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Theorem tgpt0 22656
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.
StepHypRef Expression
1 tgpt1.j . . 3 𝐽 = (TopOpen‘𝐺)
21tgpt1 22655 . 2 (𝐺 ∈ TopGrp → (𝐽 ∈ Haus ↔ 𝐽 ∈ Fre))
3 t1t0 21886 . . 3 (𝐽 ∈ Fre → 𝐽 ∈ Kol2)
4 eleq2 2901 . . . . . . . . . . . 12 (𝑤 = 𝑧 → (𝑥𝑤𝑥𝑧))
5 eleq2 2901 . . . . . . . . . . . 12 (𝑤 = 𝑧 → (𝑦𝑤𝑦𝑧))
64, 5imbi12d 346 . . . . . . . . . . 11 (𝑤 = 𝑧 → ((𝑥𝑤𝑦𝑤) ↔ (𝑥𝑧𝑦𝑧)))
76rspccva 3621 . . . . . . . . . 10 ((∀𝑤𝐽 (𝑥𝑤𝑦𝑤) ∧ 𝑧𝐽) → (𝑥𝑧𝑦𝑧))
87adantll 710 . . . . . . . . 9 ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) ∧ 𝑧𝐽) → (𝑥𝑧𝑦𝑧))
9 tgpgrp 22616 . . . . . . . . . . . . . . 15 (𝐺 ∈ TopGrp → 𝐺 ∈ Grp)
109ad3antrrr 726 . . . . . . . . . . . . . 14 ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) ∧ (𝑧𝐽𝑦𝑧)) → 𝐺 ∈ Grp)
11 simpllr 772 . . . . . . . . . . . . . . 15 ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) ∧ (𝑧𝐽𝑦𝑧)) → (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)))
1211simprd 496 . . . . . . . . . . . . . 14 ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) ∧ (𝑧𝐽𝑦𝑧)) → 𝑦 ∈ (Base‘𝐺))
13 eqid 2821 . . . . . . . . . . . . . . 15 (Base‘𝐺) = (Base‘𝐺)
14 eqid 2821 . . . . . . . . . . . . . . 15 (0g𝐺) = (0g𝐺)
15 eqid 2821 . . . . . . . . . . . . . . 15 (-g𝐺) = (-g𝐺)
1613, 14, 15grpsubid 18123 . . . . . . . . . . . . . 14 ((𝐺 ∈ Grp ∧ 𝑦 ∈ (Base‘𝐺)) → (𝑦(-g𝐺)𝑦) = (0g𝐺))
1710, 12, 16syl2anc 584 . . . . . . . . . . . . 13 ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) ∧ (𝑧𝐽𝑦𝑧)) → (𝑦(-g𝐺)𝑦) = (0g𝐺))
1817oveq1d 7160 . . . . . . . . . . . 12 ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) ∧ (𝑧𝐽𝑦𝑧)) → ((𝑦(-g𝐺)𝑦)(+g𝐺)𝑥) = ((0g𝐺)(+g𝐺)𝑥))
1911simpld 495 . . . . . . . . . . . . 13 ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) ∧ (𝑧𝐽𝑦𝑧)) → 𝑥 ∈ (Base‘𝐺))
20 eqid 2821 . . . . . . . . . . . . . 14 (+g𝐺) = (+g𝐺)
2113, 20, 14grplid 18073 . . . . . . . . . . . . 13 ((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) → ((0g𝐺)(+g𝐺)𝑥) = 𝑥)
