Theorem clsnsg 24342

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 19287	. . 3 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → 𝑆 ∈ (SubGrp‘𝐺))
2		subgntr.h	. . . 4 ⊢ 𝐽 = (TopOpen‘𝐺)
3	2	clssubg 24341	. . 3 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ((cls‘𝐽)‘𝑆) ∈ (SubGrp‘𝐺))
4	1, 3	sylan2 605	. 2 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) → ((cls‘𝐽)‘𝑆) ∈ (SubGrp‘𝐺))
5		df-ima 5672	. . . . . . 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 2762	. . . . . . . . . . . . . 14 ⊢ (Base‘𝐺) = (Base‘𝐺)
7	2, 6	tgptopon 24314	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ TopGrp → 𝐽 ∈ (TopOn‘(Base‘𝐺)))
8	7	ad2antrr 739	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → 𝐽 ∈ (TopOn‘(Base‘𝐺)))
9		topontop 23144	. . . . . . . . . . . 12 ⊢ (𝐽 ∈ (TopOn‘(Base‘𝐺)) → 𝐽 ∈ Top)
10	8, 9	syl 18	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → 𝐽 ∈ Top)
11	1	ad2antlr 740	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → 𝑆 ∈ (SubGrp‘𝐺))
12	6	subgss 19256	. . . . . . . . . . . . 13 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ (Base‘𝐺))
13	11, 12	syl 18	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → 𝑆 ⊆ (Base‘𝐺))
14		toponuni 23145	. . . . . . . . . . . . 13 ⊢ (𝐽 ∈ (TopOn‘(Base‘𝐺)) → (Base‘𝐺) = ∪ 𝐽)
15	8, 14	syl 18	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → (Base‘𝐺) = ∪ 𝐽)
16	13, 15	sseqtrd 3970	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → 𝑆 ⊆ ∪ 𝐽)
17		eqid 2762	. . . . . . . . . . . 12 ⊢ ∪ 𝐽 = ∪ 𝐽
18	17	clsss3 23290	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) → ((cls‘𝐽)‘𝑆) ⊆ ∪ 𝐽)
19	10, 16, 18	syl2anc 596	. . . . . . . . . 10 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → ((cls‘𝐽)‘𝑆) ⊆ ∪ 𝐽)
20	19, 15	sseqtrrd 3971	. . . . . . . . 9 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → ((cls‘𝐽)‘𝑆) ⊆ (Base‘𝐺))
21	20	resmptd 6040	. . . . . . . 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 5926	. . . . . . 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 2809	. . . . . 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 2762	. . . . . . . . . 10 ⊢ (+g‘𝐺) = (+g‘𝐺)
25		tgptmd 24311	. . . . . . . . . . 11 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ TopMnd)
26	25	ad2antrr 739	. . . . . . . . . 10 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → 𝐺 ∈ TopMnd)
27		simpr 490	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → 𝑥 ∈ (Base‘𝐺))
28	8, 8, 27	cnmptc 23894	. . . . . . . . . 10 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑦 ∈ (Base‘𝐺) ↦ 𝑥) ∈ (𝐽 Cn 𝐽))
29	8	cnmptid 23893	. . . . . . . . . 10 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑦 ∈ (Base‘𝐺) ↦ 𝑦) ∈ (𝐽 Cn 𝐽))
30	2, 24, 26, 8, 28, 29	cnmpt1plusg 24319	. . . . . . . . 9 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑦 ∈ (Base‘𝐺) ↦ (𝑥(+g‘𝐺)𝑦)) ∈ (𝐽 Cn 𝐽))
31		eqid 2762	. . . . . . . . . . 11 ⊢ (-g‘𝐺) = (-g‘𝐺)
32	2, 31	tgpsubcn 24322	. . . . . . . . . 10 ⊢ (𝐺 ∈ TopGrp → (-g‘𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
33	32	ad2antrr 739	. . . . . . . . 9 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → (-g‘𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
34	8, 30, 28, 33	cnmpt12f 23898	. . . . . . . 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 23503	. . . . . . . 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 596	. . . . . . 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 5672	. . . . . . . . . 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 6040	. . . . . . . . . . 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 5926	. . . . . . . . . 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 2809	. . . . . . . . 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 19288	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ 𝑆) → ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑥) ∈ 𝑆)
42	41	ad4ant234 1194	. . . . . . . . . . 11 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑦 ∈ 𝑆) → ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑥) ∈ 𝑆)
43	42	fmpttd 7112	. . . . . . . . . 10 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑦 ∈ 𝑆 ↦ ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑥)):𝑆⟶𝑆)
44	43	frnd 6715	. . . . . . . . 9 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → ran (𝑦 ∈ 𝑆 ↦ ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑥)) ⊆ 𝑆)
45	40, 44	eqsstrd 3968	. . . . . . . 8 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → ((𝑦 ∈ (Base‘𝐺) ↦ ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑥)) “ 𝑆) ⊆ 𝑆)
46	17	clsss 23285	. . . . . . . 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 1398	. . . . . . 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 3944	. . . . . 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 3969	. . . . 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 7450	. . . . . . 7 ⊢ ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑥) ∈ V
51		eqid 2762	. . . . . . 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 6679	. . . . . 6 ⊢ (𝑦 ∈ ((cls‘𝐽)‘𝑆) ↦ ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑥)) Fn ((cls‘𝐽)‘𝑆)
53		df-f 6541	. . . . . 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 722	. . . . 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 237	. . . 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 7107	. . . 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 237	. . 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 3156	. 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 19289	. 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 595	1 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) → ((cls‘𝐽)‘𝑆) ∈ (NrmSGrp‘𝐺))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3078   ⊆ wss 3902  ∪ cuni 4870   ↦ cmpt 5190  ran crn 5660   ↾ cres 5661   “ cima 5662   Fn wfn 6532  ⟶wf 6533  ‘cfv 6537  (class class class)co 7417  Basecbs 17307  +_gcplusg 17348  TopOpenctopn 17512  -_gcsg 19065  SubGrpcsubg 19249  NrmSGrpcnsg 19250  Topctop 23124  TopOnctopon 23141  clsccl 23249   Cn ccn 23455   ×_t ctx 23792  TopMndctmd 24302  TopGrpctgp 24303

