Theorem tgpt0 23552

Description: Hausdorff and T0 are equivalent for topological groups. (Contributed by Mario Carneiro, 18-Sep-2015.)

Hypothesis

Ref	Expression
tgpt1.j	⊢ 𝐽 = (TopOpen‘𝐺)

Assertion

Ref	Expression
tgpt0	⊢ (𝐺 ∈ TopGrp → (𝐽 ∈ Haus ↔ 𝐽 ∈ Kol2))

Proof of Theorem tgpt0

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

Step	Hyp	Ref	Expression
1		tgpt1.j	. . 3 ⊢ 𝐽 = (TopOpen‘𝐺)
2	1	tgpt1 23551	. 2 ⊢ (𝐺 ∈ TopGrp → (𝐽 ∈ Haus ↔ 𝐽 ∈ Fre))
3		t1t0 22781	. . 3 ⊢ (𝐽 ∈ Fre → 𝐽 ∈ Kol2)
4		eleq2 2821	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑧 → (𝑥 ∈ 𝑤 ↔ 𝑥 ∈ 𝑧))
5		eleq2 2821	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑧 → (𝑦 ∈ 𝑤 ↔ 𝑦 ∈ 𝑧))
6	4, 5	imbi12d 344	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑧 → ((𝑥 ∈ 𝑤 → 𝑦 ∈ 𝑤) ↔ (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧)))
7	6	rspccva 3608	. . . . . . . . . 10 ⊢ ((∀𝑤 ∈ 𝐽 (𝑥 ∈ 𝑤 → 𝑦 ∈ 𝑤) ∧ 𝑧 ∈ 𝐽) → (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧))
8	7	adantll 712	. . . . . . . . 9 ⊢ ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤 ∈ 𝐽 (𝑥 ∈ 𝑤 → 𝑦 ∈ 𝑤)) ∧ 𝑧 ∈ 𝐽) → (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧))
9		tgpgrp 23511	. . . . . . . . . . . . . . 15 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ Grp)
10	9	ad3antrrr 728	. . . . . . . . . . . . . 14 ⊢ ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤 ∈ 𝐽 (𝑥 ∈ 𝑤 → 𝑦 ∈ 𝑤)) ∧ (𝑧 ∈ 𝐽 ∧ 𝑦 ∈ 𝑧)) → 𝐺 ∈ Grp)
11		simpllr 774	. . . . . . . . . . . . . . 15 ⊢ ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤 ∈ 𝐽 (𝑥 ∈ 𝑤 → 𝑦 ∈ 𝑤)) ∧ (𝑧 ∈ 𝐽 ∧ 𝑦 ∈ 𝑧)) → (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)))
12	11	simprd 496	. . . . . . . . . . . . . 14 ⊢ ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤 ∈ 𝐽 (𝑥 ∈ 𝑤 → 𝑦 ∈ 𝑤)) ∧ (𝑧 ∈ 𝐽 ∧ 𝑦 ∈ 𝑧)) → 𝑦 ∈ (Base‘𝐺))
13		eqid 2731	. . . . . . . . . . . . . . 15 ⊢ (Base‘𝐺) = (Base‘𝐺)
14		eqid 2731	. . . . . . . . . . . . . . 15 ⊢ (0g‘𝐺) = (0g‘𝐺)
15		eqid 2731	. . . . . . . . . . . . . . 15 ⊢ (-g‘𝐺) = (-g‘𝐺)
