Theorem clsnsg 24023

Description: The closure of a normal subgroup is a normal subgroup. (Contributed by Mario Carneiro, 17-Sep-2015.)

Hypothesis

Ref	Expression
subgntr.h	⊢ 𝐽 = (TopOpen‘𝐺)

Assertion

Ref	Expression
clsnsg	⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) → ((cls‘𝐽)‘𝑆) ∈ (NrmSGrp‘𝐺))

Proof of Theorem clsnsg

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

Step	Hyp	Ref	Expression
1		nsgsubg 19068	. . 3 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → 𝑆 ∈ (SubGrp‘𝐺))
2		subgntr.h	. . . 4 ⊢ 𝐽 = (TopOpen‘𝐺)
3	2	clssubg 24022	. . 3 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ((cls‘𝐽)‘𝑆) ∈ (SubGrp‘𝐺))
4	1, 3	sylan2 593	. 2 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) → ((cls‘𝐽)‘𝑆) ∈ (SubGrp‘𝐺))
5		df-ima 5629	. . . . . . 7 ⊢ ((𝑦 ∈ (Base‘𝐺) ↦ ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑥)) “ ((cls‘𝐽)‘𝑆)) = ran ((𝑦 ∈ (Base‘𝐺) ↦ ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑥)) ↾ ((cls‘𝐽)‘𝑆))
6		eqid 2731	. . . . . . . . . . . . . 14 ⊢ (Base‘𝐺) = (Base‘𝐺)
7	2, 6	tgptopon 23995	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ TopGrp → 𝐽 ∈ (TopOn‘(Base‘𝐺)))
8	7	ad2antrr 726	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → 𝐽 ∈ (TopOn‘(Base‘𝐺)))
9		topontop 22826	. . . . . . . . . . . 12 ⊢ (𝐽 ∈ (TopOn‘(Base‘𝐺)) → 𝐽 ∈ Top)
10	8, 9	syl 17	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → 𝐽 ∈ Top)
11	1	ad2antlr 727	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → 𝑆 ∈ (SubGrp‘𝐺))
12	6	subgss 19037	. . . . . . . . . . . . 13 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ (Base‘𝐺))
13	11, 12	syl 17	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → 𝑆 ⊆ (Base‘𝐺))
14		toponuni 22827	. . . . . . . . . . . . 13 ⊢ (𝐽 ∈ (TopOn‘(Base‘𝐺)) → (Base‘𝐺) = ∪ 𝐽)
15	8, 14	syl 17	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → (Base‘𝐺) = ∪ 𝐽)
16	13, 15	sseqtrd 3971	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → 𝑆 ⊆ ∪ 𝐽)
17		eqid 2731	. . . . . . . . . . . 12 ⊢ ∪ 𝐽 = ∪ 𝐽
18	17	clsss3 22972	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) → ((cls‘𝐽)‘𝑆) ⊆ ∪ 𝐽)
19	10, 16, 18	syl2anc 584	. . . . . . . . . 10 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → ((cls‘𝐽)‘𝑆) ⊆ ∪ 𝐽)
20	19, 15	sseqtrrd 3972	. . . . . . . . 9 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → ((cls‘𝐽)‘𝑆) ⊆ (Base‘𝐺))
21	20	resmptd 5989	. . . . . . . 8 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → ((𝑦 ∈ (Base‘𝐺) ↦ ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑥)) ↾ ((cls‘𝐽)‘𝑆)) = (𝑦 ∈ ((cls‘𝐽)‘𝑆) ↦ ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑥)))
22	21	rneqd 5878	. . . . . . 7 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → ran ((𝑦 ∈ (Base‘𝐺) ↦ ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑥)) ↾ ((cls‘𝐽)‘𝑆)) = ran (𝑦 ∈ ((cls‘𝐽)‘𝑆) ↦ ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑥)))
23	5, 22	eqtrid 2778	. . . . . 6 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → ((𝑦 ∈ (Base‘𝐺) ↦ ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑥)) “ ((cls‘𝐽)‘𝑆)) = ran (𝑦 ∈ ((cls‘𝐽)‘𝑆) ↦ ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑥)))
24		eqid 2731	. . . . . . . . . 10 ⊢ (+g‘𝐺) = (+g‘𝐺)
25		tgptmd 23992	. . . . . . . . . . 11 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ TopMnd)
26	25	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → 𝐺 ∈ TopMnd)
27		simpr 484	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → 𝑥 ∈ (Base‘𝐺))
28	8, 8, 27	cnmptc 23575	. . . . . . . . . 10 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑦 ∈ (Base‘𝐺) ↦ 𝑥) ∈ (𝐽 Cn 𝐽))
29	8	cnmptid 23574	. . . . . . . . . 10 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑦 ∈ (Base‘𝐺) ↦ 𝑦) ∈ (𝐽 Cn 𝐽))
30	2, 24, 26, 8, 28, 29	cnmpt1plusg 24000	. . . . . . . . 9 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑦 ∈ (Base‘𝐺) ↦ (𝑥(+g‘𝐺)𝑦)) ∈ (𝐽 Cn 𝐽))
31		eqid 2731	. . . . . . . . . . 11 ⊢ (-g‘𝐺) = (-g‘𝐺)
32	2, 31	tgpsubcn 24003	. . . . . . . . . 10 ⊢ (𝐺 ∈ TopGrp → (-g‘𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
33	32	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → (-g‘𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
34	8, 30, 28, 33	cnmpt12f 23579	. . . . . . . 8 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑦 ∈ (Base‘𝐺) ↦ ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑥)) ∈ (𝐽 Cn 𝐽))
35	17	cnclsi 23185	. . . . . . . 8 ⊢ (((𝑦 ∈ (Base‘𝐺) ↦ ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑥)) ∈ (𝐽 Cn 𝐽) ∧ 𝑆 ⊆ ∪ 𝐽) → ((𝑦 ∈ (Base‘𝐺) ↦ ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑥)) “ ((cls‘𝐽)‘𝑆)) ⊆ ((cls‘𝐽)‘((𝑦 ∈ (Base‘𝐺) ↦ ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑥)) “ 𝑆)))
36	34, 16, 35	syl2anc 584	. . . . . . 7 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → ((𝑦 ∈ (Base‘𝐺) ↦ ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑥)) “ ((cls‘𝐽)‘𝑆)) ⊆ ((cls‘𝐽)‘((𝑦 ∈ (Base‘𝐺) ↦ ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑥)) “ 𝑆)))
37		df-ima 5629	. . . . . . . . . 10 ⊢ ((𝑦 ∈ (Base‘𝐺) ↦ ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑥)) “ 𝑆) = ran ((𝑦 ∈ (Base‘𝐺) ↦ ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑥)) ↾ 𝑆)
