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Theorem tgpt0 24143
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 24142 . 2 (𝐺 ∈ TopGrp → (𝐽 ∈ Haus ↔ 𝐽 ∈ Fre))
3 t1t0 23372 . . 3 (𝐽 ∈ Fre → 𝐽 ∈ Kol2)
4 eleq2 2828 . . . . . . . . . . . 12 (𝑤 = 𝑧 → (𝑥𝑤𝑥𝑧))
5 eleq2 2828 . . . . . . . . . . . 12 (𝑤 = 𝑧 → (𝑦𝑤𝑦𝑧))
64, 5imbi12d 344 . . . . . . . . . . 11 (𝑤 = 𝑧 → ((𝑥𝑤𝑦𝑤) ↔ (𝑥𝑧𝑦𝑧)))
76rspccva 3621 . . . . . . . . . 10 ((∀𝑤𝐽 (𝑥𝑤𝑦𝑤) ∧ 𝑧𝐽) → (𝑥𝑧𝑦𝑧))
87adantll 714 . . . . . . . . 9 ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) ∧ 𝑧𝐽) → (𝑥𝑧𝑦𝑧))
9 tgpgrp 24102 . . . . . . . . . . . . . . 15 (𝐺 ∈ TopGrp → 𝐺 ∈ Grp)
109ad3antrrr 730 . . . . . . . . . . . . . 14 ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) ∧ (𝑧𝐽𝑦𝑧)) → 𝐺 ∈ Grp)
11 simpllr 776 . . . . . . . . . . . . . . 15 ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) ∧ (𝑧𝐽𝑦𝑧)) → (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)))
1211simprd 495 . . . . . . . . . . . . . 14 ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) ∧ (𝑧𝐽𝑦𝑧)) → 𝑦 ∈ (Base‘𝐺))
13 eqid 2735 . . . . . . . . . . . . . . 15 (Base‘𝐺) = (Base‘𝐺)
14 eqid 2735 . . . . . . . . . . . . . . 15 (0g𝐺) = (0g𝐺)
15 eqid 2735 . . . . . . . . . . . . . . 15 (-g𝐺) = (-g𝐺)
1613, 14, 15grpsubid 19055 . . . . . . . . . . . . . 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 7446 . . . . . . . . . . . 12 ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) ∧ (𝑧𝐽𝑦𝑧)) → ((𝑦(-g𝐺)𝑦)(+g𝐺)𝑥) = ((0g𝐺)(+g𝐺)𝑥))
1911simpld 494 . . . . . . . . . . . . 13 ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) ∧ (𝑧𝐽𝑦𝑧)) → 𝑥 ∈ (Base‘𝐺))
20 eqid 2735 . . . . . . . . . . . . . 14 (+g𝐺) = (+g𝐺)
2113, 20, 14grplid 18998 . . . . . . . . . . . . 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 2775 . . . . . . . . . . 11 ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) ∧ (𝑧𝐽𝑦𝑧)) → ((𝑦(-g𝐺)𝑦)(+g𝐺)𝑥) = 𝑥)
2413, 20, 15grpnpcan 19063 . . . . . . . . . . . . . . . 16 ((𝐺 ∈ Grp ∧ 𝑦 ∈ (Base‘𝐺) ∧ 𝑥 ∈ (Base‘𝐺)) → ((𝑦(-g𝐺)𝑥)(+g𝐺)𝑥) = 𝑦)
2510, 12, 19, 24syl3anc 1370 . . . . . . . . . . . . . . 15 ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) ∧ (𝑧𝐽𝑦𝑧)) → ((𝑦(-g𝐺)𝑥)(+g𝐺)𝑥) = 𝑦)
26 simprr 773 . . . . . . . . . . . . . . 15 ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) ∧ (𝑧𝐽𝑦𝑧)) → 𝑦𝑧)
2725, 26eqeltrd 2839 . . . . . . . . . . . . . 14 ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) ∧ (𝑧𝐽𝑦𝑧)) → ((𝑦(-g𝐺)𝑥)(+g𝐺)𝑥) ∈ 𝑧)
28 oveq2 7439 . . . . . . . . . . . . . . . . 17 (𝑎 = 𝑥 → (𝑦(-g𝐺)𝑎) = (𝑦(-g𝐺)𝑥))
