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Theorem tgpt0 22142
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 22141 . 2 (𝐺 ∈ TopGrp → (𝐽 ∈ Haus ↔ 𝐽 ∈ Fre))
3 t1t0 21373 . . 3 (𝐽 ∈ Fre → 𝐽 ∈ Kol2)
4 eleq2 2839 . . . . . . . . . . . 12 (𝑤 = 𝑧 → (𝑥𝑤𝑥𝑧))
5 eleq2 2839 . . . . . . . . . . . 12 (𝑤 = 𝑧 → (𝑦𝑤𝑦𝑧))
64, 5imbi12d 333 . . . . . . . . . . 11 (𝑤 = 𝑧 → ((𝑥𝑤𝑦𝑤) ↔ (𝑥𝑧𝑦𝑧)))
76rspccva 3459 . . . . . . . . . 10 ((∀𝑤𝐽 (𝑥𝑤𝑦𝑤) ∧ 𝑧𝐽) → (𝑥𝑧𝑦𝑧))
87adantll 693 . . . . . . . . 9 ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) ∧ 𝑧𝐽) → (𝑥𝑧𝑦𝑧))
9 tgpgrp 22102 . . . . . . . . . . . . . . 15 (𝐺 ∈ TopGrp → 𝐺 ∈ Grp)
109ad3antrrr 709 . . . . . . . . . . . . . 14 ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) ∧ (𝑧𝐽𝑦𝑧)) → 𝐺 ∈ Grp)
11 simpllr 760 . . . . . . . . . . . . . . 15 ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) ∧ (𝑧𝐽𝑦𝑧)) → (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)))
1211simprd 483 . . . . . . . . . . . . . 14 ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) ∧ (𝑧𝐽𝑦𝑧)) → 𝑦 ∈ (Base‘𝐺))
13 eqid 2771 . . . . . . . . . . . . . . 15 (Base‘𝐺) = (Base‘𝐺)
14 eqid 2771 . . . . . . . . . . . . . . 15 (0g𝐺) = (0g𝐺)
15 eqid 2771 . . . . . . . . . . . . . . 15 (-g𝐺) = (-g𝐺)
1613, 14, 15grpsubid 17707 . . . . . . . . . . . . . 14 ((𝐺 ∈ Grp ∧ 𝑦 ∈ (Base‘𝐺)) → (𝑦(-g𝐺)𝑦) = (0g𝐺))
1710, 12, 16syl2anc 573 . . . . . . . . . . . . 13 ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) ∧ (𝑧𝐽𝑦𝑧)) → (𝑦(-g𝐺)𝑦) = (0g𝐺))
1817oveq1d 6808 . . . . . . . . . . . 12 ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) ∧ (𝑧𝐽𝑦𝑧)) → ((𝑦(-g𝐺)𝑦)(+g𝐺)𝑥) = ((0g𝐺)(+g𝐺)𝑥))
1911simpld 482 . . . . . . . . . . . . 13 ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) ∧ (𝑧𝐽𝑦𝑧)) → 𝑥 ∈ (Base‘𝐺))
20 eqid 2771 . . . . . . . . . . . . . 14 (+g𝐺) = (+g𝐺)
2113, 20, 14grplid 17660 . . . . . . . . . . . . 13 ((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) → ((0g𝐺)(+g𝐺)𝑥) = 𝑥)
2210, 19, 21syl2anc 573 . . . . . . . . . . . 12 ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) ∧ (𝑧𝐽𝑦𝑧)) → ((0g𝐺)(+g𝐺)𝑥) = 𝑥)
2318, 22eqtrd 2805 . . . . . . . . . . 11 ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) ∧ (𝑧𝐽𝑦𝑧)) → ((𝑦(-g𝐺)𝑦)(+g𝐺)𝑥) = 𝑥)
2413, 20, 15grpnpcan 17715 . . . . . . . . . . . . . . . 16 ((𝐺 ∈ Grp ∧ 𝑦 ∈ (Base‘𝐺) ∧ 𝑥 ∈ (Base‘𝐺)) → ((𝑦(-g𝐺)𝑥)(+g𝐺)𝑥) = 𝑦)
2510, 12, 19, 24syl3anc 1476 . . . . . . . . . . . . . . 15 ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) ∧ (𝑧𝐽𝑦𝑧)) → ((𝑦(-g𝐺)𝑥)(+g𝐺)𝑥) = 𝑦)
26 simprr 756 . . . . . . . . . . . . . . 15 ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) ∧ (𝑧𝐽𝑦𝑧)) → 𝑦𝑧)
2725, 26eqeltrd 2850 . . . . . . . . . . . . . 14 ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) ∧ (𝑧𝐽𝑦𝑧)) → ((𝑦(-g𝐺)𝑥)(+g𝐺)𝑥) ∈ 𝑧)
