Theorem clsnsg 24066

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 19099	. . 3 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → 𝑆 ∈ (SubGrp‘𝐺))
2		subgntr.h	. . . 4 ⊢ 𝐽 = (TopOpen‘𝐺)
3	2	clssubg 24065	. . 3 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ((cls‘𝐽)‘𝑆) ∈ (SubGrp‘𝐺))
4	1, 3	sylan2 594	. 2 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) → ((cls‘𝐽)‘𝑆) ∈ (SubGrp‘𝐺))
5		df-ima 5645	. . . . . . 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 2737	. . . . . . . . . . . . . 14 ⊢ (Base‘𝐺) = (Base‘𝐺)
7	2, 6	tgptopon 24038	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ TopGrp → 𝐽 ∈ (TopOn‘(Base‘𝐺)))
8	7	ad2antrr 727	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → 𝐽 ∈ (TopOn‘(Base‘𝐺)))
9		topontop 22869	. . . . . . . . . . . 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 728	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → 𝑆 ∈ (SubGrp‘𝐺))
12	6	subgss 19069	. . . . . . . . . . . . 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 22870	. . . . . . . . . . . . 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 3972	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → 𝑆 ⊆ ∪ 𝐽)
17		eqid 2737	. . . . . . . . . . . 12 ⊢ ∪ 𝐽 = ∪ 𝐽
18	17	clsss3 23015	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) → ((cls‘𝐽)‘𝑆) ⊆ ∪ 𝐽)
19	10, 16, 18	syl2anc 585	. . . . . . . . . 10 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → ((cls‘𝐽)‘𝑆) ⊆ ∪ 𝐽)
20	19, 15	sseqtrrd 3973	. . . . . . . . 9 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → ((cls‘𝐽)‘𝑆) ⊆ (Base‘𝐺))
21	20	resmptd 6007	. . . . . . . 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 5895	. . . . . . 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 2784	. . . . . 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 2737	. . . . . . . . . 10 ⊢ (+g‘𝐺) = (+g‘𝐺)
25		tgptmd 24035	. . . . . . . . . . 11 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ TopMnd)
26	25	ad2antrr 727	. . . . . . . . . 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 23618	. . . . . . . . . 10 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑦 ∈ (Base‘𝐺) ↦ 𝑥) ∈ (𝐽 Cn 𝐽))
29	8	cnmptid 23617	. . . . . . . . . 10 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑦 ∈ (Base‘𝐺) ↦ 𝑦) ∈ (𝐽 Cn 𝐽))
30	2, 24, 26, 8, 28, 29	cnmpt1plusg 24043	. . . . . . . . 9 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑦 ∈ (Base‘𝐺) ↦ (𝑥(+g‘𝐺)𝑦)) ∈ (𝐽 Cn 𝐽))
31		eqid 2737	. . . . . . . . . . 11 ⊢ (-g‘𝐺) = (-g‘𝐺)
32	2, 31	tgpsubcn 24046	. . . . . . . . . 10 ⊢ (𝐺 ∈ TopGrp → (-g‘𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
33	32	ad2antrr 727	. . . . . . . . 9 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → (-g‘𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
34	8, 30, 28, 33	cnmpt12f 23622	. . . . . . . 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 23228	. . . . . . . 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 585	. . . . . . 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 5645	. . . . . . . . . 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 6007	. . . . . . . . . . 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 5895	. . . . . . . . . 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 2784	. . . . . . . . 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 19100	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ 𝑆) → ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑥) ∈ 𝑆)
42	41	ad4ant234 1177	. . . . . . . . . . 11 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑦 ∈ 𝑆) → ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑥) ∈ 𝑆)
43	42	fmpttd 7069	. . . . . . . . . 10 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑦 ∈ 𝑆 ↦ ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑥)):𝑆⟶𝑆)
44	43	frnd 6678	. . . . . . . . 9 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → ran (𝑦 ∈ 𝑆 ↦ ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑥)) ⊆ 𝑆)
45	40, 44	eqsstrd 3970	. . . . . . . 8 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → ((𝑦 ∈ (Base‘𝐺) ↦ ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑥)) “ 𝑆) ⊆ 𝑆)
46	17	clsss 23010	. . . . . . . 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 1374	. . . . . . 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 3946	. . . . . 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 3971	. . . . 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 7401	. . . . . . 7 ⊢ ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑥) ∈ V
51		eqid 2737	. . . . . . 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 6643	. . . . . 6 ⊢ (𝑦 ∈ ((cls‘𝐽)‘𝑆) ↦ ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑥)) Fn ((cls‘𝐽)‘𝑆)
53		df-f 6504	. . . . . 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 710	. . . . 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 7064	. . . 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 3130	. 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 19101	. 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 584	1 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) → ((cls‘𝐽)‘𝑆) ∈ (NrmSGrp‘𝐺))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052   ⊆ wss 3903  ∪ cuni 4865   ↦ cmpt 5181  ran crn 5633   ↾ cres 5634   “ cima 5635   Fn wfn 6495  ⟶wf 6496  ‘cfv 6500  (class class class)co 7368  Basecbs 17148  +_gcplusg 17189  TopOpenctopn 17353  -_gcsg 18877  SubGrpcsubg 19062  NrmSGrpcnsg 19063  Topctop 22849  TopOnctopon 22866  clsccl 22974   Cn ccn 23180   ×_t ctx 23516  TopMndctmd 24026  TopGrpctgp 24027

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-iin 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-om 7819  df-1st 7943  df-2nd 7944  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-er 8645  df-map 8777  df-en 8896  df-dom 8897  df-sdom 8898  df-pnf 11180  df-mnf 11181  df-xr 11182  df-ltxr 11183  df-le 11184  df-sub 11378  df-neg 11379  df-nn 12158  df-2 12220  df-sets 17103  df-slot 17121  df-ndx 17133  df-base 17149  df-ress 17170  df-plusg 17202  df-0g 17373  df-topgen 17375  df-plusf 18576  df-mgm 18577  df-sgrp 18656  df-mnd 18672  df-grp 18878  df-minusg 18879  df-sbg 18880  df-subg 19065  df-nsg 19066  df-top 22850  df-topon 22867  df-topsp 22889  df-bases 22902  df-cld 22975  df-ntr 22976  df-cls 22977  df-cn 23183  df-cnp 23184  df-tx 23518  df-tmd 24028  df-tgp 24029

This theorem is referenced by: (None)