38	13	resmptd 5989	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → ((𝑦 ∈ (Base‘𝐺) ↦ ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑥)) ↾ 𝑆) = (𝑦 ∈ 𝑆 ↦ ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑥)))
39	38	rneqd 5878	. . . . . . . . . 10 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → ran ((𝑦 ∈ (Base‘𝐺) ↦ ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑥)) ↾ 𝑆) = ran (𝑦 ∈ 𝑆 ↦ ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑥)))
40	37, 39	eqtrid 2778	. . . . . . . . 9 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → ((𝑦 ∈ (Base‘𝐺) ↦ ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑥)) “ 𝑆) = ran (𝑦 ∈ 𝑆 ↦ ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑥)))
41	6, 24, 31	nsgconj 19069	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ 𝑆) → ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑥) ∈ 𝑆)
42	41	ad4ant234 1176	. . . . . . . . . . 11 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑦 ∈ 𝑆) → ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑥) ∈ 𝑆)
43	42	fmpttd 7048	. . . . . . . . . 10 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑦 ∈ 𝑆 ↦ ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑥)):𝑆⟶𝑆)
44	43	frnd 6659	. . . . . . . . 9 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → ran (𝑦 ∈ 𝑆 ↦ ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑥)) ⊆ 𝑆)
45	40, 44	eqsstrd 3969	. . . . . . . 8 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → ((𝑦 ∈ (Base‘𝐺) ↦ ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑥)) “ 𝑆) ⊆ 𝑆)
46	17	clsss 22967	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽 ∧ ((𝑦 ∈ (Base‘𝐺) ↦ ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑥)) “ 𝑆) ⊆ 𝑆) → ((cls‘𝐽)‘((𝑦 ∈ (Base‘𝐺) ↦ ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑥)) “ 𝑆)) ⊆ ((cls‘𝐽)‘𝑆))
47	10, 16, 45, 46	syl3anc 1373	. . . . . . 7 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → ((cls‘𝐽)‘((𝑦 ∈ (Base‘𝐺) ↦ ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑥)) “ 𝑆)) ⊆ ((cls‘𝐽)‘𝑆))
48	36, 47	sstrd 3945	. . . . . 6 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → ((𝑦 ∈ (Base‘𝐺) ↦ ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑥)) “ ((cls‘𝐽)‘𝑆)) ⊆ ((cls‘𝐽)‘𝑆))
49	23, 48	eqsstrrd 3970	. . . . 5 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → ran (𝑦 ∈ ((cls‘𝐽)‘𝑆) ↦ ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑥)) ⊆ ((cls‘𝐽)‘𝑆))
50		ovex 7379	. . . . . . 7 ⊢ ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑥) ∈ V
51		eqid 2731	. . . . . . 7 ⊢ (𝑦 ∈ ((cls‘𝐽)‘𝑆) ↦ ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑥)) = (𝑦 ∈ ((cls‘𝐽)‘𝑆) ↦ ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑥))
52	50, 51	fnmpti 6624	. . . . . 6 ⊢ (𝑦 ∈ ((cls‘𝐽)‘𝑆) ↦ ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑥)) Fn ((cls‘𝐽)‘𝑆)
53		df-f 6485	. . . . . 6 ⊢ ((𝑦 ∈ ((cls‘𝐽)‘𝑆) ↦ ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑥)):((cls‘𝐽)‘𝑆)⟶((cls‘𝐽)‘𝑆) ↔ ((𝑦 ∈ ((cls‘𝐽)‘𝑆) ↦ ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑥)) Fn ((cls‘𝐽)‘𝑆) ∧ ran (𝑦 ∈ ((cls‘𝐽)‘𝑆) ↦ ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑥)) ⊆ ((cls‘𝐽)‘𝑆)))
54	52, 53	mpbiran 709	. . . . 5 ⊢ ((𝑦 ∈ ((cls‘𝐽)‘𝑆) ↦ ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑥)):((cls‘𝐽)‘𝑆)⟶((cls‘𝐽)‘𝑆) ↔ ran (𝑦 ∈ ((cls‘𝐽)‘𝑆) ↦ ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑥)) ⊆ ((cls‘𝐽)‘𝑆))
55	49, 54	sylibr 234	. . . 4 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑦 ∈ ((cls‘𝐽)‘𝑆) ↦ ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑥)):((cls‘𝐽)‘𝑆)⟶((cls‘𝐽)‘𝑆))
56	51	fmpt 7043	. . . 4 ⊢ (∀𝑦 ∈ ((cls‘𝐽)‘𝑆)((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑥) ∈ ((cls‘𝐽)‘𝑆) ↔ (𝑦 ∈ ((cls‘𝐽)‘𝑆) ↦ ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑥)):((cls‘𝐽)‘𝑆)⟶((cls‘𝐽)‘𝑆))
57	55, 56	sylibr 234	. . 3 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → ∀𝑦 ∈ ((cls‘𝐽)‘𝑆)((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑥) ∈ ((cls‘𝐽)‘𝑆))
58	57	ralrimiva 3124	. 2 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) → ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ ((cls‘𝐽)‘𝑆)((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑥) ∈ ((cls‘𝐽)‘𝑆))
59	6, 24, 31	isnsg3 19070	. 2 ⊢ (((cls‘𝐽)‘𝑆) ∈ (NrmSGrp‘𝐺) ↔ (((cls‘𝐽)‘𝑆) ∈ (SubGrp‘𝐺) ∧ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ ((cls‘𝐽)‘𝑆)((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑥) ∈ ((cls‘𝐽)‘𝑆)))
60	4, 58, 59	sylanbrc 583	1 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) → ((cls‘𝐽)‘𝑆) ∈ (NrmSGrp‘𝐺))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111  ∀wral 3047   ⊆ wss 3902  ∪ cuni 4859   ↦ cmpt 5172  ran crn 5617   ↾ cres 5618   “ cima 5619   Fn wfn 6476  ⟶wf 6477  ‘cfv 6481  (class class class)co 7346  Basecbs 17117  +_gcplusg 17158  TopOpenctopn 17322  -_gcsg 18845  SubGrpcsubg 19030  NrmSGrpcnsg 19031  Topctop 22806  TopOnctopon 22823  clsccl 22931   Cn ccn 23137   ×_t ctx 23473  TopMndctmd 23983  TopGrpctgp 23984

