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Theorem tgpt0 24093

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 24092	. 2 ⊢ (𝐺 ∈ TopGrp → (𝐽 ∈ Haus ↔ 𝐽 ∈ Fre))
3		t1t0 23322	. . 3 ⊢ (𝐽 ∈ Fre → 𝐽 ∈ Kol2)
4		eleq2 2826	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑧 → (𝑥 ∈ 𝑤 ↔ 𝑥 ∈ 𝑧))
5		eleq2 2826	. . . . . . . . . . . 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 3564	. . . . . . . . . 10 ⊢ ((∀𝑤 ∈ 𝐽 (𝑥 ∈ 𝑤 → 𝑦 ∈ 𝑤) ∧ 𝑧 ∈ 𝐽) → (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧))
8	7	adantll 715	. . . . . . . . 9 ⊢ ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤 ∈ 𝐽 (𝑥 ∈ 𝑤 → 𝑦 ∈ 𝑤)) ∧ 𝑧 ∈ 𝐽) → (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧))
9		tgpgrp 24052	. . . . . . . . . . . . . . 15 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ Grp)
10	9	ad3antrrr 731	. . . . . . . . . . . . . 14 ⊢ ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤 ∈ 𝐽 (𝑥 ∈ 𝑤 → 𝑦 ∈ 𝑤)) ∧ (𝑧 ∈ 𝐽 ∧ 𝑦 ∈ 𝑧)) → 𝐺 ∈ Grp)
11		simpllr 776	. . . . . . . . . . . . . . 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 495	. . . . . . . . . . . . . 14 ⊢ ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤 ∈ 𝐽 (𝑥 ∈ 𝑤 → 𝑦 ∈ 𝑤)) ∧ (𝑧 ∈ 𝐽 ∧ 𝑦 ∈ 𝑧)) → 𝑦 ∈ (Base‘𝐺))
13		eqid 2737	. . . . . . . . . . . . . . 15 ⊢ (Base‘𝐺) = (Base‘𝐺)
14		eqid 2737	. . . . . . . . . . . . . . 15 ⊢ (0_g‘𝐺) = (0_g‘𝐺)
15		eqid 2737	. . . . . . . . . . . . . . 15 ⊢ (-_g‘𝐺) = (-_g‘𝐺)
16	13, 14, 15	grpsubid 18989	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ Grp ∧ 𝑦 ∈ (Base‘𝐺)) → (𝑦(-_g‘𝐺)𝑦) = (0_g‘𝐺))
17	10, 12, 16	syl2anc 585	. . . . . . . . . . . . 13 ⊢ ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤 ∈ 𝐽 (𝑥 ∈ 𝑤 → 𝑦 ∈ 𝑤)) ∧ (𝑧 ∈ 𝐽 ∧ 𝑦 ∈ 𝑧)) → (𝑦(-_g‘𝐺)𝑦) = (0_g‘𝐺))
18	17	oveq1d 7373	. . . . . . . . . . . 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) = ((0_g‘𝐺)(+_g‘𝐺)𝑥))
19	11	simpld 494	. . . . . . . . . . . . 13 ⊢ ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤 ∈ 𝐽 (𝑥 ∈ 𝑤 → 𝑦 ∈ 𝑤)) ∧ (𝑧 ∈ 𝐽 ∧ 𝑦 ∈ 𝑧)) → 𝑥 ∈ (Base‘𝐺))
20		eqid 2737	. . . . . . . . . . . . . 14 ⊢ (+_g‘𝐺) = (+_g‘𝐺)
21	13, 20, 14	grplid 18932	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) → ((0_g‘𝐺)(+_g‘𝐺)𝑥) = 𝑥)
22	10, 19, 21	syl2anc 585	. . . . . . . . . . . 12 ⊢ ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤 ∈ 𝐽 (𝑥 ∈ 𝑤 → 𝑦 ∈ 𝑤)) ∧ (𝑧 ∈ 𝐽 ∧ 𝑦 ∈ 𝑧)) → ((0_g‘𝐺)(+_g‘𝐺)𝑥) = 𝑥)
23	18, 22	eqtrd 2772	. . . . . . . . . . 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 18997	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺 ∈ Grp ∧ 𝑦 ∈ (Base‘𝐺) ∧ 𝑥 ∈ (Base‘𝐺)) → ((𝑦(-_g‘𝐺)𝑥)(+_g‘𝐺)𝑥) = 𝑦)
25	10, 12, 19, 24	syl3anc 1374	. . . . . . . . . . . . . . 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 773	. . . . . . . . . . . . . . 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 2837	. . . . . . . . . . . . . 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 7366	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 = 𝑥 → (𝑦(-_g‘𝐺)𝑎) = (𝑦(-_g‘𝐺)𝑥))
29	28	oveq1d 7373	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 = 𝑥 → ((𝑦(-_g‘𝐺)𝑎)(+_g‘𝐺)𝑥) = ((𝑦(-_g‘𝐺)𝑥)(+_g‘𝐺)𝑥))
30	29	eleq1d 2822	. . . . . . . . . . . . . . 15 ⊢ (𝑎 = 𝑥 → (((𝑦(-_g‘𝐺)𝑎)(+_g‘𝐺)𝑥) ∈ 𝑧 ↔ ((𝑦(-_g‘𝐺)𝑥)(+_g‘𝐺)𝑥) ∈ 𝑧))
31		eqid 2737	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 ∈ (Base‘𝐺) ↦ ((𝑦(-_g‘𝐺)𝑎)(+_g‘𝐺)𝑥)) = (𝑎 ∈ (Base‘𝐺) ↦ ((𝑦(-_g‘𝐺)𝑎)(+_g‘𝐺)𝑥))
32	31	mptpreima 6194	. . . . . . . . . . . . . . 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 3638	. . . . . . . . . . . . . 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 584	. . . . . . . . . . . . 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 2826	. . . . . . . . . . . . . . 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 2826	. . . . . . . . . . . . . . 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 769	. . . . . . . . . . . . . 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 24053	. . . . . . . . . . . . . . . . 17 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ TopMnd)
