Theorem subgdmdprd 18433

Description: A direct product in a subgroup. (Contributed by Mario Carneiro, 27-Apr-2016.)

Hypothesis

Ref	Expression
subgdprd.1	|- H = ( G ↾s A )

Assertion

Ref	Expression
subgdmdprd	|- ( A e. (SubGrp ` G ) -> ( H dom DProd S <-> ( G dom DProd S /\ ran S C_ ~P A ) ) )

Proof of Theorem subgdmdprd

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

Step	Hyp	Ref	Expression
1		reldmdprd 18396	. . . 4 |- Rel dom DProd
2	1	brrelex2i 5159	. . 3 |- ( H dom DProd S -> S e. _V )
3	2	a1i 11	. 2 |- ( A e. (SubGrp ` G ) -> ( H dom DProd S -> S e. _V ) )
4	1	brrelex2i 5159	. . . 4 |- ( G dom DProd S -> S e. _V )
5	4	adantr 481	. . 3 |- ( ( G dom DProd S /\ ran S C_ ~P A ) -> S e. _V )
6	5	a1i 11	. 2 |- ( A e. (SubGrp ` G ) -> ( ( G dom DProd S /\ ran S C_ ~P A ) -> S e. _V ) )
7		ffvelrn 6357	. . . . . . . . . . . . . . . 16 |- ( ( S : dom S --> (SubGrp ` H ) /\ x e. dom S ) -> ( S ` x ) e. (SubGrp ` H ) )
8	7	ad2ant2lr 784	. . . . . . . . . . . . . . 15 |- ( ( ( A e. (SubGrp ` G ) /\ S : dom S --> (SubGrp ` H ) ) /\ ( x e. dom S /\ y e. ( dom S \ { x } ) ) ) -> ( S ` x ) e. (SubGrp ` H ) )
9		eqid 2622	. . . . . . . . . . . . . . . 16 |- ( Base ` H ) = ( Base ` H )
10	9	subgss 17595	. . . . . . . . . . . . . . 15 |- ( ( S ` x ) e. (SubGrp ` H ) -> ( S ` x ) C_ ( Base ` H ) )
11	8, 10	syl 17	. . . . . . . . . . . . . 14 |- ( ( ( A e. (SubGrp ` G ) /\ S : dom S --> (SubGrp ` H ) ) /\ ( x e. dom S /\ y e. ( dom S \ { x } ) ) ) -> ( S ` x ) C_ ( Base ` H ) )
12		subgdprd.1	. . . . . . . . . . . . . . . 16 |- H = ( G ↾s A )
13	12	subgbas 17598	. . . . . . . . . . . . . . 15 |- ( A e. (SubGrp ` G ) -> A = ( Base ` H ) )
14	13	ad2antrr 762	. . . . . . . . . . . . . 14 |- ( ( ( A e. (SubGrp ` G ) /\ S : dom S --> (SubGrp ` H ) ) /\ ( x e. dom S /\ y e. ( dom S \ { x } ) ) ) -> A = ( Base ` H ) )
15	11, 14	sseqtr4d 3642	. . . . . . . . . . . . 13 |- ( ( ( A e. (SubGrp ` G ) /\ S : dom S --> (SubGrp ` H ) ) /\ ( x e. dom S /\ y e. ( dom S \ { x } ) ) ) -> ( S ` x ) C_ A )
16	15	biantrud 528	. . . . . . . . . . . 12 |- ( ( ( A e. (SubGrp ` G ) /\ S : dom S --> (SubGrp ` H ) ) /\ ( x e. dom S /\ y e. ( dom S \ { x } ) ) ) -> ( ( S ` x ) C_ ( (Cntz ` G ) ` ( S ` y ) ) <-> ( ( S ` x ) C_ ( (Cntz ` G ) ` ( S ` y ) ) /\ ( S ` x ) C_ A ) ) )
17		simpll 790	. . . . . . . . . . . . . . 15 |- ( ( ( A e. (SubGrp ` G ) /\ S : dom S --> (SubGrp ` H ) ) /\ ( x e. dom S /\ y e. ( dom S \ { x } ) ) ) -> A e. (SubGrp ` G ) )
18		simplr 792	. . . . . . . . . . . . . . . . . 18 |- ( ( ( A e. (SubGrp ` G ) /\ S : dom S --> (SubGrp ` H ) ) /\ ( x e. dom S /\ y e. ( dom S \ { x } ) ) ) -> S : dom S --> (SubGrp ` H ) )
19		eldifi 3732	. . . . . . . . . . . . . . . . . . 19 |- ( y e. ( dom S \ { x } ) -> y e. dom S )
20	19	ad2antll 765	. . . . . . . . . . . . . . . . . 18 |- ( ( ( A e. (SubGrp ` G ) /\ S : dom S --> (SubGrp ` H ) ) /\ ( x e. dom S /\ y e. ( dom S \ { x } ) ) ) -> y e. dom S )
21	18, 20	ffvelrnd 6360	. . . . . . . . . . . . . . . . 17 |- ( ( ( A e. (SubGrp ` G ) /\ S : dom S --> (SubGrp ` H ) ) /\ ( x e. dom S /\ y e. ( dom S \ { x } ) ) ) -> ( S ` y ) e. (SubGrp ` H ) )
22	9	subgss 17595	. . . . . . . . . . . . . . . . 17 |- ( ( S ` y ) e. (SubGrp ` H ) -> ( S ` y ) C_ ( Base ` H ) )
23	21, 22	syl 17	. . . . . . . . . . . . . . . 16 |- ( ( ( A e. (SubGrp ` G ) /\ S : dom S --> (SubGrp ` H ) ) /\ ( x e. dom S /\ y e. ( dom S \ { x } ) ) ) -> ( S ` y ) C_ ( Base ` H ) )