2210, 19, 21syl2anc 584 . . . . . . . . . . . 12 ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) ∧ (𝑧𝐽𝑦𝑧)) → ((0g𝐺)(+g𝐺)𝑥) = 𝑥)
2318, 22eqtrd 2856 . . . . . . . . . . 11 ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) ∧ (𝑧𝐽𝑦𝑧)) → ((𝑦(-g𝐺)𝑦)(+g𝐺)𝑥) = 𝑥)
2413, 20, 15grpnpcan 18131 . . . . . . . . . . . . . . . 16 ((𝐺 ∈ Grp ∧ 𝑦 ∈ (Base‘𝐺) ∧ 𝑥 ∈ (Base‘𝐺)) → ((𝑦(-g𝐺)𝑥)(+g𝐺)𝑥) = 𝑦)
2510, 12, 19, 24syl3anc 1363 . . . . . . . . . . . . . . 15 ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) ∧ (𝑧𝐽𝑦𝑧)) → ((𝑦(-g𝐺)𝑥)(+g𝐺)𝑥) = 𝑦)
26 simprr 769 . . . . . . . . . . . . . . 15 ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) ∧ (𝑧𝐽𝑦𝑧)) → 𝑦𝑧)
2725, 26eqeltrd 2913 . . . . . . . . . . . . . 14 ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) ∧ (𝑧𝐽𝑦𝑧)) → ((𝑦(-g𝐺)𝑥)(+g𝐺)𝑥) ∈ 𝑧)
28 oveq2 7153 . . . . . . . . . . . . . . . . 17 (𝑎 = 𝑥 → (𝑦(-g𝐺)𝑎) = (𝑦(-g𝐺)𝑥))
2928oveq1d 7160 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑥 → ((𝑦(-g𝐺)𝑎)(+g𝐺)𝑥) = ((𝑦(-g𝐺)𝑥)(+g𝐺)𝑥))
3029eleq1d 2897 . . . . . . . . . . . . . . 15 (𝑎 = 𝑥 → (((𝑦(-g𝐺)𝑎)(+g𝐺)𝑥) ∈ 𝑧 ↔ ((𝑦(-g𝐺)𝑥)(+g𝐺)𝑥) ∈ 𝑧))
31 eqid 2821 . . . . . . . . . . . . . . . 16 (𝑎 ∈ (Base‘𝐺) ↦ ((𝑦(-g𝐺)𝑎)(+g𝐺)𝑥)) = (𝑎 ∈ (Base‘𝐺) ↦ ((𝑦(-g𝐺)𝑎)(+g𝐺)𝑥))
3231mptpreima 6086 . . . . . . . . . . . . . . 15 ((𝑎 ∈ (Base‘𝐺) ↦ ((𝑦(-g𝐺)𝑎)(+g𝐺)𝑥)) “ 𝑧) = {𝑎 ∈ (Base‘𝐺) ∣ ((𝑦(-g𝐺)𝑎)(+g𝐺)𝑥) ∈ 𝑧}
3330, 32elrab2 3682 . . . . . . . . . . . . . 14 (𝑥 ∈ ((𝑎 ∈ (Base‘𝐺) ↦ ((𝑦(-g𝐺)𝑎)(+g𝐺)𝑥)) “ 𝑧) ↔ (𝑥 ∈ (Base‘𝐺) ∧ ((𝑦(-g𝐺)𝑥)(+g𝐺)𝑥) ∈ 𝑧))
3419, 27, 33sylanbrc 583 . . . . . . . . . . . . 13 ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) ∧ (𝑧𝐽𝑦𝑧)) → 𝑥 ∈ ((𝑎 ∈ (Base‘𝐺) ↦ ((𝑦(-g𝐺)𝑎)(+g𝐺)𝑥)) “ 𝑧))
35 eleq2 2901 . . . . . . . . . . . . . . 15 (𝑤 = ((𝑎 ∈ (Base‘𝐺) ↦ ((𝑦(-g𝐺)𝑎)(+g𝐺)𝑥)) “ 𝑧) → (𝑥𝑤𝑥 ∈ ((𝑎 ∈ (Base‘𝐺) ↦ ((𝑦(-g𝐺)𝑎)(+g𝐺)𝑥)) “ 𝑧)))
36 eleq2 2901 . . . . . . . . . . . . . . 15 (𝑤 = ((𝑎 ∈ (Base‘𝐺) ↦ ((𝑦(-g𝐺)𝑎)(+g𝐺)𝑥)) “ 𝑧) → (𝑦𝑤𝑦 ∈ ((𝑎 ∈ (Base‘𝐺) ↦ ((𝑦(-g𝐺)𝑎)(+g𝐺)𝑥)) “ 𝑧)))