40	39	ad3antrrr 731	. . . . . . . . . . . . . . . 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 24056	. . . . . . . . . . . . . . . . 17 ⊢ (𝐺 ∈ TopGrp → 𝐽 ∈ (TopOn‘(Base‘𝐺)))
42	41	ad3antrrr 731	. . . . . . . . . . . . . . . 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 23636	. . . . . . . . . . . . . . . . 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 23635	. . . . . . . . . . . . . . . . 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 24064	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐺 ∈ TopGrp → (-_g‘𝐺) ∈ ((𝐽 ×_t 𝐽) Cn 𝐽))
46	45	ad3antrrr 731	. . . . . . . . . . . . . . . . 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 23640	. . . . . . . . . . . . . . . 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 23636	. . . . . . . . . . . . . . . 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 24061	. . . . . . . . . . . . . . 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 771	. . . . . . . . . . . . . . 15 ⊢ ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤 ∈ 𝐽 (𝑥 ∈ 𝑤 → 𝑦 ∈ 𝑤)) ∧ (𝑧 ∈ 𝐽 ∧ 𝑦 ∈ 𝑧)) → 𝑧 ∈ 𝐽)
51		cnima 23239	. . . . . . . . . . . . . . 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 585	. . . . . . . . . . . . . 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 3566	. . . . . . . . . . . . 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 7366	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 = 𝑦 → (𝑦(-_g‘𝐺)𝑎) = (𝑦(-_g‘𝐺)𝑦))
56	55	oveq1d 7373	. . . . . . . . . . . . . . 15 ⊢ (𝑎 = 𝑦 → ((𝑦(-_g‘𝐺)𝑎)(+_g‘𝐺)𝑥) = ((𝑦(-_g‘𝐺)𝑦)(+_g‘𝐺)𝑥))
57	56	eleq1d 2822	. . . . . . . . . . . . . 14 ⊢ (𝑎 = 𝑦 → (((𝑦(-_g‘𝐺)𝑎)(+_g‘𝐺)𝑥) ∈ 𝑧 ↔ ((𝑦(-_g‘𝐺)𝑦)(+_g‘𝐺)𝑥) ∈ 𝑧))
58	57, 32	elrab2 3638	. . . . . . . . . . . . 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 2838	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤 ∈ 𝐽 (𝑥 ∈ 𝑤 → 𝑦 ∈ 𝑤)) ∧ (𝑧 ∈ 𝐽 ∧ 𝑦 ∈ 𝑧)) → 𝑥 ∈ 𝑧)
62	61	expr 456	. . . . . . . . 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 212	. . . . . . . 8 ⊢ ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤 ∈ 𝐽 (𝑥 ∈ 𝑤 → 𝑦 ∈ 𝑤)) ∧ 𝑧 ∈ 𝐽) → (𝑥 ∈ 𝑧 ↔ 𝑦 ∈ 𝑧))
64	63	ralrimiva 3130	. . . . . . 7 ⊢ (((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤 ∈ 𝐽 (𝑥 ∈ 𝑤 → 𝑦 ∈ 𝑤)) → ∀𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 ↔ 𝑦 ∈ 𝑧))
65	64	ex 412	. . . . . 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 3185	. . . 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 23318	. . . . 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 23321	. . . . 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 294	. . 3 ⊢ (𝐺 ∈ TopGrp → (𝐽 ∈ Kol2 → 𝐽 ∈ Fre))
73	3, 72	impbid2 226	. 2 ⊢ (𝐺 ∈ TopGrp → (𝐽 ∈ Fre ↔ 𝐽 ∈ Kol2))
74	2, 73	bitrd 279	1 ⊢ (𝐺 ∈ TopGrp → (𝐽 ∈ Haus ↔ 𝐽 ∈ Kol2))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ∀wral 3052 ↦ cmpt 5167 ^◡ccnv 5621 “ cima 5625 ‘cfv 6490 (class class class)co 7358 Basecbs 17168 +_gcplusg 17209 TopOpenctopn 17373 0_gc0g 17391 Grpcgrp 18898 -_gcsg 18900 TopOnctopon 22884 Cn ccn 23198 Kol2ct0 23280 Frect1 23281 Hauscha 23282 ×_t ctx 23534 TopMndctmd 24044 TopGrpctgp 24045

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-sep 5231 ax-nul 5241 ax-pow 5300 ax-pr 5368 ax-un 7680

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 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-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-id 5517 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-1st 7933 df-2nd 7934 df-map 8766 df-0g 17393 df-topgen 17395 df-plusf 18596 df-mgm 18597 df-sgrp 18676 df-mnd 18692 df-grp 18901 df-minusg 18902 df-sbg 18903 df-top 22868 df-topon 22885 df-topsp 22907 df-bases 22920 df-cld 22993 df-cn 23201 df-cnp 23202 df-t0 23287 df-t1 23288 df-haus 23289 df-tx 23536 df-tmd 24046 df-tgp 24047

This theorem is referenced by: (None)

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