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7740  ax-cnex 11184  ax-resscn 11185  ax-1cn 11186  ax-icn 11187  ax-addcl 11188  ax-addrcl 11189  ax-mulcl 11190  ax-mulrcl 11191  ax-mulcom 11192  ax-addass 11193  ax-mulass 11194  ax-distr 11195  ax-i2m1 11196  ax-1ne0 11197  ax-1rid 11198  ax-rnegex 11199  ax-rrecex 11200  ax-cnre 11201  ax-pre-lttri 11202  ax-pre-lttrn 11203  ax-pre-ltadd 11204  ax-pre-mulgt0 11205

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3064  df-ral 3079  df-rex 3089  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-int 4911  df-iun 4956  df-iin 4957  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7374  df-ov 7420  df-oprab 7421  df-mpo 7422  df-om 7867  df-1st 7990  df-2nd 7991  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-er 8700  df-map 8832  df-en 8957  df-dom 8958  df-sdom 8959  df-pnf 11273  df-mnf 11274  df-xr 11275  df-ltxr 11276  df-le 11277  df-sub 11471  df-neg 11472  df-nn 12262  df-2 12331  df-sets 17262  df-slot 17280  df-ndx 17292  df-base 17308  df-ress 17329  df-plusg 17361  df-0g 17532  df-topgen 17534  df-plusf 18735  df-mgm 18736  df-sgrp 18827  df-mnd 18843  df-grp 19066  df-minusg 19067  df-sbg 19068  df-subg 19252  df-nsg 19253  df-top 23125  df-topon 23142  df-topsp 23164  df-bases 23177  df-cld 23250  df-ntr 23251  df-cls 23252  df-cn 23458  df-cnp 23459  df-tx 23794  df-tmd 24304  df-tgp 24305

This theorem is used by: (None)