3735, 36imbi12d 346 . . . . . . . . . . . . . 14 (𝑤 = ((𝑎 ∈ (Base‘𝐺) ↦ ((𝑦(-g𝐺)𝑎)(+g𝐺)𝑥)) “ 𝑧) → ((𝑥𝑤𝑦𝑤) ↔ (𝑥 ∈ ((𝑎 ∈ (Base‘𝐺) ↦ ((𝑦(-g𝐺)𝑎)(+g𝐺)𝑥)) “ 𝑧) → 𝑦 ∈ ((𝑎 ∈ (Base‘𝐺) ↦ ((𝑦(-g𝐺)𝑎)(+g𝐺)𝑥)) “ 𝑧))))
38 simplr 765 . . . . . . . . . . . . . 14 ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) ∧ (𝑧𝐽𝑦𝑧)) → ∀𝑤𝐽 (𝑥𝑤𝑦𝑤))
39 tgptmd 22617 . . . . . . . . . . . . . . . . 17 (𝐺 ∈ TopGrp → 𝐺 ∈ TopMnd)
4039ad3antrrr 726 . . . . . . . . . . . . . . . 16 ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) ∧ (𝑧𝐽𝑦𝑧)) → 𝐺 ∈ TopMnd)
411, 13tgptopon 22620 . . . . . . . . . . . . . . . . 17 (𝐺 ∈ TopGrp → 𝐽 ∈ (TopOn‘(Base‘𝐺)))
4241ad3antrrr 726 . . . . . . . . . . . . . . . 16 ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) ∧ (𝑧𝐽𝑦𝑧)) → 𝐽 ∈ (TopOn‘(Base‘𝐺)))
4342, 42, 12cnmptc 22200 . . . . . . . . . . . . . . . . 17 ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) ∧ (𝑧𝐽𝑦𝑧)) → (𝑎 ∈ (Base‘𝐺) ↦ 𝑦) ∈ (𝐽 Cn 𝐽))
4442cnmptid 22199 . . . . . . . . . . . . . . . . 17 ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) ∧ (𝑧𝐽𝑦𝑧)) → (𝑎 ∈ (Base‘𝐺) ↦ 𝑎) ∈ (𝐽 Cn 𝐽))
451, 15tgpsubcn 22628 . . . . . . . . . . . . . . . . . 18 (𝐺 ∈ TopGrp → (-g𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
4645ad3antrrr 726 . . . . . . . . . . . . . . . . 17 ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) ∧ (𝑧𝐽𝑦𝑧)) → (-g𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
4742, 43, 44, 46cnmpt12f 22204 . . . . . . . . . . . . . . . 16 ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) ∧ (𝑧𝐽𝑦𝑧)) → (𝑎 ∈ (Base‘𝐺) ↦ (𝑦(-g𝐺)𝑎)) ∈ (𝐽 Cn 𝐽))
4842, 42, 19cnmptc 22200 . . . . . . . . . . . . . . . 16 ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) ∧ (𝑧𝐽𝑦𝑧)) → (𝑎 ∈ (Base‘𝐺) ↦ 𝑥) ∈ (𝐽 Cn 𝐽))
491, 20, 40, 42, 47, 48cnmpt1plusg 22625 . . . . . . . . . . . . . . 15 ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) ∧ (𝑧𝐽𝑦𝑧)) → (𝑎 ∈ (Base‘𝐺) ↦ ((𝑦(-g𝐺)𝑎)(+g𝐺)𝑥)) ∈ (𝐽 Cn 𝐽))
50 simprl 767 . . . . . . . . . . . . . . 15 ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) ∧ (𝑧𝐽𝑦𝑧)) → 𝑧𝐽)