16	13, 14, 15	grpsubid 18881	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ Grp ∧ 𝑦 ∈ (Base‘𝐺)) → (𝑦(-g‘𝐺)𝑦) = (0g‘𝐺))
17	10, 12, 16	syl2anc 584	. . . . . . . . . . . . 13 ⊢ ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤 ∈ 𝐽 (𝑥 ∈ 𝑤 → 𝑦 ∈ 𝑤)) ∧ (𝑧 ∈ 𝐽 ∧ 𝑦 ∈ 𝑧)) → (𝑦(-g‘𝐺)𝑦) = (0g‘𝐺))
18	17	oveq1d 7408	. . . . . . . . . . . 12 ⊢ ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤 ∈ 𝐽 (𝑥 ∈ 𝑤 → 𝑦 ∈ 𝑤)) ∧ (𝑧 ∈ 𝐽 ∧ 𝑦 ∈ 𝑧)) → ((𝑦(-g‘𝐺)𝑦)(+g‘𝐺)𝑥) = ((0g‘𝐺)(+g‘𝐺)𝑥))
19	11	simpld 495	. . . . . . . . . . . . 13 ⊢ ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤 ∈ 𝐽 (𝑥 ∈ 𝑤 → 𝑦 ∈ 𝑤)) ∧ (𝑧 ∈ 𝐽 ∧ 𝑦 ∈ 𝑧)) → 𝑥 ∈ (Base‘𝐺))
20		eqid 2731	. . . . . . . . . . . . . 14 ⊢ (+g‘𝐺) = (+g‘𝐺)
21	13, 20, 14	grplid 18827	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) → ((0g‘𝐺)(+g‘𝐺)𝑥) = 𝑥)
22	10, 19, 21	syl2anc 584	. . . . . . . . . . . 12 ⊢ ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤 ∈ 𝐽 (𝑥 ∈ 𝑤 → 𝑦 ∈ 𝑤)) ∧ (𝑧 ∈ 𝐽 ∧ 𝑦 ∈ 𝑧)) → ((0g‘𝐺)(+g‘𝐺)𝑥) = 𝑥)
23	18, 22	eqtrd 2771	. . . . . . . . . . 11 ⊢ ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤 ∈ 𝐽 (𝑥 ∈ 𝑤 → 𝑦 ∈ 𝑤)) ∧ (𝑧 ∈ 𝐽 ∧ 𝑦 ∈ 𝑧)) → ((𝑦(-g‘𝐺)𝑦)(+g‘𝐺)𝑥) = 𝑥)
24	13, 20, 15	grpnpcan 18889	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺 ∈ Grp ∧ 𝑦 ∈ (Base‘𝐺) ∧ 𝑥 ∈ (Base‘𝐺)) → ((𝑦(-g‘𝐺)𝑥)(+g‘𝐺)𝑥) = 𝑦)
25	10, 12, 19, 24	syl3anc 1371	. . . . . . . . . . . . . . 15 ⊢ ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤 ∈ 𝐽 (𝑥 ∈ 𝑤 → 𝑦 ∈ 𝑤)) ∧ (𝑧 ∈ 𝐽 ∧ 𝑦 ∈ 𝑧)) → ((𝑦(-g‘𝐺)𝑥)(+g‘𝐺)𝑥) = 𝑦)
26		simprr 771	. . . . . . . . . . . . . . 15 ⊢ ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤 ∈ 𝐽 (𝑥 ∈ 𝑤 → 𝑦 ∈ 𝑤)) ∧ (𝑧 ∈ 𝐽 ∧ 𝑦 ∈ 𝑧)) → 𝑦 ∈ 𝑧)
27	25, 26	eqeltrd 2832	. . . . . . . . . . . . . 14 ⊢ ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤 ∈ 𝐽 (𝑥 ∈ 𝑤 → 𝑦 ∈ 𝑤)) ∧ (𝑧 ∈ 𝐽 ∧ 𝑦 ∈ 𝑧)) → ((𝑦(-g‘𝐺)𝑥)(+g‘𝐺)𝑥) ∈ 𝑧)
28		oveq2 7401	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 = 𝑥 → (𝑦(-g‘𝐺)𝑎) = (𝑦(-g‘𝐺)𝑥))
29	28	oveq1d 7408	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 = 𝑥 → ((𝑦(-g‘𝐺)𝑎)(+g‘𝐺)𝑥) = ((𝑦(-g‘𝐺)𝑥)(+g‘𝐺)𝑥))