24	23, 14	sseqtr4d 3642	. . . . . . . . . . . . . . 15 |- ( ( ( A e. (SubGrp ` G ) /\ S : dom S --> (SubGrp ` H ) ) /\ ( x e. dom S /\ y e. ( dom S \ { x } ) ) ) -> ( S ` y ) C_ A )
25		eqid 2622	. . . . . . . . . . . . . . . 16 |- (Cntz ` G ) = (Cntz ` G )
26		eqid 2622	. . . . . . . . . . . . . . . 16 |- (Cntz ` H ) = (Cntz ` H )
27	12, 25, 26	resscntz 17764	. . . . . . . . . . . . . . 15 |- ( ( A e. (SubGrp ` G ) /\ ( S ` y ) C_ A ) -> ( (Cntz ` H ) ` ( S ` y ) ) = ( ( (Cntz ` G ) ` ( S ` y ) ) i^i A ) )
28	17, 24, 27	syl2anc 693	. . . . . . . . . . . . . 14 |- ( ( ( A e. (SubGrp ` G ) /\ S : dom S --> (SubGrp ` H ) ) /\ ( x e. dom S /\ y e. ( dom S \ { x } ) ) ) -> ( (Cntz ` H ) ` ( S ` y ) ) = ( ( (Cntz ` G ) ` ( S ` y ) ) i^i A ) )
29	28	sseq2d 3633	. . . . . . . . . . . . 13 |- ( ( ( A e. (SubGrp ` G ) /\ S : dom S --> (SubGrp ` H ) ) /\ ( x e. dom S /\ y e. ( dom S \ { x } ) ) ) -> ( ( S ` x ) C_ ( (Cntz ` H ) ` ( S ` y ) ) <-> ( S ` x ) C_ ( ( (Cntz ` G ) ` ( S ` y ) ) i^i A ) ) )
30		ssin 3835	. . . . . . . . . . . . 13 |- ( ( ( S ` x ) C_ ( (Cntz ` G ) ` ( S ` y ) ) /\ ( S ` x ) C_ A ) <-> ( S ` x ) C_ ( ( (Cntz ` G ) ` ( S ` y ) ) i^i A ) )
31	29, 30	syl6bbr 278	. . . . . . . . . . . 12 |- ( ( ( A e. (SubGrp ` G ) /\ S : dom S --> (SubGrp ` H ) ) /\ ( x e. dom S /\ y e. ( dom S \ { x } ) ) ) -> ( ( S ` x ) C_ ( (Cntz ` H ) ` ( S ` y ) ) <-> ( ( S ` x ) C_ ( (Cntz ` G ) ` ( S ` y ) ) /\ ( S ` x ) C_ A ) ) )
32	16, 31	bitr4d 271	. . . . . . . . . . 11 |- ( ( ( A e. (SubGrp ` G ) /\ S : dom S --> (SubGrp ` H ) ) /\ ( x e. dom S /\ y e. ( dom S \ { x } ) ) ) -> ( ( S ` x ) C_ ( (Cntz ` G ) ` ( S ` y ) ) <-> ( S ` x ) C_ ( (Cntz ` H ) ` ( S ` y ) ) ) )
33	32	anassrs 680	. . . . . . . . . 10 |- ( ( ( ( A e. (SubGrp ` G ) /\ S : dom S --> (SubGrp ` H ) ) /\ x e. dom S ) /\ y e. ( dom S \ { x } ) ) -> ( ( S ` x ) C_ ( (Cntz ` G ) ` ( S ` y ) ) <-> ( S ` x ) C_ ( (Cntz ` H ) ` ( S ` y ) ) ) )
34	33	ralbidva 2985	. . . . . . . . 9 |- ( ( ( A e. (SubGrp ` G ) /\ S : dom S --> (SubGrp ` H ) ) /\ x e. dom S ) -> ( A. y e. ( dom S \ { x } ) ( S ` x ) C_ ( (Cntz ` G ) ` ( S ` y ) ) <-> A. y e. ( dom S \ { x } ) ( S ` x ) C_ ( (Cntz ` H ) ` ( S ` y ) ) ) )
35		subgrcl 17599	. . . . . . . . . . . . . . 15 |- ( A e. (SubGrp ` G ) -> G e. Grp )
36	35	ad2antrr 762	. . . . . . . . . . . . . 14 |- ( ( ( A e. (SubGrp ` G ) /\ S : dom S --> (SubGrp ` H ) ) /\ x e. dom S ) -> G e. Grp )
37		eqid 2622	. . . . . . . . . . . . . . 15 |- ( Base ` G ) = ( Base ` G )
38	37	subgacs 17629	. . . . . . . . . . . . . 14 |- ( G e. Grp -> (SubGrp ` G ) e. (ACS ` ( Base ` G ) ) )
39		acsmre 16313	. . . . . . . . . . . . . 14 |- ( (SubGrp ` G ) e. (ACS ` ( Base ` G ) ) -> (SubGrp ` G ) e. (Moore ` ( Base ` G ) ) )
40	36, 38, 39	3syl 18	. . . . . . . . . . . . 13 |- ( ( ( A e. (SubGrp ` G ) /\ S : dom S --> (SubGrp ` H ) ) /\ x e. dom S ) -> (SubGrp ` G ) e. (Moore ` ( Base ` G ) ) )
41	12	subggrp 17597	. . . . . . . . . . . . . . . 16 |- ( A e. (SubGrp ` G ) -> H e. Grp )
42	41	ad2antrr 762	. . . . . . . . . . . . . . 15 |- ( ( ( A e. (SubGrp ` G ) /\ S : dom S --> (SubGrp ` H ) ) /\ x e. dom S ) -> H e. Grp )
43	9	subgacs 17629	. . . . . . . . . . . . . . 15 |- ( H e. Grp -> (SubGrp ` H ) e. (ACS ` ( Base ` H ) ) )
44		acsmre 16313	. . . . . . . . . . . . . . 15 |- ( (SubGrp ` H ) e. (ACS ` ( Base ` H ) ) -> (SubGrp ` H ) e. (Moore ` ( Base ` H ) ) )
45	42, 43, 44	3syl 18	. . . . . . . . . . . . . 14 |- ( ( ( A e. (SubGrp ` G ) /\ S : dom S --> (SubGrp ` H ) ) /\ x e. dom S ) -> (SubGrp ` H ) e. (Moore ` ( Base ` H ) ) )
