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Theorem tgpt0 21832
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 21831 . 2 (𝐺 ∈ TopGrp → (𝐽 ∈ Haus ↔ 𝐽 ∈ Fre))
3 t1t0 21062 . . 3 (𝐽 ∈ Fre → 𝐽 ∈ Kol2)
4 eleq2 2687 . . . . . . . . . . . 12 (𝑤 = 𝑧 → (𝑥𝑤𝑥𝑧))
5 eleq2 2687 . . . . . . . . . . . 12 (𝑤 = 𝑧 → (𝑦𝑤𝑦𝑧))
64, 5imbi12d 334 . . . . . . . . . . 11 (𝑤 = 𝑧 → ((𝑥𝑤𝑦𝑤) ↔ (𝑥𝑧𝑦𝑧)))
76rspccva 3294 . . . . . . . . . 10 ((∀𝑤𝐽 (𝑥𝑤𝑦𝑤) ∧ 𝑧𝐽) → (𝑥𝑧𝑦𝑧))
87adantll 749 . . . . . . . . 9 ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) ∧ 𝑧𝐽) → (𝑥𝑧𝑦𝑧))
9 tgpgrp 21792 . . . . . . . . . . . . . . 15 (𝐺 ∈ TopGrp → 𝐺 ∈ Grp)
109ad3antrrr 765 . . . . . . . . . . . . . 14 ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) ∧ (𝑧𝐽𝑦𝑧)) → 𝐺 ∈ Grp)
11 simpllr 798 . . . . . . . . . . . . . . 15 ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) ∧ (𝑧𝐽𝑦𝑧)) → (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)))
1211simprd 479 . . . . . . . . . . . . . 14 ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) ∧ (𝑧𝐽𝑦𝑧)) → 𝑦 ∈ (Base‘𝐺))
13 eqid 2621 . . . . . . . . . . . . . . 15 (Base‘𝐺) = (Base‘𝐺)
14 eqid 2621 . . . . . . . . . . . . . . 15 (0g𝐺) = (0g𝐺)
15 eqid 2621 . . . . . . . . . . . . . . 15 (-g𝐺) = (-g𝐺)
1613, 14, 15grpsubid 17420 . . . . . . . . . . . . . 14 ((𝐺 ∈ Grp ∧ 𝑦 ∈ (Base‘𝐺)) → (𝑦(-g𝐺)𝑦) = (0g𝐺))
1710, 12, 16syl2anc 692 . . . . . . . . . . . . 13 ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) ∧ (𝑧𝐽𝑦𝑧)) → (𝑦(-g𝐺)𝑦) = (0g𝐺))
1817oveq1d 6619 . . . . . . . . . . . 12 ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) ∧ (𝑧𝐽𝑦𝑧)) → ((𝑦(-g𝐺)𝑦)(+g𝐺)𝑥) = ((0g𝐺)(+g𝐺)𝑥))
1911simpld 475 . . . . . . . . . . . . 13 ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) ∧ (𝑧𝐽𝑦𝑧)) → 𝑥 ∈ (Base‘𝐺))
20 eqid 2621 . . . . . . . . . . . . . 14 (+g𝐺) = (+g𝐺)
2113, 20, 14grplid 17373 . . . . . . . . . . . . 13 ((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) → ((0g𝐺)(+g𝐺)𝑥) = 𝑥)
2210, 19, 21syl2anc 692 . . . . . . . . . . . 12 ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) ∧ (𝑧𝐽𝑦𝑧)) → ((0g𝐺)(+g𝐺)𝑥) = 𝑥)
2318, 22eqtrd 2655 . . . . . . . . . . 11 ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) ∧ (𝑧𝐽𝑦𝑧)) → ((𝑦(-g𝐺)𝑦)(+g𝐺)𝑥) = 𝑥)
24 tgptmd 21793 . . . . . . . . . . . . . . . 16 (𝐺 ∈ TopGrp → 𝐺 ∈ TopMnd)
