Theorem clsnsg 24105

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 19152	. . 3 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → 𝑆 ∈ (SubGrp‘𝐺))
2		subgntr.h	. . . 4 ⊢ 𝐽 = (TopOpen‘𝐺)
3	2	clssubg 24104	. . 3 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ((cls‘𝐽)‘𝑆) ∈ (SubGrp‘𝐺))
4	1, 3	sylan2 591	. 2 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) → ((cls‘𝐽)‘𝑆) ∈ (SubGrp‘𝐺))
5		df-ima 5695	. . . . . . 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 2726	. . . . . . . . . . . . . 14 ⊢ (Base‘𝐺) = (Base‘𝐺)
7	2, 6	tgptopon 24077	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ TopGrp → 𝐽 ∈ (TopOn‘(Base‘𝐺)))
8	7	ad2antrr 724	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → 𝐽 ∈ (TopOn‘(Base‘𝐺)))
9		topontop 22906	. . . . . . . . . . . 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 725	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → 𝑆 ∈ (SubGrp‘𝐺))
12	6	subgss 19121	. . . . . . . . . . . . 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 22907	. . . . . . . . . . . . 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 4020	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → 𝑆 ⊆ ∪ 𝐽)
17		eqid 2726	. . . . . . . . . . . 12 ⊢ ∪ 𝐽 = ∪ 𝐽
18	17	clsss3 23054	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) → ((cls‘𝐽)‘𝑆) ⊆ ∪ 𝐽)
19	10, 16, 18	syl2anc 582	. . . . . . . . . 10 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → ((cls‘𝐽)‘𝑆) ⊆ ∪ 𝐽)
20	19, 15	sseqtrrd 4021	. . . . . . . . 9 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → ((cls‘𝐽)‘𝑆) ⊆ (Base‘𝐺))
21	20	resmptd 6049	. . . . . . . 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 5944	. . . . . . 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 2726	. . . . . . . . . 10 ⊢ (+g‘𝐺) = (+g‘𝐺)
25		tgptmd 24074	. . . . . . . . . . 11 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ TopMnd)
26	25	ad2antrr 724	. . . . . . . . . 10 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → 𝐺 ∈ TopMnd)
27		simpr 483	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → 𝑥 ∈ (Base‘𝐺))
28	8, 8, 27	cnmptc 23657	. . . . . . . . . 10 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑦 ∈ (Base‘𝐺) ↦ 𝑥) ∈ (𝐽 Cn 𝐽))
29	8	cnmptid 23656	. . . . . . . . . 10 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑦 ∈ (Base‘𝐺) ↦ 𝑦) ∈ (𝐽 Cn 𝐽))
30	2, 24, 26, 8, 28, 29	cnmpt1plusg 24082	. . . . . . . . 9 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑦 ∈ (Base‘𝐺) ↦ (𝑥(+g‘𝐺)𝑦)) ∈ (𝐽 Cn 𝐽))
31		eqid 2726	. . . . . . . . . . 11 ⊢ (-g‘𝐺) = (-g‘𝐺)
32	2, 31	tgpsubcn 24085	. . . . . . . . . 10 ⊢ (𝐺 ∈ TopGrp → (-g‘𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
33	32	ad2antrr 724	. . . . . . . . 9 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → (-g‘𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
34	8, 30, 28, 33	cnmpt12f 23661	. . . . . . . 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 23267	. . . . . . . 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 582	. . . . . . 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 5695	. . . . . . . . . 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 6049	. . . . . . . . . . 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 5944	. . . . . . . . . 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 19153	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ 𝑆) → ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑥) ∈ 𝑆)
42	41	ad4ant234 1172	. . . . . . . . . . 11 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑦 ∈ 𝑆) → ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑥) ∈ 𝑆)
43	42	fmpttd 7129	. . . . . . . . . 10 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑦 ∈ 𝑆 ↦ ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑥)):𝑆⟶𝑆)
44	43	frnd 6736	. . . . . . . . 9 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → ran (𝑦 ∈ 𝑆 ↦ ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑥)) ⊆ 𝑆)
45	40, 44	eqsstrd 4018	. . . . . . . 8 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → ((𝑦 ∈ (Base‘𝐺) ↦ ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑥)) “ 𝑆) ⊆ 𝑆)
46	17	clsss 23049	. . . . . . . 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 1368	. . . . . . 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 3990	. . . . . 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 4019	. . . . 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 7457	. . . . . . 7 ⊢ ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑥) ∈ V
51		eqid 2726	. . . . . . 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 6704	. . . . . 6 ⊢ (𝑦 ∈ ((cls‘𝐽)‘𝑆) ↦ ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑥)) Fn ((cls‘𝐽)‘𝑆)
53		df-f 6558	. . . . . 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 707	. . . . 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 233	. . . 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 7124	. . . 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 233	. . 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 3136	. 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 19154	. 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 581	1 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) → ((cls‘𝐽)‘𝑆) ∈ (NrmSGrp‘𝐺))

Syntax hints:   → wi 4   ∧ wa 394   = wceq 1534   ∈ wcel 2099  ∀wral 3051   ⊆ wss 3947  ∪ cuni 4913   ↦ cmpt 5236  ran crn 5683   ↾ cres 5684   “ cima 5685   Fn wfn 6549  ⟶wf 6550  ‘cfv 6554  (class class class)co 7424  Basecbs 17213  +_gcplusg 17266  TopOpenctopn 17436  -_gcsg 18930  SubGrpcsubg 19114  NrmSGrpcnsg 19115  Topctop 22886  TopOnctopon 22903  clsccl 23013   Cn ccn 23219   ×_t ctx 23555  TopMndctmd 24065  TopGrpctgp 24066

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5290  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-cnex 11214  ax-resscn 11215  ax-1cn 11216  ax-icn 11217  ax-addcl 11218  ax-addrcl 11219  ax-mulcl 11220  ax-mulrcl 11221  ax-mulcom 11222  ax-addass 11223  ax-mulass 11224  ax-distr 11225  ax-i2m1 11226  ax-1ne0 11227  ax-1rid 11228  ax-rnegex 11229  ax-rrecex 11230  ax-cnre 11231  ax-pre-lttri 11232  ax-pre-lttrn 11233  ax-pre-ltadd 11234  ax-pre-mulgt0 11235

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-int 4955  df-iun 5003  df-iin 5004  df-br 5154  df-opab 5216  df-mpt 5237  df-tr 5271  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6312  df-ord 6379  df-on 6380  df-lim 6381  df-suc 6382  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-riota 7380  df-ov 7427  df-oprab 7428  df-mpo 7429  df-om 7877  df-1st 8003  df-2nd 8004  df-frecs 8296  df-wrecs 8327  df-recs 8401  df-rdg 8440  df-er 8734  df-map 8857  df-en 8975  df-dom 8976  df-sdom 8977  df-pnf 11300  df-mnf 11301  df-xr 11302  df-ltxr 11303  df-le 11304  df-sub 11496  df-neg 11497  df-nn 12265  df-2 12327  df-sets 17166  df-slot 17184  df-ndx 17196  df-base 17214  df-ress 17243  df-plusg 17279  df-0g 17456  df-topgen 17458  df-plusf 18632  df-mgm 18633  df-sgrp 18712  df-mnd 18728  df-grp 18931  df-minusg 18932  df-sbg 18933  df-subg 19117  df-nsg 19118  df-top 22887  df-topon 22904  df-topsp 22926  df-bases 22940  df-cld 23014  df-ntr 23015  df-cls 23016  df-cn 23222  df-cnp 23223  df-tx 23557  df-tmd 24067  df-tgp 24068

This theorem is referenced by: (None)