28 oveq2 6801 . . . . . . . . . . . . . . . . 17 (𝑎 = 𝑥 → (𝑦(-g𝐺)𝑎) = (𝑦(-g𝐺)𝑥))
2928oveq1d 6808 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑥 → ((𝑦(-g𝐺)𝑎)(+g𝐺)𝑥) = ((𝑦(-g𝐺)𝑥)(+g𝐺)𝑥))
3029eleq1d 2835 . . . . . . . . . . . . . . 15 (𝑎 = 𝑥 → (((𝑦(-g𝐺)𝑎)(+g𝐺)𝑥) ∈ 𝑧 ↔ ((𝑦(-g𝐺)𝑥)(+g𝐺)𝑥) ∈ 𝑧))
31 eqid 2771 . . . . . . . . . . . . . . . 16 (𝑎 ∈ (Base‘𝐺) ↦ ((𝑦(-g𝐺)𝑎)(+g𝐺)𝑥)) = (𝑎 ∈ (Base‘𝐺) ↦ ((𝑦(-g𝐺)𝑎)(+g𝐺)𝑥))
3231mptpreima 5772 . . . . . . . . . . . . . . 15 ((𝑎 ∈ (Base‘𝐺) ↦ ((𝑦(-g𝐺)𝑎)(+g𝐺)𝑥)) “ 𝑧) = {𝑎 ∈ (Base‘𝐺) ∣ ((𝑦(-g𝐺)𝑎)(+g𝐺)𝑥) ∈ 𝑧}
3330, 32elrab2 3518 . . . . . . . . . . . . . 14 (𝑥 ∈ ((𝑎 ∈ (Base‘𝐺) ↦ ((𝑦(-g𝐺)𝑎)(+g𝐺)𝑥)) “ 𝑧) ↔ (𝑥 ∈ (Base‘𝐺) ∧ ((𝑦(-g𝐺)𝑥)(+g𝐺)𝑥) ∈ 𝑧))
3419, 27, 33sylanbrc 572 . . . . . . . . . . . . 13 ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) ∧ (𝑧𝐽𝑦𝑧)) → 𝑥 ∈ ((𝑎 ∈ (Base‘𝐺) ↦ ((𝑦(-g𝐺)𝑎)(+g𝐺)𝑥)) “ 𝑧))
35 eleq2 2839 . . . . . . . . . . . . . . 15 (𝑤 = ((𝑎 ∈ (Base‘𝐺) ↦ ((𝑦(-g𝐺)𝑎)(+g𝐺)𝑥)) “ 𝑧) → (𝑥𝑤𝑥 ∈ ((𝑎 ∈ (Base‘𝐺) ↦ ((𝑦(-g𝐺)𝑎)(+g𝐺)𝑥)) “ 𝑧)))
36 eleq2 2839 . . . . . . . . . . . . . . 15 (𝑤 = ((𝑎 ∈ (Base‘𝐺) ↦ ((𝑦(-g𝐺)𝑎)(+g𝐺)𝑥)) “ 𝑧) → (𝑦𝑤𝑦 ∈ ((𝑎 ∈ (Base‘𝐺) ↦ ((𝑦(-g𝐺)𝑎)(+g𝐺)𝑥)) “ 𝑧)))
3735, 36imbi12d 333 . . . . . . . . . . . . . 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 752 . . . . . . . . . . . . . 14 ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) ∧ (𝑧𝐽𝑦𝑧)) → ∀𝑤𝐽 (𝑥𝑤𝑦𝑤))
39 tgptmd 22103 . . . . . . . . . . . . . . . . 17 (𝐺 ∈ TopGrp → 𝐺 ∈ TopMnd)
4039ad3antrrr 709 . . . . . . . . . . . . . . . 16 ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) ∧ (𝑧𝐽𝑦𝑧)) → 𝐺 ∈ TopMnd)
411, 13tgptopon 22106 . . . . . . . . . . . . . . . . 17 (𝐺 ∈ TopGrp → 𝐽 ∈ (TopOn‘(Base‘𝐺)))
4241ad3antrrr 709 . . . . . . . . . . . . . . . 16 ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) ∧ (𝑧𝐽𝑦𝑧)) → 𝐽 ∈ (TopOn‘(Base‘𝐺)))
4342, 42, 12cnmptc 21686 . . . . . . . . . . . . . . . . 17 ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) ∧ (𝑧𝐽𝑦𝑧)) → (𝑎 ∈ (Base‘𝐺) ↦ 𝑦) ∈ (𝐽 Cn 𝐽))
4442cnmptid 21685 . . . . . . . . . . . . . . . . 17 ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) ∧ (𝑧𝐽𝑦𝑧)) → (𝑎 ∈ (Base‘𝐺) ↦ 𝑎) ∈ (𝐽 Cn 𝐽))
451, 15tgpsubcn 22114 . . . . . . . . . . . . . . . . . 18 (𝐺 ∈ TopGrp → (-g𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
4645ad3antrrr 709 . . . . . . . . . . . . . . . . 17 ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) ∧ (𝑧𝐽𝑦𝑧)) → (-g𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
4742, 43, 44, 46cnmpt12f 21690 . . . . . . . . . . . . . . . 16 ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) ∧ (𝑧𝐽𝑦𝑧)) → (𝑎 ∈ (Base‘𝐺) ↦ (𝑦(-g𝐺)𝑎)) ∈ (𝐽 Cn 𝐽))