46		eqid 2622	. . . . . . . . . . . . . 14 |- (mrCls ` (SubGrp ` H ) ) = (mrCls ` (SubGrp ` H ) )
47		imassrn 5477	. . . . . . . . . . . . . . . . 17 |- ( S " ( dom S \ { x } ) ) C_ ran S
48		frn 6053	. . . . . . . . . . . . . . . . . 18 |- ( S : dom S --> (SubGrp ` H ) -> ran S C_ (SubGrp ` H ) )
49	48	ad2antlr 763	. . . . . . . . . . . . . . . . 17 |- ( ( ( A e. (SubGrp ` G ) /\ S : dom S --> (SubGrp ` H ) ) /\ x e. dom S ) -> ran S C_ (SubGrp ` H ) )
50	47, 49	syl5ss 3614	. . . . . . . . . . . . . . . 16 |- ( ( ( A e. (SubGrp ` G ) /\ S : dom S --> (SubGrp ` H ) ) /\ x e. dom S ) -> ( S " ( dom S \ { x } ) ) C_ (SubGrp ` H ) )
51		mresspw 16252	. . . . . . . . . . . . . . . . 17 |- ( (SubGrp ` H ) e. (Moore ` ( Base ` H ) ) -> (SubGrp ` H ) C_ ~P ( Base ` H ) )
52	45, 51	syl 17	. . . . . . . . . . . . . . . 16 |- ( ( ( A e. (SubGrp ` G ) /\ S : dom S --> (SubGrp ` H ) ) /\ x e. dom S ) -> (SubGrp ` H ) C_ ~P ( Base ` H ) )
53	50, 52	sstrd 3613	. . . . . . . . . . . . . . 15 |- ( ( ( A e. (SubGrp ` G ) /\ S : dom S --> (SubGrp ` H ) ) /\ x e. dom S ) -> ( S " ( dom S \ { x } ) ) C_ ~P ( Base ` H ) )
54		sspwuni 4611	. . . . . . . . . . . . . . 15 |- ( ( S " ( dom S \ { x } ) ) C_ ~P ( Base ` H ) <-> U. ( S " ( dom S \ { x } ) ) C_ ( Base ` H ) )
55	53, 54	sylib 208	. . . . . . . . . . . . . 14 |- ( ( ( A e. (SubGrp ` G ) /\ S : dom S --> (SubGrp ` H ) ) /\ x e. dom S ) -> U. ( S " ( dom S \ { x } ) ) C_ ( Base ` H ) )
56	45, 46, 55	mrcssidd 16285	. . . . . . . . . . . . 13 |- ( ( ( A e. (SubGrp ` G ) /\ S : dom S --> (SubGrp ` H ) ) /\ x e. dom S ) -> U. ( S " ( dom S \ { x } ) ) C_ ( (mrCls ` (SubGrp ` H ) ) ` U. ( S " ( dom S \ { x } ) ) ) )
57	46	mrccl 16271	. . . . . . . . . . . . . . . 16 |- ( ( (SubGrp ` H ) e. (Moore ` ( Base ` H ) ) /\ U. ( S " ( dom S \ { x } ) ) C_ ( Base ` H ) ) -> ( (mrCls ` (SubGrp ` H ) ) ` U. ( S " ( dom S \ { x } ) ) ) e. (SubGrp ` H ) )
58	45, 55, 57	syl2anc 693	. . . . . . . . . . . . . . 15 |- ( ( ( A e. (SubGrp ` G ) /\ S : dom S --> (SubGrp ` H ) ) /\ x e. dom S ) -> ( (mrCls ` (SubGrp ` H ) ) ` U. ( S " ( dom S \ { x } ) ) ) e. (SubGrp ` H ) )
59	12	subsubg 17617	. . . . . . . . . . . . . . . 16 |- ( A e. (SubGrp ` G ) -> ( ( (mrCls ` (SubGrp ` H ) ) ` U. ( S " ( dom S \ { x } ) ) ) e. (SubGrp ` H ) <-> ( ( (mrCls ` (SubGrp ` H ) ) ` U. ( S " ( dom S \ { x } ) ) ) e. (SubGrp ` G ) /\ ( (mrCls ` (SubGrp ` H ) ) ` U. ( S " ( dom S \ { x } ) ) ) C_ A ) ) )
60	59	ad2antrr 762	. . . . . . . . . . . . . . 15 |- ( ( ( A e. (SubGrp ` G ) /\ S : dom S --> (SubGrp ` H ) ) /\ x e. dom S ) -> ( ( (mrCls ` (SubGrp ` H ) ) ` U. ( S " ( dom S \ { x } ) ) ) e. (SubGrp ` H ) <-> ( ( (mrCls ` (SubGrp ` H ) ) ` U. ( S " ( dom S \ { x } ) ) ) e. (SubGrp ` G ) /\ ( (mrCls ` (SubGrp ` H ) ) ` U. ( S " ( dom S \ { x } ) ) ) C_ A ) ) )
61	58, 60	mpbid 222	. . . . . . . . . . . . . 14 |- ( ( ( A e. (SubGrp ` G ) /\ S : dom S --> (SubGrp ` H ) ) /\ x e. dom S ) -> ( ( (mrCls ` (SubGrp ` H ) ) ` U. ( S " ( dom S \ { x } ) ) ) e. (SubGrp ` G ) /\ ( (mrCls ` (SubGrp ` H ) ) ` U. ( S " ( dom S \ { x } ) ) ) C_ A ) )
62	61	simpld 475	. . . . . . . . . . . . 13 |- ( ( ( A e. (SubGrp ` G ) /\ S : dom S --> (SubGrp ` H ) ) /\ x e. dom S ) -> ( (mrCls ` (SubGrp ` H ) ) ` U. ( S " ( dom S \ { x } ) ) ) e. (SubGrp ` G ) )
63		eqid 2622	. . . . . . . . . . . . . 14 |- (mrCls ` (SubGrp ` G ) ) = (mrCls ` (SubGrp ` G ) )