2928oveq1d 7446 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑥 → ((𝑦(-g𝐺)𝑎)(+g𝐺)𝑥) = ((𝑦(-g𝐺)𝑥)(+g𝐺)𝑥))
3029eleq1d 2824 . . . . . . . . . . . . . . 15 (𝑎 = 𝑥 → (((𝑦(-g𝐺)𝑎)(+g𝐺)𝑥) ∈ 𝑧 ↔ ((𝑦(-g𝐺)𝑥)(+g𝐺)𝑥) ∈ 𝑧))
31 eqid 2735 . . . . . . . . . . . . . . . 16 (𝑎 ∈ (Base‘𝐺) ↦ ((𝑦(-g𝐺)𝑎)(+g𝐺)𝑥)) = (𝑎 ∈ (Base‘𝐺) ↦ ((𝑦(-g𝐺)𝑎)(+g𝐺)𝑥))
3231mptpreima 6260 . . . . . . . . . . . . . . 15 ((𝑎 ∈ (Base‘𝐺) ↦ ((𝑦(-g𝐺)𝑎)(+g𝐺)𝑥)) “ 𝑧) = {𝑎 ∈ (Base‘𝐺) ∣ ((𝑦(-g𝐺)𝑎)(+g𝐺)𝑥) ∈ 𝑧}
3330, 32elrab2 3698 . . . . . . . . . . . . . 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 2828 . . . . . . . . . . . . . . 15 (𝑤 = ((𝑎 ∈ (Base‘𝐺) ↦ ((𝑦(-g𝐺)𝑎)(+g𝐺)𝑥)) “ 𝑧) → (𝑥𝑤𝑥 ∈ ((𝑎 ∈ (Base‘𝐺) ↦ ((𝑦(-g𝐺)𝑎)(+g𝐺)𝑥)) “ 𝑧)))
36 eleq2 2828 . . . . . . . . . . . . . . 15 (𝑤 = ((𝑎 ∈ (Base‘𝐺) ↦ ((𝑦(-g𝐺)𝑎)(+g𝐺)𝑥)) “ 𝑧) → (𝑦𝑤𝑦 ∈ ((𝑎 ∈ (Base‘𝐺) ↦ ((𝑦(-g𝐺)𝑎)(+g𝐺)𝑥)) “ 𝑧)))
3735, 36imbi12d 344 . . . . . . . . . . . . . 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 769 . . . . . . . . . . . . . 14 ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) ∧ (𝑧𝐽𝑦𝑧)) → ∀𝑤𝐽 (𝑥𝑤𝑦𝑤))
39 tgptmd 24103 . . . . . . . . . . . . . . . . 17 (𝐺 ∈ TopGrp → 𝐺 ∈ TopMnd)
4039ad3antrrr 730 . . . . . . . . . . . . . . . 16 ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) ∧ (𝑧𝐽𝑦𝑧)) → 𝐺 ∈ TopMnd)
411, 13tgptopon 24106 . . . . . . . . . . . . . . . . 17 (𝐺 ∈ TopGrp → 𝐽 ∈ (TopOn‘(Base‘𝐺)))
4241ad3antrrr 730 . . . . . . . . . . . . . . . 16 ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) ∧ (𝑧𝐽𝑦𝑧)) → 𝐽 ∈ (TopOn‘(Base‘𝐺)))
4342, 42, 12cnmptc 23686 . . . . . . . . . . . . . . . . 17 ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) ∧ (𝑧𝐽𝑦𝑧)) → (𝑎 ∈ (Base‘𝐺) ↦ 𝑦) ∈ (𝐽 Cn 𝐽))
4442cnmptid 23685 . . . . . . . . . . . . . . . . 17 ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) ∧ (𝑧𝐽𝑦𝑧)) → (𝑎 ∈ (Base‘𝐺) ↦ 𝑎) ∈ (𝐽 Cn 𝐽))
451, 15tgpsubcn 24114 . . . . . . . . . . . . . . . . . 18 (𝐺 ∈ TopGrp → (-g𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
4645ad3antrrr 730 . . . . . . . . . . . . . . . . 17 ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) ∧ (𝑧𝐽𝑦𝑧)) → (-g𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
4742, 43, 44, 46cnmpt12f 23690 . . . . . . . . . . . . . . . 16 ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) ∧ (𝑧𝐽𝑦𝑧)) → (𝑎 ∈ (Base‘𝐺) ↦ (𝑦(-g𝐺)𝑎)) ∈ (𝐽 Cn 𝐽))
4842, 42, 19cnmptc 23686 . . . . . . . . . . . . . . . 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 24111 . . . . . . . . . . . . . . 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 771 . . . . . . . . . . . . . . 15 ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) ∧ (𝑧𝐽𝑦𝑧)) → 𝑧𝐽)