4842, 42, 19cnmptc 21686 . . . . . . . . . . . . . . . 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 22111 . . . . . . . . . . . . . . 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 754 . . . . . . . . . . . . . . 15 ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) ∧ (𝑧𝐽𝑦𝑧)) → 𝑧𝐽)
51 cnima 21290 . . . . . . . . . . . . . . 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 573 . . . . . . . . . . . . . 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 3466 . . . . . . . . . . . . 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 6801 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑦 → (𝑦(-g𝐺)𝑎) = (𝑦(-g𝐺)𝑦))
5655oveq1d 6808 . . . . . . . . . . . . . . 15 (𝑎 = 𝑦 → ((𝑦(-g𝐺)𝑎)(+g𝐺)𝑥) = ((𝑦(-g𝐺)𝑦)(+g𝐺)𝑥))
5756eleq1d 2835 . . . . . . . . . . . . . 14 (𝑎 = 𝑦 → (((𝑦(-g𝐺)𝑎)(+g𝐺)𝑥) ∈ 𝑧 ↔ ((𝑦(-g𝐺)𝑦)(+g𝐺)𝑥) ∈ 𝑧))
5857, 32elrab2 3518 . . . . . . . . . . . . 13 (𝑦 ∈ ((𝑎 ∈ (Base‘𝐺) ↦ ((𝑦(-g𝐺)𝑎)(+g𝐺)𝑥)) “ 𝑧) ↔ (𝑦 ∈ (Base‘𝐺) ∧ ((𝑦(-g𝐺)𝑦)(+g𝐺)𝑥) ∈ 𝑧))
5958simprbi 484 . . . . . . . . . . . 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 2851 . . . . . . . . . 10 ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) ∧ (𝑧𝐽𝑦𝑧)) → 𝑥𝑧)
6261expr 444 . . . . . . . . 9 ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) ∧ 𝑧𝐽) → (𝑦𝑧𝑥𝑧))
638, 62impbid 202 . . . . . . . 8 ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) ∧ 𝑧𝐽) → (𝑥𝑧𝑦𝑧))
6463ralrimiva 3115 . . . . . . 7 (((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) → ∀𝑧𝐽 (𝑥𝑧𝑦𝑧))
6564ex 397 . . . . . 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 3113 . . . 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 21369 . . . . 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 21372 . . . . 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 283 . . 3 (𝐺 ∈ TopGrp → (𝐽 ∈ Kol2 → 𝐽 ∈ Fre))
733, 72impbid2 216 . 2 (𝐺 ∈ TopGrp → (𝐽 ∈ Fre ↔ 𝐽 ∈ Kol2))
742, 73bitrd 268 1 (𝐺 ∈ TopGrp → (𝐽 ∈ Haus ↔ 𝐽 ∈ Kol2))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1631  wcel 2145  wral 3061  cmpt 4863  ccnv 5248  cima 5252  cfv 6031  (class class class)co 6793  Basecbs 16064  +gcplusg 16149  TopOpenctopn 16290  0gc0g 16308  Grpcgrp 17630  -gcsg 17632  TopOnctopon 20935   Cn ccn 21249  Kol2ct0 21331  Frect1 21332  Hauscha 21333   ×t ctx 21584  TopMndctmd 22094  TopGrpctgp 22095
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-1st 7315  df-2nd 7316  df-map 8011  df-0g 16310  df-topgen 16312  df-plusf 17449  df-mgm 17450  df-sgrp 17492  df-mnd 17503  df-grp 17633  df-minusg 17634  df-sbg 17635  df-top 20919  df-topon 20936  df-topsp 20958  df-bases 20971  df-cld 21044  df-cn 21252  df-cnp 21253  df-t0 21338  df-t1 21339  df-haus 21340  df-tx 21586  df-tmd 22096  df-tgp 22097
This theorem is referenced by: (None)
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