64	63	mrcsscl 16280	. . . . . . . . . . . . 13 |- ( ( (SubGrp ` G ) e. (Moore ` ( Base ` G ) ) /\ U. ( S " ( dom S \ { x } ) ) C_ ( (mrCls ` (SubGrp ` H ) ) ` U. ( S " ( dom S \ { x } ) ) ) /\ ( (mrCls ` (SubGrp ` H ) ) ` U. ( S " ( dom S \ { x } ) ) ) e. (SubGrp ` G ) ) -> ( (mrCls ` (SubGrp ` G ) ) ` U. ( S " ( dom S \ { x } ) ) ) C_ ( (mrCls ` (SubGrp ` H ) ) ` U. ( S " ( dom S \ { x } ) ) ) )
65	40, 56, 62, 64	syl3anc 1326	. . . . . . . . . . . 12 |- ( ( ( A e. (SubGrp ` G ) /\ S : dom S --> (SubGrp ` H ) ) /\ x e. dom S ) -> ( (mrCls ` (SubGrp ` G ) ) ` U. ( S " ( dom S \ { x } ) ) ) C_ ( (mrCls ` (SubGrp ` H ) ) ` U. ( S " ( dom S \ { x } ) ) ) )
66	13	ad2antrr 762	. . . . . . . . . . . . . . . 16 |- ( ( ( A e. (SubGrp ` G ) /\ S : dom S --> (SubGrp ` H ) ) /\ x e. dom S ) -> A = ( Base ` H ) )
67	55, 66	sseqtr4d 3642	. . . . . . . . . . . . . . 15 |- ( ( ( A e. (SubGrp ` G ) /\ S : dom S --> (SubGrp ` H ) ) /\ x e. dom S ) -> U. ( S " ( dom S \ { x } ) ) C_ A )
68	37	subgss 17595	. . . . . . . . . . . . . . . 16 |- ( A e. (SubGrp ` G ) -> A C_ ( Base ` G ) )
69	68	ad2antrr 762	. . . . . . . . . . . . . . 15 |- ( ( ( A e. (SubGrp ` G ) /\ S : dom S --> (SubGrp ` H ) ) /\ x e. dom S ) -> A C_ ( Base ` G ) )
70	67, 69	sstrd 3613	. . . . . . . . . . . . . 14 |- ( ( ( A e. (SubGrp ` G ) /\ S : dom S --> (SubGrp ` H ) ) /\ x e. dom S ) -> U. ( S " ( dom S \ { x } ) ) C_ ( Base ` G ) )
71	40, 63, 70	mrcssidd 16285	. . . . . . . . . . . . 13 |- ( ( ( A e. (SubGrp ` G ) /\ S : dom S --> (SubGrp ` H ) ) /\ x e. dom S ) -> U. ( S " ( dom S \ { x } ) ) C_ ( (mrCls ` (SubGrp ` G ) ) ` U. ( S " ( dom S \ { x } ) ) ) )
72	63	mrccl 16271	. . . . . . . . . . . . . . 15 |- ( ( (SubGrp ` G ) e. (Moore ` ( Base ` G ) ) /\ U. ( S " ( dom S \ { x } ) ) C_ ( Base ` G ) ) -> ( (mrCls ` (SubGrp ` G ) ) ` U. ( S " ( dom S \ { x } ) ) ) e. (SubGrp ` G ) )
73	40, 70, 72	syl2anc 693	. . . . . . . . . . . . . 14 |- ( ( ( A e. (SubGrp ` G ) /\ S : dom S --> (SubGrp ` H ) ) /\ x e. dom S ) -> ( (mrCls ` (SubGrp ` G ) ) ` U. ( S " ( dom S \ { x } ) ) ) e. (SubGrp ` G ) )
74		simpll 790	. . . . . . . . . . . . . . 15 |- ( ( ( A e. (SubGrp ` G ) /\ S : dom S --> (SubGrp ` H ) ) /\ x e. dom S ) -> A e. (SubGrp ` G ) )
75	63	mrcsscl 16280	. . . . . . . . . . . . . . 15 |- ( ( (SubGrp ` G ) e. (Moore ` ( Base ` G ) ) /\ U. ( S " ( dom S \ { x } ) ) C_ A /\ A e. (SubGrp ` G ) ) -> ( (mrCls ` (SubGrp ` G ) ) ` U. ( S " ( dom S \ { x } ) ) ) C_ A )
76	40, 67, 74, 75	syl3anc 1326	. . . . . . . . . . . . . 14 |- ( ( ( A e. (SubGrp ` G ) /\ S : dom S --> (SubGrp ` H ) ) /\ x e. dom S ) -> ( (mrCls ` (SubGrp ` G ) ) ` U. ( S " ( dom S \ { x } ) ) ) C_ A )
77	12	subsubg 17617	. . . . . . . . . . . . . . 15 |- ( A e. (SubGrp ` G ) -> ( ( (mrCls ` (SubGrp ` G ) ) ` U. ( S " ( dom S \ { x } ) ) ) e. (SubGrp ` H ) <-> ( ( (mrCls ` (SubGrp ` G ) ) ` U. ( S " ( dom S \ { x } ) ) ) e. (SubGrp ` G ) /\ ( (mrCls ` (SubGrp ` G ) ) ` U. ( S " ( dom S \ { x } ) ) ) C_ A ) ) )
78	77	ad2antrr 762	. . . . . . . . . . . . . 14 |- ( ( ( A e. (SubGrp ` G ) /\ S : dom S --> (SubGrp ` H ) ) /\ x e. dom S ) -> ( ( (mrCls ` (SubGrp ` G ) ) ` U. ( S " ( dom S \ { x } ) ) ) e. (SubGrp ` H ) <-> ( ( (mrCls ` (SubGrp ` G ) ) ` U. ( S " ( dom S \ { x } ) ) ) e. (SubGrp ` G ) /\ ( (mrCls ` (SubGrp ` G ) ) ` U. ( S " ( dom S \ { x } ) ) ) C_ A ) ) )
79	73, 76, 78	mpbir2and 957	. . . . . . . . . . . . 13 |- ( ( ( A e. (SubGrp ` G ) /\ S : dom S --> (SubGrp ` H ) ) /\ x e. dom S ) -> ( (mrCls ` (SubGrp ` G ) ) ` U. ( S " ( dom S \ { x } ) ) ) e. (SubGrp ` H ) )