51 cnima 23289 . . . . . . . . . . . . . . 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 3623 . . . . . . . . . . . . 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 7439 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑦 → (𝑦(-g𝐺)𝑎) = (𝑦(-g𝐺)𝑦))
5655oveq1d 7446 . . . . . . . . . . . . . . 15 (𝑎 = 𝑦 → ((𝑦(-g𝐺)𝑎)(+g𝐺)𝑥) = ((𝑦(-g𝐺)𝑦)(+g𝐺)𝑥))
5756eleq1d 2824 . . . . . . . . . . . . . 14 (𝑎 = 𝑦 → (((𝑦(-g𝐺)𝑎)(+g𝐺)𝑥) ∈ 𝑧 ↔ ((𝑦(-g𝐺)𝑦)(+g𝐺)𝑥) ∈ 𝑧))
5857, 32elrab2 3698 . . . . . . . . . . . . 13 (𝑦 ∈ ((𝑎 ∈ (Base‘𝐺) ↦ ((𝑦(-g𝐺)𝑎)(+g𝐺)𝑥)) “ 𝑧) ↔ (𝑦 ∈ (Base‘𝐺) ∧ ((𝑦(-g𝐺)𝑦)(+g𝐺)𝑥) ∈ 𝑧))
5958simprbi 496 . . . . . . . . . . . 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 2840 . . . . . . . . . 10 ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) ∧ (𝑧𝐽𝑦𝑧)) → 𝑥𝑧)
6261expr 456 . . . . . . . . 9 ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) ∧ 𝑧𝐽) → (𝑦𝑧𝑥𝑧))
638, 62impbid 212 . . . . . . . 8 ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) ∧ 𝑧𝐽) → (𝑥𝑧𝑦𝑧))
6463ralrimiva 3144 . . . . . . 7 (((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) → ∀𝑧𝐽 (𝑥𝑧𝑦𝑧))
6564ex 412 . . . . . 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 3204 . . . 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 23368 . . . . 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 23371 . . . . 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 294 . . 3 (𝐺 ∈ TopGrp → (𝐽 ∈ Kol2 → 𝐽 ∈ Fre))
733, 72impbid2 226 . 2 (𝐺 ∈ TopGrp → (𝐽 ∈ Fre ↔ 𝐽 ∈ Kol2))
742, 73bitrd 279 1 (𝐺 ∈ TopGrp → (𝐽 ∈ Haus ↔ 𝐽 ∈ Kol2))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  wral 3059  cmpt 5231  ccnv 5688  cima 5692  cfv 6563  (class class class)co 7431  Basecbs 17245  +gcplusg 17298  TopOpenctopn 17468  0gc0g 17486  Grpcgrp 18964  -gcsg 18966  TopOnctopon 22932   Cn ccn 23248  Kol2ct0 23330  Frect1 23331  Hauscha 23332   ×t ctx 23584  TopMndctmd 24094  TopGrpctgp 24095
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-map 8867  df-0g 17488  df-topgen 17490  df-plusf 18665  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-grp 18967  df-minusg 18968  df-sbg 18969  df-top 22916  df-topon 22933  df-topsp 22955  df-bases 22969  df-cld 23043  df-cn 23251  df-cnp 23252  df-t0 23337  df-t1 23338  df-haus 23339  df-tx 23586  df-tmd 24096  df-tgp 24097
This theorem is referenced by: (None)
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