G ) -> ran S C_ (SubGrp ` G ) )
53	9, 52	syl 17	. . . . . . . . . . . . . . 15 |- ( ph -> ran S C_ (SubGrp ` G ) )
54		mresspw 16252	. . . . . . . . . . . . . . . 16 |- ( (SubGrp ` G ) e. (Moore ` ( Base ` G ) ) -> (SubGrp ` G ) C_ ~P ( Base ` G ) )
55	7, 54	syl 17	. . . . . . . . . . . . . . 15 |- ( ph -> (SubGrp ` G ) C_ ~P ( Base ` G ) )
56	53, 55	sstrd 3613	. . . . . . . . . . . . . 14 |- ( ph -> ran S C_ ~P ( Base ` G ) )
57	51, 56	syl5ss 3614	. . . . . . . . . . . . 13 |- ( ph -> ( S " ( A |` C ) ) C_ ~P ( Base ` G ) )
58		sspwuni 4611	. . . . . . . . . . . . 13 |- ( ( S " ( A |` C ) ) C_ ~P ( Base ` G ) <-> U. ( S " ( A |` C ) ) C_ ( Base ` G ) )
59	57, 58	sylib 208	. . . . . . . . . . . 12 |- ( ph -> U. ( S " ( A |` C ) ) C_ ( Base ` G ) )
60	8	mrccl 16271	. . . . . . . . . . . 12 |- ( ( (SubGrp ` G ) e. (Moore ` ( Base ` G ) ) /\ U. ( S " ( A |` C ) ) C_ ( Base ` G ) ) -> ( K ` U. ( S " ( A |` C ) ) ) e. (SubGrp ` G ) )
61	7, 59, 60	syl2anc 693	. . . . . . . . . . 11 |- ( ph -> ( K ` U. ( S " ( A |` C ) ) ) e. (SubGrp ` G ) )
62	61	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ i e. C ) -> ( K ` U. ( S " ( A |` C ) ) ) e. (SubGrp ` G ) )
63		oveq2 6658	. . . . . . . . . . . . 13 |- ( j = k -> ( i S j ) = ( i S k ) )
64	63, 48, 47	fvmpt3i 6287	. . . . . . . . . . . 12 |- ( k e. ( A " { i } ) -> ( ( j e. ( A " { i } ) |-> ( i S j ) ) ` k ) = ( i S k ) )
65	64	adantl 482	. . . . . . . . . . 11 |- ( ( ( ph /\ i e. C ) /\ k e. ( A " { i } ) ) -> ( ( j e. ( A " { i } ) |-> ( i S j ) ) ` k ) = ( i S k ) )
66		df-ov 6653	. . . . . . . . . . . . . 14 |- ( i S k ) = ( S ` <. i , k >. )
67		ffn 6045	. . . . . . . . . . . . . . . . 17 |- ( S : A --> (SubGrp ` G ) -> S Fn A )
68	9, 67	syl 17	. . . . . . . . . . . . . . . 16 |- ( ph -> S Fn A )
69	68	ad2antrr 762	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ i e. C ) /\ k e. ( A " { i } ) ) -> S Fn A )
70	13	a1i 11	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ i e. C ) /\ k e. ( A " { i } ) ) -> ( A |` C ) C_ A )
71		elrelimasn 5489	. . . . . . . . . . . . . . . . . . . 20 |- ( Rel A -> ( k e. ( A " { i } ) <-> i A k ) )
72	15, 71	syl 17	. . . . . . . . . . . . . . . . . . 19 |- ( ph -> ( k e. ( A " { i } ) <-> i A k ) )
73	72	adantr 481	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ i e. C ) -> ( k e. ( A " { i } ) <-> i A k ) )
74	73	biimpa 501	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ i e. C ) /\ k e. ( A " { i } ) ) -> i A k )
75		df-br 4654	. . . . . . . . . . . . . . . . 17 |- ( i A k <-> <. i , k >. e. A )
76	74, 75	sylib 208	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ i e. C ) /\ k e. ( A " { i } ) ) -> <. i , k >. e. A )
77		simplr 792	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ i e. C ) /\ k e. ( A " { i } ) ) -> i e. C )
78		vex 3203	. . . . . . . . . . . . . . . . 17 |- k e. _V
79	78	opelres 5401	. . . . . . . . . . . . . . . 16 |- ( <. i , k >. e. ( A |` C ) <-> ( <. i , k >. e. A /\ i e. C ) )
80	76, 77, 79	sylanbrc 698	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ i e. C ) /\ k e. ( A " { i } ) ) -> <. i , k >. e. ( A |` C ) )