80	46	mrcsscl 16280	. . . . . . . . . . . . 13 |- ( ( (SubGrp ` H ) e. (Moore ` ( Base ` H ) ) /\ U. ( S " ( dom S \ { x } ) ) C_ ( (mrCls ` (SubGrp ` G ) ) ` U. ( S " ( dom S \ { x } ) ) ) /\ ( (mrCls ` (SubGrp ` G ) ) ` U. ( S " ( dom S \ { x } ) ) ) e. (SubGrp ` H ) ) -> ( (mrCls ` (SubGrp ` H ) ) ` U. ( S " ( dom S \ { x } ) ) ) C_ ( (mrCls ` (SubGrp ` G ) ) ` U. ( S " ( dom S \ { x } ) ) ) )
81	45, 71, 79, 80	syl3anc 1326	. . . . . . . . . . . 12 |- ( ( ( A e. (SubGrp ` G ) /\ S : dom S --> (SubGrp ` H ) ) /\ x e. dom S ) -> ( (mrCls ` (SubGrp ` H ) ) ` U. ( S " ( dom S \ { x } ) ) ) C_ ( (mrCls ` (SubGrp ` G ) ) ` U. ( S " ( dom S \ { x } ) ) ) )
82	65, 81	eqssd 3620	. . . . . . . . . . 11 |- ( ( ( A e. (SubGrp ` G ) /\ S : dom S --> (SubGrp ` H ) ) /\ x e. dom S ) -> ( (mrCls ` (SubGrp ` G ) ) ` U. ( S " ( dom S \ { x } ) ) ) = ( (mrCls ` (SubGrp ` H ) ) ` U. ( S " ( dom S \ { x } ) ) ) )
83	82	ineq2d 3814	. . . . . . . . . 10 |- ( ( ( A e. (SubGrp ` G ) /\ S : dom S --> (SubGrp ` H ) ) /\ x e. dom S ) -> ( ( S ` x ) i^i ( (mrCls ` (SubGrp ` G ) ) ` U. ( S " ( dom S \ { x } ) ) ) ) = ( ( S ` x ) i^i ( (mrCls ` (SubGrp ` H ) ) ` U. ( S " ( dom S \ { x } ) ) ) ) )
84		eqid 2622	. . . . . . . . . . . . 13 |- ( 0g ` G ) = ( 0g ` G )
85	12, 84	subg0 17600	. . . . . . . . . . . 12 |- ( A e. (SubGrp ` G ) -> ( 0g ` G ) = ( 0g ` H ) )
86	85	ad2antrr 762	. . . . . . . . . . 11 |- ( ( ( A e. (SubGrp ` G ) /\ S : dom S --> (SubGrp ` H ) ) /\ x e. dom S ) -> ( 0g ` G ) = ( 0g ` H ) )
87	86	sneqd 4189	. . . . . . . . . 10 |- ( ( ( A e. (SubGrp ` G ) /\ S : dom S --> (SubGrp ` H ) ) /\ x e. dom S ) -> { ( 0g ` G ) } = { ( 0g ` H ) } )
88	83, 87	eqeq12d 2637	. . . . . . . . 9 |- ( ( ( A e. (SubGrp ` G ) /\ S : dom S --> (SubGrp ` H ) ) /\ x e. dom S ) -> ( ( ( S ` x ) i^i ( (mrCls ` (SubGrp ` G ) ) ` U. ( S " ( dom S \ { x } ) ) ) ) = { ( 0g ` G ) } <-> ( ( S ` x ) i^i ( (mrCls ` (SubGrp ` H ) ) ` U. ( S " ( dom S \ { x } ) ) ) ) = { ( 0g ` H ) } ) )
89	34, 88	anbi12d 747	. . . . . . . 8 |- ( ( ( A e. (SubGrp ` G ) /\ S : dom S --> (SubGrp ` H ) ) /\ x e. dom S ) -> ( ( A. y e. ( dom S \ { x } ) ( S ` x ) C_ ( (Cntz ` G ) ` ( S ` y ) ) /\ ( ( S ` x ) i^i ( (mrCls ` (SubGrp ` G ) ) ` U. ( S " ( dom S \ { x } ) ) ) ) = { ( 0g ` G ) } ) <-> ( A. y e. ( dom S \ { x } ) ( S ` x ) C_ ( (Cntz ` H ) ` ( S ` y ) ) /\ ( ( S ` x ) i^i ( (mrCls ` (SubGrp ` H ) ) ` U. ( S " ( dom S \ { x } ) ) ) ) = { ( 0g ` H ) } ) ) )
90	89	ralbidva 2985	. . . . . . 7 |- ( ( A e. (SubGrp ` G ) /\ S : dom S --> (SubGrp ` H ) ) -> ( A. x e. dom S ( A. y e. ( dom S \ { x } ) ( S ` x ) C_ ( (Cntz ` G ) ` ( S ` y ) ) /\ ( ( S ` x ) i^i ( (mrCls ` (SubGrp ` G ) ) ` U. ( S " ( dom S \ { x } ) ) ) ) = { ( 0g ` G ) } ) <-> A. x e. dom S ( A. y e. ( dom S \ { x } ) ( S ` x ) C_ ( (Cntz ` H ) ` ( S ` y ) ) /\ ( ( S ` x ) i^i ( (mrCls ` (SubGrp ` H ) ) ` U. ( S " ( dom S \ { x } ) ) ) ) = { ( 0g ` H ) } ) ) )
91	90	pm5.32da 673	. . . . . 6 |- ( A e. (SubGrp ` G ) -> ( ( S : dom S --> (SubGrp ` H ) /\ A. x e. dom S ( A. y e. ( dom S \ { x } ) ( S ` x ) C_ ( (Cntz ` G ) ` ( S ` y ) ) /\ ( ( S ` x ) i^i ( (mrCls ` (SubGrp ` G ) ) ` U. ( S " ( dom S \ { x } ) ) ) ) = { ( 0g ` G ) } ) ) <-> ( S : dom S --> (SubGrp ` H ) /\ A. x e. dom S ( A. y e. ( dom S \ { x } ) ( S ` x ) C_ ( (Cntz ` H ) ` ( S ` y ) ) /\ ( ( S ` x ) i^i ( (mrCls ` (SubGrp ` H ) ) ` U. ( S " ( dom S \ { x } ) ) ) ) = { ( 0g ` H ) } ) ) ) )
