Theorem clsnsg 24100

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 19131	. . 3 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → 𝑆 ∈ (SubGrp‘𝐺))
2		subgntr.h	. . . 4 ⊢ 𝐽 = (TopOpen‘𝐺)
3	2	clssubg 24099	. . 3 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ((cls‘𝐽)‘𝑆) ∈ (SubGrp‘𝐺))
4	1, 3	sylan2 599	. 2 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) → ((cls‘𝐽)‘𝑆) ∈ (SubGrp‘𝐺))
5		df-ima 5638	. . . . . . 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 2740	. . . . . . . . . . . . . 14 ⊢ (Base‘𝐺) = (Base‘𝐺)
7	2, 6	tgptopon 24072	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ TopGrp → 𝐽 ∈ (TopOn‘(Base‘𝐺)))
8	7	ad2antrr 732	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → 𝐽 ∈ (TopOn‘(Base‘𝐺)))
9		topontop 22903	. . . . . . . . . . . 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 733	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → 𝑆 ∈ (SubGrp‘𝐺))
12	6	subgss 19101	. . . . . . . . . . . . 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 22904	. . . . . . . . . . . . 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 3958	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → 𝑆 ⊆ ∪ 𝐽)
17		eqid 2740	. . . . . . . . . . . 12 ⊢ ∪ 𝐽 = ∪ 𝐽
18	17	clsss3 23049	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) → ((cls‘𝐽)‘𝑆) ⊆ ∪ 𝐽)
19	10, 16, 18	syl2anc 590	. . . . . . . . . 10 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → ((cls‘𝐽)‘𝑆) ⊆ ∪ 𝐽)
20	19, 15	sseqtrrd 3959	. . . . . . . . 9 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → ((cls‘𝐽)‘𝑆) ⊆ (Base‘𝐺))
21	20	resmptd 5999	. . . . . . . 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 5887	. . . . . . 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 2787	. . . . . 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 2740	. . . . . . . . . 10 ⊢ (+g‘𝐺) = (+g‘𝐺)
25		tgptmd 24069	. . . . . . . . . . 11 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ TopMnd)
26	25	ad2antrr 732	. . . . . . . . . 10 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → 𝐺 ∈ TopMnd)
27		simpr 485	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → 𝑥 ∈ (Base‘𝐺))
28	8, 8, 27	cnmptc 23652	. . . . . . . . . 10 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑦 ∈ (Base‘𝐺) ↦ 𝑥) ∈ (𝐽 Cn 𝐽))
29	8	cnmptid 23651	. . . . . . . . . 10 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑦 ∈ (Base‘𝐺) ↦ 𝑦) ∈ (𝐽 Cn 𝐽))
30	2, 24, 26, 8, 28, 29	cnmpt1plusg 24077	. . . . . . . . 9 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑦 ∈ (Base‘𝐺) ↦ (𝑥(+g‘𝐺)𝑦)) ∈ (𝐽 Cn 𝐽))
31		eqid 2740	. . . . . . . . . . 11 ⊢ (-g‘𝐺) = (-g‘𝐺)
32	2, 31	tgpsubcn 24080	. . . . . . . . . 10 ⊢ (𝐺 ∈ TopGrp → (-g‘𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
33	32	ad2antrr 732	. . . . . . . . 9 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → (-g‘𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
34	8, 30, 28, 33	cnmpt12f 23656	. . . . . . . 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 23262	. . . . . . . 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 590	. . . . . . 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 5638	. . . . . . . . . 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 5999	. . . . . . . . . . 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 5887	. . . . . . . . . 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 2787	. . . . . . . . 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 19132	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ 𝑆) → ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑥) ∈ 𝑆)
42	41	ad4ant234 1182	. . . . . . . . . . 11 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑦 ∈ 𝑆) → ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑥) ∈ 𝑆)
43	42	fmpttd 7063	. . . . . . . . . 10 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑦 ∈ 𝑆 ↦ ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑥)):𝑆⟶𝑆)
44	43	frnd 6670	. . . . . . . . 9 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → ran (𝑦 ∈ 𝑆 ↦ ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑥)) ⊆ 𝑆)
45	40, 44	eqsstrd 3956	. . . . . . . 8 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → ((𝑦 ∈ (Base‘𝐺) ↦ ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑥)) “ 𝑆) ⊆ 𝑆)
46	17	clsss 23044	. . . . . . . 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 1379	. . . . . . 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 3932	. . . . . 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 3957	. . . . 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 7396	. . . . . . 7 ⊢ ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑥) ∈ V
51		eqid 2740	. . . . . . 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 6635	. . . . . 6 ⊢ (𝑦 ∈ ((cls‘𝐽)‘𝑆) ↦ ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑥)) Fn ((cls‘𝐽)‘𝑆)
53		df-f 6496	. . . . . 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 715	. . . . 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 235	. . . 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 7058	. . . 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 235	. . 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 3132	. 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 19133	. 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 589	1 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) → ((cls‘𝐽)‘𝑆) ∈ (NrmSGrp‘𝐺))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119  ∀wral 3054   ⊆ wss 3890  ∪ cuni 4845   ↦ cmpt 5160  ran crn 5626   ↾ cres 5627   “ cima 5628   Fn wfn 6487  ⟶wf 6488  ‘cfv 6492  (class class class)co 7363  Basecbs 17177  +_gcplusg 17218  TopOpenctopn 17382  -_gcsg 18909  SubGrpcsubg 19094  NrmSGrpcnsg 19095  Topctop 22883  TopOnctopon 22900  clsccl 23008   Cn ccn 23214   ×_t ctx 23550  TopMndctmd 24060  TopGrpctgp 24061

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685  ax-cnex 11092  ax-resscn 11093  ax-1cn 11094  ax-icn 11095  ax-addcl 11096  ax-addrcl 11097  ax-mulcl 11098  ax-mulrcl 11099  ax-mulcom 11100  ax-addass 11101  ax-mulass 11102  ax-distr 11103  ax-i2m1 11104  ax-1ne0 11105  ax-1rid 11106  ax-rnegex 11107  ax-rrecex 11108  ax-cnre 11109  ax-pre-lttri 11110  ax-pre-lttrn 11111  ax-pre-ltadd 11112  ax-pre-mulgt0 11113

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-nel 3040  df-ral 3055  df-rex 3065  df-rmo 3345  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-int 4885  df-iun 4930  df-iin 4931  df-br 5080  df-opab 5142  df-mpt 5161  df-tr 5187  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7320  df-ov 7366  df-oprab 7367  df-mpo 7368  df-om 7814  df-1st 7938  df-2nd 7939  df-frecs 8228  df-wrecs 8259  df-recs 8308  df-rdg 8346  df-er 8640  df-map 8772  df-en 8891  df-dom 8892  df-sdom 8893  df-pnf 11179  df-mnf 11180  df-xr 11181  df-ltxr 11182  df-le 11183  df-sub 11377  df-neg 11378  df-nn 12173  df-2 12242  df-sets 17132  df-slot 17150  df-ndx 17162  df-base 17178  df-ress 17199  df-plusg 17231  df-0g 17402  df-topgen 17404  df-plusf 18605  df-mgm 18606  df-sgrp 18685  df-mnd 18701  df-grp 18910  df-minusg 18911  df-sbg 18912  df-subg 19097  df-nsg 19098  df-top 22884  df-topon 22901  df-topsp 22923  df-bases 22936  df-cld 23009  df-ntr 23010  df-cls 23011  df-cn 23217  df-cnp 23218  df-tx 23552  df-tmd 24062  df-tgp 24063

This theorem is referenced by: (None)