81		fnfvima 6496	. . . . . . . . . . . . . . 15 |- ( ( S Fn A /\ ( A |` C ) C_ A /\ <. i , k >. e. ( A |` C ) ) -> ( S ` <. i , k >. ) e. ( S " ( A |` C ) ) )
82	69, 70, 80, 81	syl3anc 1326	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ i e. C ) /\ k e. ( A " { i } ) ) -> ( S ` <. i , k >. ) e. ( S " ( A |` C ) ) )
83	66, 82	syl5eqel 2705	. . . . . . . . . . . . 13 |- ( ( ( ph /\ i e. C ) /\ k e. ( A " { i } ) ) -> ( i S k ) e. ( S " ( A |` C ) ) )
84		elssuni 4467	. . . . . . . . . . . . 13 |- ( ( i S k ) e. ( S " ( A |` C ) ) -> ( i S k ) C_ U. ( S " ( A |` C ) ) )
85	83, 84	syl 17	. . . . . . . . . . . 12 |- ( ( ( ph /\ i e. C ) /\ k e. ( A " { i } ) ) -> ( i S k ) C_ U. ( S " ( A |` C ) ) )
86	7, 8, 59	mrcssidd 16285	. . . . . . . . . . . . 13 |- ( ph -> U. ( S " ( A |` C ) ) C_ ( K ` U. ( S " ( A |` C ) ) ) )
87	86	ad2antrr 762	. . . . . . . . . . . 12 |- ( ( ( ph /\ i e. C ) /\ k e. ( A " { i } ) ) -> U. ( S " ( A |` C ) ) C_ ( K ` U. ( S " ( A |` C ) ) ) )
88	85, 87	sstrd 3613	. . . . . . . . . . 11 |- ( ( ( ph /\ i e. C ) /\ k e. ( A " { i } ) ) -> ( i S k ) C_ ( K ` U. ( S " ( A |` C ) ) ) )
89	65, 88	eqsstrd 3639	. . . . . . . . . 10 |- ( ( ( ph /\ i e. C ) /\ k e. ( A " { i } ) ) -> ( ( j e. ( A " { i } ) |-> ( i S j ) ) ` k ) C_ ( K ` U. ( S " ( A |` C ) ) ) )
90	46, 50, 62, 89	dprdlub 18425	. . . . . . . . 9 |- ( ( ph /\ i e. C ) -> ( G DProd ( j e. ( A " { i } ) |-> ( i S j ) ) ) C_ ( K ` U. ( S " ( A |` C ) ) ) )
91		ovex 6678	. . . . . . . . . 10 |- ( G DProd ( j e. ( A " { i } ) |-> ( i S j ) ) ) e. _V
92	91	elpw 4164	. . . . . . . . 9 |- ( ( G DProd ( j e. ( A " { i } ) |-> ( i S j ) ) ) e. ~P ( K ` U. ( S " ( A |` C ) ) ) <-> ( G DProd ( j e. ( A " { i } ) |-> ( i S j ) ) ) C_ ( K ` U. ( S " ( A |` C ) ) ) )
93	90, 92	sylibr 224	. . . . . . . 8 |- ( ( ph /\ i e. C ) -> ( G DProd ( j e. ( A " { i } ) |-> ( i S j ) ) ) e. ~P ( K ` U. ( S " ( A |` C ) ) ) )
94	93, 29	fmptd 6385	. . . . . . 7 |- ( ph -> ( i e. C |-> ( G DProd ( j e. ( A " { i } ) |-> ( i S j ) ) ) ) : C --> ~P ( K ` U. ( S " ( A |` C ) ) ) )
95		frn 6053	. . . . . . 7 |- ( ( i e. C |-> ( G DProd ( j e. ( A " { i } ) |-> ( i S j ) ) ) ) : C --> ~P ( K ` U. ( S " ( A |` C ) ) ) -> ran ( i e. C |-> ( G DProd ( j e. ( A " { i } ) |-> ( i S j ) ) ) ) C_ ~P ( K ` U. ( S " ( A |` C ) ) ) )
96	94, 95	syl 17	. . . . . 6 |- ( ph -> ran ( i e. C |-> ( G DProd ( j e. ( A " { i } ) |-> ( i S j ) ) ) ) C_ ~P ( K ` U. ( S " ( A |` C ) ) ) )
97		sspwuni 4611	. . . . . 6 |- ( ran ( i e. C |-> ( G DProd ( j e. ( A " { i } ) |-> ( i S j ) ) ) ) C_ ~P ( K ` U. ( S " ( A |` C ) ) ) <-> U. ran ( i e. C |-> ( G DProd ( j e. ( A " { i } ) |-> ( i S j ) ) ) ) C_ ( K ` U. ( S " ( A |` C ) ) ) )
98	96, 97	sylib 208	. . . . 5 |- ( ph -> U. ran ( i e. C |-> ( G DProd ( j e. ( A " { i } ) |-> ( i S j ) ) ) ) C_ ( K ` U. ( S " ( A |` C ) ) ) )
99	7, 8	mrcssvd 16283	. . . . 5 |- ( ph -> ( K ` U. ( S " ( A |` C ) ) ) C_ ( Base ` G ) )
100	98, 99	sstrd 3613	. . . 4 |- ( ph -> U. ran ( i e. C |-> ( G DProd ( j e. ( A " { i } ) |-> ( i S j ) ) ) ) C_ ( Base ` G ) )