92	12	subsubg 17617	. . . . . . . . . . . . 13 |- ( A e. (SubGrp ` G ) -> ( x e. (SubGrp ` H ) <-> ( x e. (SubGrp ` G ) /\ x C_ A ) ) )
93		elin 3796	. . . . . . . . . . . . . 14 |- ( x e. ( (SubGrp ` G ) i^i ~P A ) <-> ( x e. (SubGrp ` G ) /\ x e. ~P A ) )
94		selpw 4165	. . . . . . . . . . . . . . 15 |- ( x e. ~P A <-> x C_ A )
95	94	anbi2i 730	. . . . . . . . . . . . . 14 |- ( ( x e. (SubGrp ` G ) /\ x e. ~P A ) <-> ( x e. (SubGrp ` G ) /\ x C_ A ) )
96	93, 95	bitri 264	. . . . . . . . . . . . 13 |- ( x e. ( (SubGrp ` G ) i^i ~P A ) <-> ( x e. (SubGrp ` G ) /\ x C_ A ) )
97	92, 96	syl6bbr 278	. . . . . . . . . . . 12 |- ( A e. (SubGrp ` G ) -> ( x e. (SubGrp ` H ) <-> x e. ( (SubGrp ` G ) i^i ~P A ) ) )
98	97	eqrdv 2620	. . . . . . . . . . 11 |- ( A e. (SubGrp ` G ) -> (SubGrp ` H ) = ( (SubGrp ` G ) i^i ~P A ) )
99	98	sseq2d 3633	. . . . . . . . . 10 |- ( A e. (SubGrp ` G ) -> ( ran S C_ (SubGrp ` H ) <-> ran S C_ ( (SubGrp ` G ) i^i ~P A ) ) )
100		ssin 3835	. . . . . . . . . 10 |- ( ( ran S C_ (SubGrp ` G ) /\ ran S C_ ~P A ) <-> ran S C_ ( (SubGrp ` G ) i^i ~P A ) )
101	99, 100	syl6bbr 278	. . . . . . . . 9 |- ( A e. (SubGrp ` G ) -> ( ran S C_ (SubGrp ` H ) <-> ( ran S C_ (SubGrp ` G ) /\ ran S C_ ~P A ) ) )
102	101	anbi2d 740	. . . . . . . 8 |- ( A e. (SubGrp ` G ) -> ( ( S Fn dom S /\ ran S C_ (SubGrp ` H ) ) <-> ( S Fn dom S /\ ( ran S C_ (SubGrp ` G ) /\ ran S C_ ~P A ) ) ) )
103		df-f 5892	. . . . . . . 8 |- ( S : dom S --> (SubGrp ` H ) <-> ( S Fn dom S /\ ran S C_ (SubGrp ` H ) ) )
104		df-f 5892	. . . . . . . . . 10 |- ( S : dom S --> (SubGrp ` G ) <-> ( S Fn dom S /\ ran S C_ (SubGrp ` G ) ) )
105	104	anbi1i 731	. . . . . . . . 9 |- ( ( S : dom S --> (SubGrp ` G ) /\ ran S C_ ~P A ) <-> ( ( S Fn dom S /\ ran S C_ (SubGrp ` G ) ) /\ ran S C_ ~P A ) )
106		anass 681	. . . . . . . . 9 |- ( ( ( S Fn dom S /\ ran S C_ (SubGrp ` G ) ) /\ ran S C_ ~P A ) <-> ( S Fn dom S /\ ( ran S C_ (SubGrp ` G ) /\ ran S C_ ~P A ) ) )
107	105, 106	bitri 264	. . . . . . . 8 |- ( ( S : dom S --> (SubGrp ` G ) /\ ran S C_ ~P A ) <-> ( S Fn dom S /\ ( ran S C_ (SubGrp ` G ) /\ ran S C_ ~P A ) ) )
108	102, 103, 107	3bitr4g 303	. . . . . . 7 |- ( A e. (SubGrp ` G ) -> ( S : dom S --> (SubGrp ` H ) <-> ( S : dom S --> (SubGrp ` G ) /\ ran S C_ ~P A ) ) )
109	108	anbi1d 741	. . . . . 6 |- ( A e. (SubGrp ` G ) -> ( ( S : dom S --> (SubGrp ` H ) /\ A. x e. dom S ( A. y e. ( dom S \ { x } ) ( S ` x ) C_ ( (Cntz ` G ) ` ( S ` y ) ) /\ ( ( S ` x ) i^i ( (mrCls ` (SubGrp ` G ) ) ` U. ( S " ( dom S \ { x } ) ) ) ) = { ( 0g ` G ) } ) ) <-> ( ( S : dom S --> (SubGrp ` G ) /\ ran S C_ ~P A ) /\ A. x e. dom S ( A. y e. ( dom S \ { x } ) ( S ` x ) C_ ( (Cntz ` G ) ` ( S ` y ) ) /\ ( ( S ` x ) i^i ( (mrCls ` (SubGrp ` G ) ) ` U. ( S " ( dom S \ { x } ) ) ) ) = { ( 0g ` G ) } ) ) ) )
110	91, 109	bitr3d 270	. . . . 5 |- ( A e. (SubGrp ` G ) -> ( ( S : dom S --> (SubGrp ` H ) /\ A. x e. dom S ( A. y e. ( dom S \ { x } ) ( S ` x ) C_ ( (Cntz ` H ) ` ( S ` y ) ) /\ ( ( S ` x ) i^i ( (mrCls ` (SubGrp ` H ) ) ` U. ( S " ( dom S \ { x } ) ) ) ) = { ( 0g ` H ) } ) ) <-> ( ( S : dom S --> (SubGrp ` G ) /\ ran S C_ ~P A ) /\ A. x e. dom S ( A. y e. ( dom S \ { x } ) ( S ` x ) C_ ( (Cntz ` G ) ` ( S ` y ) ) /\ ( ( S ` x ) i^i ( (mrCls ` (SubGrp ` G ) ) ` U. ( S " ( dom S \ { x } ) ) ) ) = { ( 0g ` G ) } ) ) ) )