30	29	eleq1d 2817	. . . . . . . . . . . . . . 15 ⊢ (𝑎 = 𝑥 → (((𝑦(-g‘𝐺)𝑎)(+g‘𝐺)𝑥) ∈ 𝑧 ↔ ((𝑦(-g‘𝐺)𝑥)(+g‘𝐺)𝑥) ∈ 𝑧))
31		eqid 2731	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 ∈ (Base‘𝐺) ↦ ((𝑦(-g‘𝐺)𝑎)(+g‘𝐺)𝑥)) = (𝑎 ∈ (Base‘𝐺) ↦ ((𝑦(-g‘𝐺)𝑎)(+g‘𝐺)𝑥))
32	31	mptpreima 6226	. . . . . . . . . . . . . . 15 ⊢ (◡(𝑎 ∈ (Base‘𝐺) ↦ ((𝑦(-g‘𝐺)𝑎)(+g‘𝐺)𝑥)) “ 𝑧) = {𝑎 ∈ (Base‘𝐺) ∣ ((𝑦(-g‘𝐺)𝑎)(+g‘𝐺)𝑥) ∈ 𝑧}
33	30, 32	elrab2 3682	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (◡(𝑎 ∈ (Base‘𝐺) ↦ ((𝑦(-g‘𝐺)𝑎)(+g‘𝐺)𝑥)) “ 𝑧) ↔ (𝑥 ∈ (Base‘𝐺) ∧ ((𝑦(-g‘𝐺)𝑥)(+g‘𝐺)𝑥) ∈ 𝑧))
34	19, 27, 33	sylanbrc 583	. . . . . . . . . . . . 13 ⊢ ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤 ∈ 𝐽 (𝑥 ∈ 𝑤 → 𝑦 ∈ 𝑤)) ∧ (𝑧 ∈ 𝐽 ∧ 𝑦 ∈ 𝑧)) → 𝑥 ∈ (◡(𝑎 ∈ (Base‘𝐺) ↦ ((𝑦(-g‘𝐺)𝑎)(+g‘𝐺)𝑥)) “ 𝑧))
35		eleq2 2821	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = (◡(𝑎 ∈ (Base‘𝐺) ↦ ((𝑦(-g‘𝐺)𝑎)(+g‘𝐺)𝑥)) “ 𝑧) → (𝑥 ∈ 𝑤 ↔ 𝑥 ∈ (◡(𝑎 ∈ (Base‘𝐺) ↦ ((𝑦(-g‘𝐺)𝑎)(+g‘𝐺)𝑥)) “ 𝑧)))
36		eleq2 2821	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = (◡(𝑎 ∈ (Base‘𝐺) ↦ ((𝑦(-g‘𝐺)𝑎)(+g‘𝐺)𝑥)) “ 𝑧) → (𝑦 ∈ 𝑤 ↔ 𝑦 ∈ (◡(𝑎 ∈ (Base‘𝐺) ↦ ((𝑦(-g‘𝐺)𝑎)(+g‘𝐺)𝑥)) “ 𝑧)))
37	35, 36	imbi12d 344	. . . . . . . . . . . . . 14 ⊢ (𝑤 = (◡(𝑎 ∈ (Base‘𝐺) ↦ ((𝑦(-g‘𝐺)𝑎)(+g‘𝐺)𝑥)) “ 𝑧) → ((𝑥 ∈ 𝑤 → 𝑦 ∈ 𝑤) ↔ (𝑥 ∈ (◡(𝑎 ∈ (Base‘𝐺) ↦ ((𝑦(-g‘𝐺)𝑎)(+g‘𝐺)𝑥)) “ 𝑧) → 𝑦 ∈ (◡(𝑎 ∈ (Base‘𝐺) ↦ ((𝑦(-g‘𝐺)𝑎)(+g‘𝐺)𝑥)) “ 𝑧))))
38		simplr 767	. . . . . . . . . . . . . 14 ⊢ ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤 ∈ 𝐽 (𝑥 ∈ 𝑤 → 𝑦 ∈ 𝑤)) ∧ (𝑧 ∈ 𝐽 ∧ 𝑦 ∈ 𝑧)) → ∀𝑤 ∈ 𝐽 (𝑥 ∈ 𝑤 → 𝑦 ∈ 𝑤))
39		tgptmd 23512	. . . . . . . . . . . . . . . . 17 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ TopMnd)
40	39	ad3antrrr 728	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤 ∈ 𝐽 (𝑥 ∈ 𝑤 → 𝑦 ∈ 𝑤)) ∧ (𝑧 ∈ 𝐽 ∧ 𝑦 ∈ 𝑧)) → 𝐺 ∈ TopMnd)
41	1, 13	tgptopon 23515	. . . . . . . . . . . . . . . . 17 ⊢ (𝐺 ∈ TopGrp → 𝐽 ∈ (TopOn‘(Base‘𝐺)))
42	41	ad3antrrr 728	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤 ∈ 𝐽 (𝑥 ∈ 𝑤 → 𝑦 ∈ 𝑤)) ∧ (𝑧 ∈ 𝐽 ∧ 𝑦 ∈ 𝑧)) → 𝐽 ∈ (TopOn‘(Base‘𝐺)))