101	7, 8, 43, 100	mrcssd 16284	. . 3 |- ( ph -> ( K ` U. ( S " ( A |` C ) ) ) C_ ( K ` U. ran ( i e. C |-> ( G DProd ( j e. ( A " { i } ) |-> ( i S j ) ) ) ) ) )
102	8	mrcsscl 16280	. . . 4 |- ( ( (SubGrp ` G ) e. (Moore ` ( Base ` G ) ) /\ U. ran ( i e. C |-> ( G DProd ( j e. ( A " { i } ) |-> ( i S j ) ) ) ) C_ ( K ` U. ( S " ( A |` C ) ) ) /\ ( K ` U. ( S " ( A |` C ) ) ) e. (SubGrp ` G ) ) -> ( K ` U. ran ( i e. C |-> ( G DProd ( j e. ( A " { i } ) |-> ( i S j ) ) ) ) ) C_ ( K ` U. ( S " ( A |` C ) ) ) )
103	7, 98, 61, 102	syl3anc 1326	. . 3 |- ( ph -> ( K ` U. ran ( i e. C |-> ( G DProd ( j e. ( A " { i } ) |-> ( i S j ) ) ) ) ) C_ ( K ` U. ( S " ( A |` C ) ) ) )
104	101, 103	eqssd 3620	. 2 |- ( ph -> ( K ` U. ( S " ( A |` C ) ) ) = ( K ` U. ran ( i e. C |-> ( G DProd ( j e. ( A " { i } ) |-> ( i S j ) ) ) ) ) )
105		eqid 2622	. . . . . . . 8 |- ( i e. I |-> ( G DProd ( j e. ( A " { i } ) |-> ( i S j ) ) ) ) = ( i e. I |-> ( G DProd ( j e. ( A " { i } ) |-> ( i S j ) ) ) )
106	91, 105	dmmpti 6023	. . . . . . 7 |- dom ( i e. I |-> ( G DProd ( j e. ( A " { i } ) |-> ( i S j ) ) ) ) = I
107	106	a1i 11	. . . . . 6 |- ( ph -> dom ( i e. I |-> ( G DProd ( j e. ( A " { i } ) |-> ( i S j ) ) ) ) = I )
108	1, 107, 44	dprdres 18427	. . . . 5 |- ( ph -> ( G dom DProd ( ( i e. I |-> ( G DProd ( j e. ( A " { i } ) |-> ( i S j ) ) ) ) |` C ) /\ ( G DProd ( ( i e. I |-> ( G DProd ( j e. ( A " { i } ) |-> ( i S j ) ) ) ) |` C ) ) C_ ( G DProd ( i e. I |-> ( G DProd ( j e. ( A " { i } ) |-> ( i S j ) ) ) ) ) ) )
109	108	simpld 475	. . . 4 |- ( ph -> G dom DProd ( ( i e. I |-> ( G DProd ( j e. ( A " { i } ) |-> ( i S j ) ) ) ) |` C ) )
110	44	resmptd 5452	. . . 4 |- ( ph -> ( ( i e. I |-> ( G DProd ( j e. ( A " { i } ) |-> ( i S j ) ) ) ) |` C ) = ( i e. C |-> ( G DProd ( j e. ( A " { i } ) |-> ( i S j ) ) ) ) )
111	109, 110	breqtrd 4679	. . 3 |- ( ph -> G dom DProd ( i e. C |-> ( G DProd ( j e. ( A " { i } ) |-> ( i S j ) ) ) ) )
112	8	dprdspan 18426	. . 3 |- ( G dom DProd ( i e. C |-> ( G DProd ( j e. ( A " { i } ) |-> ( i S j ) ) ) ) -> ( G DProd ( i e. C |-> ( G DProd ( j e. ( A " { i } ) |-> ( i S j ) ) ) ) ) = ( K ` U. ran ( i e. C |-> ( G DProd ( j e. ( A " { i } ) |-> ( i S j ) ) ) ) ) )
113	111, 112	syl 17	. 2 |- ( ph -> ( G DProd ( i e. C |-> ( G DProd ( j e. ( A " { i } ) |-> ( i S j ) ) ) ) ) = ( K ` U. ran ( i e. C |-> ( G DProd ( j e. ( A " { i } ) |-> ( i S j ) ) ) ) ) )
114	104, 113	eqtr4d 2659	1 |- ( ph -> ( K ` U. ( S " ( A |` C ) ) ) = ( G DProd ( i e. C |-> ( G DProd ( j e. ( A " { i } ) |-> ( i S j ) ) ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   C_ wss 3574   ~Pcpw 4158   {csn 4177   <.cop 4183   U.cuni 4436   U_ciun 4520   class class class wbr 4653   |-> cmpt 4729   dom cdm 5114   ran crn 5115   |` cres 5116   "cima 5117   Rel wrel 5119   Fun wfun 5882   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   1stc1st 7166   2ndc2nd 7167   Basecbs 15857  Moorecmre 16242  mrClscmrc 16243  ACScacs 16245   Grpcgrp 17422  SubGrpcsubg 17588   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:  dprd2da  18441  dprd2db  18442