111	110	adantr 481	. . . 4 |- ( ( A e. (SubGrp ` G ) /\ S e. _V ) -> ( ( S : dom S --> (SubGrp ` H ) /\ A. x e. dom S ( A. y e. ( dom S \ { x } ) ( S ` x ) C_ ( (Cntz ` H ) ` ( S ` y ) ) /\ ( ( S ` x ) i^i ( (mrCls ` (SubGrp ` H ) ) ` U. ( S " ( dom S \ { x } ) ) ) ) = { ( 0g ` H ) } ) ) <-> ( ( S : dom S --> (SubGrp ` G ) /\ ran S C_ ~P A ) /\ A. x e. dom S ( A. y e. ( dom S \ { x } ) ( S ` x ) C_ ( (Cntz ` G ) ` ( S ` y ) ) /\ ( ( S ` x ) i^i ( (mrCls ` (SubGrp ` G ) ) ` U. ( S " ( dom S \ { x } ) ) ) ) = { ( 0g ` G ) } ) ) ) )
112		dmexg 7097	. . . . . 6 |- ( S e. _V -> dom S e. _V )
113	112	adantl 482	. . . . 5 |- ( ( A e. (SubGrp ` G ) /\ S e. _V ) -> dom S e. _V )
114		eqidd 2623	. . . . 5 |- ( ( A e. (SubGrp ` G ) /\ S e. _V ) -> dom S = dom S )
115	41	adantr 481	. . . . 5 |- ( ( A e. (SubGrp ` G ) /\ S e. _V ) -> H e. Grp )
116		eqid 2622	. . . . . . . 8 |- ( 0g ` H ) = ( 0g ` H )
117	26, 116, 46	dmdprd 18397	. . . . . . 7 |- ( ( dom S e. _V /\ dom S = dom S ) -> ( H dom DProd S <-> ( H e. Grp /\ S : dom S --> (SubGrp ` H ) /\ A. x e. dom S ( A. y e. ( dom S \ { x } ) ( S ` x ) C_ ( (Cntz ` H ) ` ( S ` y ) ) /\ ( ( S ` x ) i^i ( (mrCls ` (SubGrp ` H ) ) ` U. ( S " ( dom S \ { x } ) ) ) ) = { ( 0g ` H ) } ) ) ) )
118		3anass 1042	. . . . . . 7 |- ( ( H e. Grp /\ S : dom S --> (SubGrp ` H ) /\ A. x e. dom S ( A. y e. ( dom S \ { x } ) ( S ` x ) C_ ( (Cntz ` H ) ` ( S ` y ) ) /\ ( ( S ` x ) i^i ( (mrCls ` (SubGrp ` H ) ) ` U. ( S " ( dom S \ { x } ) ) ) ) = { ( 0g ` H ) } ) ) <-> ( H e. Grp /\ ( S : dom S --> (SubGrp ` H ) /\ A. x e. dom S ( A. y e. ( dom S \ { x } ) ( S ` x ) C_ ( (Cntz ` H ) ` ( S ` y ) ) /\ ( ( S ` x ) i^i ( (mrCls ` (SubGrp ` H ) ) ` U. ( S " ( dom S \ { x } ) ) ) ) = { ( 0g ` H ) } ) ) ) )
119	117, 118	syl6bb 276	. . . . . 6 |- ( ( dom S e. _V /\ dom S = dom S ) -> ( H dom DProd S <-> ( H e. Grp /\ ( S : dom S --> (SubGrp ` H ) /\ A. x e. dom S ( A. y e. ( dom S \ { x } ) ( S ` x ) C_ ( (Cntz ` H ) ` ( S ` y ) ) /\ ( ( S ` x ) i^i ( (mrCls ` (SubGrp ` H ) ) ` U. ( S " ( dom S \ { x } ) ) ) ) = { ( 0g ` H ) } ) ) ) ) )
120	119	baibd 948	. . . . 5 |- ( ( ( dom S e. _V /\ dom S = dom S ) /\ H e. Grp ) -> ( H dom DProd S <-> ( S : dom S --> (SubGrp ` H ) /\ A. x e. dom S ( A. y e. ( dom S \ { x } ) ( S ` x ) C_ ( (Cntz ` H ) ` ( S ` y ) ) /\ ( ( S ` x ) i^i ( (mrCls ` (SubGrp ` H ) ) ` U. ( S " ( dom S \ { x } ) ) ) ) = { ( 0g ` H ) } ) ) ) )
121	113, 114, 115, 120	syl21anc 1325	. . . 4 |- ( ( A e. (SubGrp ` G ) /\ S e. _V ) -> ( H dom DProd S <-> ( S : dom S --> (SubGrp ` H ) /\ A. x e. dom S ( A. y e. ( dom S \ { x } ) ( S ` x ) C_ ( (Cntz ` H ) ` ( S ` y ) ) /\ ( ( S ` x ) i^i ( (mrCls ` (SubGrp ` H ) ) ` U. ( S " ( dom S \ { x } ) ) ) ) = { ( 0g ` H ) } ) ) ) )
122	35	adantr 481	. . . . . . 7 |- ( ( A e. (SubGrp ` G ) /\ S e. _V ) -> G e. Grp )
123	25, 84, 63	dmdprd 18397	. . . . . . . . 9 |- ( ( dom S e. _V /\ dom S = dom S ) -> ( G dom DProd S <-> ( G e. Grp /\ S : dom S --> (SubGrp ` G ) /\ A. x e. dom S ( A. y e. ( dom S \ { x } ) ( S ` x ) C_ ( (Cntz ` G ) ` ( S ` y ) ) /\ ( ( S ` x ) i^i ( (mrCls ` (SubGrp ` G ) ) ` U. ( S " ( dom S \ { x } ) ) ) ) = { ( 0g ` G ) } ) ) ) )
124		3anass 1042	. . . . . . . . 9 |- ( ( G e. Grp /\ S : dom S --> (SubGrp ` G ) /\ A. x e. dom S ( A. y e. ( dom S \ { x } ) ( S ` x ) C_ ( (Cntz ` G ) ` ( S ` y ) ) /\ ( ( S ` x ) i^i ( (mrCls ` (SubGrp ` G ) ) ` U. ( S " ( dom S \ { x } ) ) ) ) = { ( 0g ` G ) } ) ) <-> ( G e. Grp /\ ( S : dom S --> (SubGrp ` G ) /\ A. x e. dom S ( A. y e. ( dom S \ { x } ) ( S ` x ) C_ ( (Cntz ` G ) ` ( S ` y ) ) /\ ( ( S ` x ) i^i ( (mrCls ` (SubGrp ` G ) ) ` U. ( S " ( dom S \ { x } ) ) ) ) = { ( 0g ` G ) } ) ) ) )