43	42, 42, 12	cnmptc 23095	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤 ∈ 𝐽 (𝑥 ∈ 𝑤 → 𝑦 ∈ 𝑤)) ∧ (𝑧 ∈ 𝐽 ∧ 𝑦 ∈ 𝑧)) → (𝑎 ∈ (Base‘𝐺) ↦ 𝑦) ∈ (𝐽 Cn 𝐽))
44	42	cnmptid 23094	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤 ∈ 𝐽 (𝑥 ∈ 𝑤 → 𝑦 ∈ 𝑤)) ∧ (𝑧 ∈ 𝐽 ∧ 𝑦 ∈ 𝑧)) → (𝑎 ∈ (Base‘𝐺) ↦ 𝑎) ∈ (𝐽 Cn 𝐽))
45	1, 15	tgpsubcn 23523	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐺 ∈ TopGrp → (-g‘𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
46	45	ad3antrrr 728	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤 ∈ 𝐽 (𝑥 ∈ 𝑤 → 𝑦 ∈ 𝑤)) ∧ (𝑧 ∈ 𝐽 ∧ 𝑦 ∈ 𝑧)) → (-g‘𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
47	42, 43, 44, 46	cnmpt12f 23099	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤 ∈ 𝐽 (𝑥 ∈ 𝑤 → 𝑦 ∈ 𝑤)) ∧ (𝑧 ∈ 𝐽 ∧ 𝑦 ∈ 𝑧)) → (𝑎 ∈ (Base‘𝐺) ↦ (𝑦(-g‘𝐺)𝑎)) ∈ (𝐽 Cn 𝐽))
48	42, 42, 19	cnmptc 23095	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤 ∈ 𝐽 (𝑥 ∈ 𝑤 → 𝑦 ∈ 𝑤)) ∧ (𝑧 ∈ 𝐽 ∧ 𝑦 ∈ 𝑧)) → (𝑎 ∈ (Base‘𝐺) ↦ 𝑥) ∈ (𝐽 Cn 𝐽))
49	1, 20, 40, 42, 47, 48	cnmpt1plusg 23520	. . . . . . . . . . . . . . 15 ⊢ ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤 ∈ 𝐽 (𝑥 ∈ 𝑤 → 𝑦 ∈ 𝑤)) ∧ (𝑧 ∈ 𝐽 ∧ 𝑦 ∈ 𝑧)) → (𝑎 ∈ (Base‘𝐺) ↦ ((𝑦(-g‘𝐺)𝑎)(+g‘𝐺)𝑥)) ∈ (𝐽 Cn 𝐽))
50		simprl 769	. . . . . . . . . . . . . . 15 ⊢ ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤 ∈ 𝐽 (𝑥 ∈ 𝑤 → 𝑦 ∈ 𝑤)) ∧ (𝑧 ∈ 𝐽 ∧ 𝑦 ∈ 𝑧)) → 𝑧 ∈ 𝐽)
51		cnima 22698	. . . . . . . . . . . . . . 15 ⊢ (((𝑎 ∈ (Base‘𝐺) ↦ ((𝑦(-g‘𝐺)𝑎)(+g‘𝐺)𝑥)) ∈ (𝐽 Cn 𝐽) ∧ 𝑧 ∈ 𝐽) → (◡(𝑎 ∈ (Base‘𝐺) ↦ ((𝑦(-g‘𝐺)𝑎)(+g‘𝐺)𝑥)) “ 𝑧) ∈ 𝐽)
52	49, 50, 51	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤 ∈ 𝐽 (𝑥 ∈ 𝑤 → 𝑦 ∈ 𝑤)) ∧ (𝑧 ∈ 𝐽 ∧ 𝑦 ∈ 𝑧)) → (◡(𝑎 ∈ (Base‘𝐺) ↦ ((𝑦(-g‘𝐺)𝑎)(+g‘𝐺)𝑥)) “ 𝑧) ∈ 𝐽)
53	37, 38, 52	rspcdva 3610	. . . . . . . . . . . . 13 ⊢ ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤 ∈ 𝐽 (𝑥 ∈ 𝑤 → 𝑦 ∈ 𝑤)) ∧ (𝑧 ∈ 𝐽 ∧ 𝑦 ∈ 𝑧)) → (𝑥 ∈ (◡(𝑎 ∈ (Base‘𝐺) ↦ ((𝑦(-g‘𝐺)𝑎)(+g‘𝐺)𝑥)) “ 𝑧) → 𝑦 ∈ (◡(𝑎 ∈ (Base‘𝐺) ↦ ((𝑦(-g‘𝐺)𝑎)(+g‘𝐺)𝑥)) “ 𝑧)))
54	34, 53	mpd 15	. . . . . . . . . . . 12 ⊢ ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤 ∈ 𝐽 (𝑥 ∈ 𝑤 → 𝑦 ∈ 𝑤)) ∧ (𝑧 ∈ 𝐽 ∧ 𝑦 ∈ 𝑧)) → 𝑦 ∈ (◡(𝑎 ∈ (Base‘𝐺) ↦ ((𝑦(-g‘𝐺)𝑎)(+g‘𝐺)𝑥)) “ 𝑧))