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5217  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668  ax-cnex 11059  ax-resscn 11060  ax-1cn 11061  ax-icn 11062  ax-addcl 11063  ax-addrcl 11064  ax-mulcl 11065  ax-mulrcl 11066  ax-mulcom 11067  ax-addass 11068  ax-mulass 11069  ax-distr 11070  ax-i2m1 11071  ax-1ne0 11072  ax-1rid 11073  ax-rnegex 11074  ax-rrecex 11075  ax-cnre 11076  ax-pre-lttri 11077  ax-pre-lttrn 11078  ax-pre-ltadd 11079  ax-pre-mulgt0 11080

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-int 4898  df-iun 4943  df-iin 4944  df-br 5092  df-opab 5154  df-mpt 5173  df-tr 5199  df-id 5511  df-eprel 5516  df-po 5524  df-so 5525  df-fr 5569  df-we 5571  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7797  df-1st 7921  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-er 8622  df-map 8752  df-en 8870  df-dom 8871  df-sdom 8872  df-pnf 11145  df-mnf 11146  df-xr 11147  df-ltxr 11148  df-le 11149  df-sub 11343  df-neg 11344  df-nn 12123  df-2 12185  df-sets 17072  df-slot 17090  df-ndx 17102  df-base 17118  df-ress 17139  df-plusg 17171  df-0g 17342  df-topgen 17344  df-plusf 18544  df-mgm 18545  df-sgrp 18624  df-mnd 18640  df-grp 18846  df-minusg 18847  df-sbg 18848  df-subg 19033  df-nsg 19034  df-top 22807  df-topon 22824  df-topsp 22846  df-bases 22859  df-cld 22932  df-ntr 22933  df-cls 22934  df-cn 23140  df-cnp 23141  df-tx 23475  df-tmd 23985  df-tgp 23986

This theorem is referenced by: (None)