125	123, 124	syl6bb 276	. . . . . . . 8 |- ( ( dom S e. _V /\ dom S = dom S ) -> ( G dom DProd S <-> ( G e. Grp /\ ( S : dom S --> (SubGrp ` G ) /\ A. x e. dom S ( A. y e. ( dom S \ { x } ) ( S ` x ) C_ ( (Cntz ` G ) ` ( S ` y ) ) /\ ( ( S ` x ) i^i ( (mrCls ` (SubGrp ` G ) ) ` U. ( S " ( dom S \ { x } ) ) ) ) = { ( 0g ` G ) } ) ) ) ) )
126	125	baibd 948	. . . . . . 7 |- ( ( ( dom S e. _V /\ dom S = dom S ) /\ G e. Grp ) -> ( G dom DProd S <-> ( S : dom S --> (SubGrp ` G ) /\ A. x e. dom S ( A. y e. ( dom S \ { x } ) ( S ` x ) C_ ( (Cntz ` G ) ` ( S ` y ) ) /\ ( ( S ` x ) i^i ( (mrCls ` (SubGrp ` G ) ) ` U. ( S " ( dom S \ { x } ) ) ) ) = { ( 0g ` G ) } ) ) ) )
127	113, 114, 122, 126	syl21anc 1325	. . . . . 6 |- ( ( A e. (SubGrp ` G ) /\ S e. _V ) -> ( G dom DProd S <-> ( S : dom S --> (SubGrp ` G ) /\ A. x e. dom S ( A. y e. ( dom S \ { x } ) ( S ` x ) C_ ( (Cntz ` G ) ` ( S ` y ) ) /\ ( ( S ` x ) i^i ( (mrCls ` (SubGrp ` G ) ) ` U. ( S " ( dom S \ { x } ) ) ) ) = { ( 0g ` G ) } ) ) ) )
128	127	anbi1d 741	. . . . 5 |- ( ( A e. (SubGrp ` G ) /\ S e. _V ) -> ( ( G dom DProd S /\ ran S C_ ~P A ) <-> ( ( S : dom S --> (SubGrp ` G ) /\ A. x e. dom S ( A. y e. ( dom S \ { x } ) ( S ` x ) C_ ( (Cntz ` G ) ` ( S ` y ) ) /\ ( ( S ` x ) i^i ( (mrCls ` (SubGrp ` G ) ) ` U. ( S " ( dom S \ { x } ) ) ) ) = { ( 0g ` G ) } ) ) /\ ran S C_ ~P A ) ) )
129		an32 839	. . . . 5 |- ( ( ( S : dom S --> (SubGrp ` G ) /\ A. x e. dom S ( A. y e. ( dom S \ { x } ) ( S ` x ) C_ ( (Cntz ` G ) ` ( S ` y ) ) /\ ( ( S ` x ) i^i ( (mrCls ` (SubGrp ` G ) ) ` U. ( S " ( dom S \ { x } ) ) ) ) = { ( 0g ` G ) } ) ) /\ ran S C_ ~P A ) <-> ( ( S : dom S --> (SubGrp ` G ) /\ ran S C_ ~P A ) /\ A. x e. dom S ( A. y e. ( dom S \ { x } ) ( S ` x ) C_ ( (Cntz ` G ) ` ( S ` y ) ) /\ ( ( S ` x ) i^i ( (mrCls ` (SubGrp ` G ) ) ` U. ( S " ( dom S \ { x } ) ) ) ) = { ( 0g ` G ) } ) ) )
130	128, 129	syl6bb 276	. . . 4 |- ( ( A e. (SubGrp ` G ) /\ S e. _V ) -> ( ( G dom DProd S /\ ran S C_ ~P A ) <-> ( ( S : dom S --> (SubGrp ` G ) /\ ran S C_ ~P A ) /\ A. x e. dom S ( A. y e. ( dom S \ { x } ) ( S ` x ) C_ ( (Cntz ` G ) ` ( S ` y ) ) /\ ( ( S ` x ) i^i ( (mrCls ` (SubGrp ` G ) ) ` U. ( S " ( dom S \ { x } ) ) ) ) = { ( 0g ` G ) } ) ) ) )
131	111, 121, 130	3bitr4d 300	. . 3 |- ( ( A e. (SubGrp ` G ) /\ S e. _V ) -> ( H dom DProd S <-> ( G dom DProd S /\ ran S C_ ~P A ) ) )
132	131	ex 450	. 2 |- ( A e. (SubGrp ` G ) -> ( S e. _V -> ( H dom DProd S <-> ( G dom DProd S /\ ran S C_ ~P A ) ) ) )
133	3, 6, 132	pm5.21ndd 369	1 |- ( A e. (SubGrp ` G ) -> ( H dom DProd S <-> ( G dom DProd S /\ ran S C_ ~P A ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   \ cdif 3571   i^i cin 3573   C_ wss 3574   ~Pcpw 4158   {csn 4177   U.cuni 4436   class class class wbr 4653   dom cdm 5114   ran crn 5115   "cima 5117   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   Basecbs 15857   ↾_s cress 15858   0gc0g 16100  Moorecmre 16242  mrClscmrc 16243  ACScacs 16245   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  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-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-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  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-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-0g 16102  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-grp 17425  df-minusg 17426  df-subg 17591  df-cntz 17750  df-dprd 18394

This theorem is referenced by:  subgdprd  18434  ablfaclem3  18486