55		oveq2 7401	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 = 𝑦 → (𝑦(-g‘𝐺)𝑎) = (𝑦(-g‘𝐺)𝑦))
56	55	oveq1d 7408	. . . . . . . . . . . . . . 15 ⊢ (𝑎 = 𝑦 → ((𝑦(-g‘𝐺)𝑎)(+g‘𝐺)𝑥) = ((𝑦(-g‘𝐺)𝑦)(+g‘𝐺)𝑥))
57	56	eleq1d 2817	. . . . . . . . . . . . . 14 ⊢ (𝑎 = 𝑦 → (((𝑦(-g‘𝐺)𝑎)(+g‘𝐺)𝑥) ∈ 𝑧 ↔ ((𝑦(-g‘𝐺)𝑦)(+g‘𝐺)𝑥) ∈ 𝑧))
58	57, 32	elrab2 3682	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (◡(𝑎 ∈ (Base‘𝐺) ↦ ((𝑦(-g‘𝐺)𝑎)(+g‘𝐺)𝑥)) “ 𝑧) ↔ (𝑦 ∈ (Base‘𝐺) ∧ ((𝑦(-g‘𝐺)𝑦)(+g‘𝐺)𝑥) ∈ 𝑧))
59	58	simprbi 497	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ (◡(𝑎 ∈ (Base‘𝐺) ↦ ((𝑦(-g‘𝐺)𝑎)(+g‘𝐺)𝑥)) “ 𝑧) → ((𝑦(-g‘𝐺)𝑦)(+g‘𝐺)𝑥) ∈ 𝑧)
60	54, 59	syl 17	. . . . . . . . . . 11 ⊢ ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤 ∈ 𝐽 (𝑥 ∈ 𝑤 → 𝑦 ∈ 𝑤)) ∧ (𝑧 ∈ 𝐽 ∧ 𝑦 ∈ 𝑧)) → ((𝑦(-g‘𝐺)𝑦)(+g‘𝐺)𝑥) ∈ 𝑧)
61	23, 60	eqeltrrd 2833	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤 ∈ 𝐽 (𝑥 ∈ 𝑤 → 𝑦 ∈ 𝑤)) ∧ (𝑧 ∈ 𝐽 ∧ 𝑦 ∈ 𝑧)) → 𝑥 ∈ 𝑧)
62	61	expr 457	. . . . . . . . 9 ⊢ ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤 ∈ 𝐽 (𝑥 ∈ 𝑤 → 𝑦 ∈ 𝑤)) ∧ 𝑧 ∈ 𝐽) → (𝑦 ∈ 𝑧 → 𝑥 ∈ 𝑧))
63	8, 62	impbid 211	. . . . . . . 8 ⊢ ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤 ∈ 𝐽 (𝑥 ∈ 𝑤 → 𝑦 ∈ 𝑤)) ∧ 𝑧 ∈ 𝐽) → (𝑥 ∈ 𝑧 ↔ 𝑦 ∈ 𝑧))
64	63	ralrimiva 3145	. . . . . . 7 ⊢ (((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤 ∈ 𝐽 (𝑥 ∈ 𝑤 → 𝑦 ∈ 𝑤)) → ∀𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 ↔ 𝑦 ∈ 𝑧))
65	64	ex 413	. . . . . 6 ⊢ ((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) → (∀𝑤 ∈ 𝐽 (𝑥 ∈ 𝑤 → 𝑦 ∈ 𝑤) → ∀𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 ↔ 𝑦 ∈ 𝑧)))
66	65	imim1d 82	. . . . 5 ⊢ ((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) → ((∀𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 ↔ 𝑦 ∈ 𝑧) → 𝑥 = 𝑦) → (∀𝑤 ∈ 𝐽 (𝑥 ∈ 𝑤 → 𝑦 ∈ 𝑤) → 𝑥 = 𝑦)))
67	66	ralimdvva 3203	. . . 4 ⊢ (𝐺 ∈ TopGrp → (∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(∀𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 ↔ 𝑦 ∈ 𝑧) → 𝑥 = 𝑦) → ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(∀𝑤 ∈ 𝐽 (𝑥 ∈ 𝑤 → 𝑦 ∈ 𝑤) → 𝑥 = 𝑦)))