51 cnima 21803 . . . . . . . . . . . . . . 15 (((𝑎 ∈ (Base‘𝐺) ↦ ((𝑦(-g𝐺)𝑎)(+g𝐺)𝑥)) ∈ (𝐽 Cn 𝐽) ∧ 𝑧𝐽) → ((𝑎 ∈ (Base‘𝐺) ↦ ((𝑦(-g𝐺)𝑎)(+g𝐺)𝑥)) “ 𝑧) ∈ 𝐽)
5249, 50, 51syl2anc 584 . . . . . . . . . . . . . 14 ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) ∧ (𝑧𝐽𝑦𝑧)) → ((𝑎 ∈ (Base‘𝐺) ↦ ((𝑦(-g𝐺)𝑎)(+g𝐺)𝑥)) “ 𝑧) ∈ 𝐽)
5337, 38, 52rspcdva 3624 . . . . . . . . . . . . 13 ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) ∧ (𝑧𝐽𝑦𝑧)) → (𝑥 ∈ ((𝑎 ∈ (Base‘𝐺) ↦ ((𝑦(-g𝐺)𝑎)(+g𝐺)𝑥)) “ 𝑧) → 𝑦 ∈ ((𝑎 ∈ (Base‘𝐺) ↦ ((𝑦(-g𝐺)𝑎)(+g𝐺)𝑥)) “ 𝑧)))
5434, 53mpd 15 . . . . . . . . . . . 12 ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) ∧ (𝑧𝐽𝑦𝑧)) → 𝑦 ∈ ((𝑎 ∈ (Base‘𝐺) ↦ ((𝑦(-g𝐺)𝑎)(+g𝐺)𝑥)) “ 𝑧))
55 oveq2 7153 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑦 → (𝑦(-g𝐺)𝑎) = (𝑦(-g𝐺)𝑦))
5655oveq1d 7160 . . . . . . . . . . . . . . 15 (𝑎 = 𝑦 → ((𝑦(-g𝐺)𝑎)(+g𝐺)𝑥) = ((𝑦(-g𝐺)𝑦)(+g𝐺)𝑥))
5756eleq1d 2897 . . . . . . . . . . . . . 14 (𝑎 = 𝑦 → (((𝑦(-g𝐺)𝑎)(+g𝐺)𝑥) ∈ 𝑧 ↔ ((𝑦(-g𝐺)𝑦)(+g𝐺)𝑥) ∈ 𝑧))
5857, 32elrab2 3682 . . . . . . . . . . . . 13 (𝑦 ∈ ((𝑎 ∈ (Base‘𝐺) ↦ ((𝑦(-g𝐺)𝑎)(+g𝐺)𝑥)) “ 𝑧) ↔ (𝑦 ∈ (Base‘𝐺) ∧ ((𝑦(-g𝐺)𝑦)(+g𝐺)𝑥) ∈ 𝑧))
5958simprbi 497 . . . . . . . . . . . 12 (𝑦 ∈ ((𝑎 ∈ (Base‘𝐺) ↦ ((𝑦(-g𝐺)𝑎)(+g𝐺)𝑥)) “ 𝑧) → ((𝑦(-g𝐺)𝑦)(+g𝐺)𝑥) ∈ 𝑧)
6054, 59syl 17 . . . . . . . . . . 11 ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) ∧ (𝑧𝐽𝑦𝑧)) → ((𝑦(-g𝐺)𝑦)(+g𝐺)𝑥) ∈ 𝑧)
6123, 60eqeltrrd 2914 . . . . . . . . . 10 ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) ∧ (𝑧𝐽𝑦𝑧)) → 𝑥𝑧)
6261expr 457 . . . . . . . . 9 ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) ∧ 𝑧𝐽) → (𝑦𝑧𝑥𝑧))
638, 62impbid 213 . . . . . . . 8 ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) ∧ 𝑧𝐽) → (𝑥𝑧𝑦𝑧))
6463ralrimiva 3182 . . . . . . 7 (((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) → ∀𝑧𝐽 (𝑥𝑧𝑦𝑧))
6564ex 413 . . . . . 6 ((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) → (∀𝑤𝐽 (𝑥𝑤𝑦𝑤) → ∀𝑧𝐽 (𝑥𝑧𝑦𝑧)))