68		ist0-2 22777	. . . . 5 ⊢ (𝐽 ∈ (TopOn‘(Base‘𝐺)) → (𝐽 ∈ Kol2 ↔ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(∀𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 ↔ 𝑦 ∈ 𝑧) → 𝑥 = 𝑦)))
69	41, 68	syl 17	. . . 4 ⊢ (𝐺 ∈ TopGrp → (𝐽 ∈ Kol2 ↔ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(∀𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 ↔ 𝑦 ∈ 𝑧) → 𝑥 = 𝑦)))
70		ist1-2 22780	. . . . 5 ⊢ (𝐽 ∈ (TopOn‘(Base‘𝐺)) → (𝐽 ∈ Fre ↔ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(∀𝑤 ∈ 𝐽 (𝑥 ∈ 𝑤 → 𝑦 ∈ 𝑤) → 𝑥 = 𝑦)))
71	41, 70	syl 17	. . . 4 ⊢ (𝐺 ∈ TopGrp → (𝐽 ∈ Fre ↔ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(∀𝑤 ∈ 𝐽 (𝑥 ∈ 𝑤 → 𝑦 ∈ 𝑤) → 𝑥 = 𝑦)))
72	67, 69, 71	3imtr4d 293	. . 3 ⊢ (𝐺 ∈ TopGrp → (𝐽 ∈ Kol2 → 𝐽 ∈ Fre))
73	3, 72	impbid2 225	. 2 ⊢ (𝐺 ∈ TopGrp → (𝐽 ∈ Fre ↔ 𝐽 ∈ Kol2))
74	2, 73	bitrd 278	1 ⊢ (𝐺 ∈ TopGrp → (𝐽 ∈ Haus ↔ 𝐽 ∈ Kol2))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  ∀wral 3060   ↦ cmpt 5224  ^◡ccnv 5668   “ cima 5672  ‘cfv 6532  (class class class)co 7393  Basecbs 17126  +_gcplusg 17179  TopOpenctopn 17349  0_gc0g 17367  Grpcgrp 18794  -_gcsg 18796  TopOnctopon 22341   Cn ccn 22657  Kol2ct0 22739  Frect1 22740  Hauscha 22741   ×_t ctx 22993  TopMndctmd 23503  TopGrpctgp 23504

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7708

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6484  df-fun 6534  df-fn 6535  df-f 6536  df-f1 6537  df-fo 6538  df-f1o 6539  df-fv 6540  df-riota 7349  df-ov 7396  df-oprab 7397  df-mpo 7398  df-1st 7957  df-2nd 7958  df-map 8805  df-0g 17369  df-topgen 17371  df-plusf 18542  df-mgm 18543  df-sgrp 18592  df-mnd 18603  df-grp 18797  df-minusg 18798  df-sbg 18799  df-top 22325  df-topon 22342  df-topsp 22364  df-bases 22378  df-cld 22452  df-cn 22660  df-cnp 22661  df-t0 22746  df-t1 22747  df-haus 22748  df-tx 22995  df-tmd 23505  df-tgp 23506

This theorem is referenced by: (None)