6665imim1d 82 . . . . 5 ((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) → ((∀𝑧𝐽 (𝑥𝑧𝑦𝑧) → 𝑥 = 𝑦) → (∀𝑤𝐽 (𝑥𝑤𝑦𝑤) → 𝑥 = 𝑦)))
6766ralimdvva 3179 . . . 4 (𝐺 ∈ TopGrp → (∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(∀𝑧𝐽 (𝑥𝑧𝑦𝑧) → 𝑥 = 𝑦) → ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(∀𝑤𝐽 (𝑥𝑤𝑦𝑤) → 𝑥 = 𝑦)))
68 ist0-2 21882 . . . . 5 (𝐽 ∈ (TopOn‘(Base‘𝐺)) → (𝐽 ∈ Kol2 ↔ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(∀𝑧𝐽 (𝑥𝑧𝑦𝑧) → 𝑥 = 𝑦)))
6941, 68syl 17 . . . 4 (𝐺 ∈ TopGrp → (𝐽 ∈ Kol2 ↔ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(∀𝑧𝐽 (𝑥𝑧𝑦𝑧) → 𝑥 = 𝑦)))
70 ist1-2 21885 . . . . 5 (𝐽 ∈ (TopOn‘(Base‘𝐺)) → (𝐽 ∈ Fre ↔ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(∀𝑤𝐽 (𝑥𝑤𝑦𝑤) → 𝑥 = 𝑦)))
7141, 70syl 17 . . . 4 (𝐺 ∈ TopGrp → (𝐽 ∈ Fre ↔ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(∀𝑤𝐽 (𝑥𝑤𝑦𝑤) → 𝑥 = 𝑦)))
7267, 69, 713imtr4d 295 . . 3 (𝐺 ∈ TopGrp → (𝐽 ∈ Kol2 → 𝐽 ∈ Fre))
733, 72impbid2 227 . 2 (𝐺 ∈ TopGrp → (𝐽 ∈ Fre ↔ 𝐽 ∈ Kol2))
742, 73bitrd 280 1 (𝐺 ∈ TopGrp → (𝐽 ∈ Haus ↔ 𝐽 ∈ Kol2))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  wral 3138  cmpt 5138  ccnv 5548  cima 5552  cfv 6349  (class class class)co 7145  Basecbs 16473  +gcplusg 16555  TopOpenctopn 16685  0gc0g 16703  Grpcgrp 18043  -gcsg 18045  TopOnctopon 21448   Cn ccn 21762  Kol2ct0 21844  Frect1 21845  Hauscha 21846   ×t ctx 22098  TopMndctmd 22608  TopGrpctgp 22609
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3497  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4833  df-iun 4914  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-1st 7680  df-2nd 7681  df-map 8398  df-0g 16705  df-topgen 16707  df-plusf 17841  df-mgm 17842  df-sgrp 17891  df-mnd 17902  df-grp 18046  df-minusg 18047  df-sbg 18048  df-top 21432  df-topon 21449  df-topsp 21471  df-bases 21484  df-cld 21557  df-cn 21765  df-cnp 21766  df-t0 21851  df-t1 21852  df-haus 21853  df-tx 22100  df-tmd 22610  df-tgp 22611
This theorem is referenced by: (None)
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