2524ad3antrrr 765 . . . . . . . . . . . . . . 15 ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) ∧ (𝑧𝐽𝑦𝑧)) → 𝐺 ∈ TopMnd)
261, 13tgptopon 21796 . . . . . . . . . . . . . . . 16 (𝐺 ∈ TopGrp → 𝐽 ∈ (TopOn‘(Base‘𝐺)))
2726ad3antrrr 765 . . . . . . . . . . . . . . 15 ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) ∧ (𝑧𝐽𝑦𝑧)) → 𝐽 ∈ (TopOn‘(Base‘𝐺)))
2827, 27, 12cnmptc 21375 . . . . . . . . . . . . . . . 16 ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) ∧ (𝑧𝐽𝑦𝑧)) → (𝑎 ∈ (Base‘𝐺) ↦ 𝑦) ∈ (𝐽 Cn 𝐽))
2927cnmptid 21374 . . . . . . . . . . . . . . . 16 ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) ∧ (𝑧𝐽𝑦𝑧)) → (𝑎 ∈ (Base‘𝐺) ↦ 𝑎) ∈ (𝐽 Cn 𝐽))
301, 15tgpsubcn 21804 . . . . . . . . . . . . . . . . 17 (𝐺 ∈ TopGrp → (-g𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
3130ad3antrrr 765 . . . . . . . . . . . . . . . 16 ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) ∧ (𝑧𝐽𝑦𝑧)) → (-g𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
3227, 28, 29, 31cnmpt12f 21379 . . . . . . . . . . . . . . 15 ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) ∧ (𝑧𝐽𝑦𝑧)) → (𝑎 ∈ (Base‘𝐺) ↦ (𝑦(-g𝐺)𝑎)) ∈ (𝐽 Cn 𝐽))
3327, 27, 19cnmptc 21375 . . . . . . . . . . . . . . 15 ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) ∧ (𝑧𝐽𝑦𝑧)) → (𝑎 ∈ (Base‘𝐺) ↦ 𝑥) ∈ (𝐽 Cn 𝐽))
341, 20, 25, 27, 32, 33cnmpt1plusg 21801 . . . . . . . . . . . . . 14 ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) ∧ (𝑧𝐽𝑦𝑧)) → (𝑎 ∈ (Base‘𝐺) ↦ ((𝑦(-g𝐺)𝑎)(+g𝐺)𝑥)) ∈ (𝐽 Cn 𝐽))
35 simprl 793 . . . . . . . . . . . . . 14 ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) ∧ (𝑧𝐽𝑦𝑧)) → 𝑧𝐽)
36 cnima 20979 . . . . . . . . . . . . . 14 (((𝑎 ∈ (Base‘𝐺) ↦ ((𝑦(-g𝐺)𝑎)(+g𝐺)𝑥)) ∈ (𝐽 Cn 𝐽) ∧ 𝑧𝐽) → ((𝑎 ∈ (Base‘𝐺) ↦ ((𝑦(-g𝐺)𝑎)(+g𝐺)𝑥)) “ 𝑧) ∈ 𝐽)
3734, 35, 36syl2anc 692 . . . . . . . . . . . . 13 ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) ∧ (𝑧𝐽𝑦𝑧)) → ((𝑎 ∈ (Base‘𝐺) ↦ ((𝑦(-g𝐺)𝑎)(+g𝐺)𝑥)) “ 𝑧) ∈ 𝐽)
38 simplr 791 . . . . . . . . . . . . 13 ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) ∧ (𝑧𝐽𝑦𝑧)) → ∀𝑤𝐽 (𝑥𝑤𝑦𝑤))
3913, 20, 15grpnpcan 17428 . . . . . . . . . . . . . . . 16 ((𝐺 ∈ Grp ∧ 𝑦 ∈ (Base‘𝐺) ∧ 𝑥 ∈ (Base‘𝐺)) → ((𝑦(-g𝐺)𝑥)(+g𝐺)𝑥) = 𝑦)
4010, 12, 19, 39syl3anc 1323 . . . . . . . . . . . . . . 15 ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) ∧ (𝑧𝐽𝑦𝑧)) → ((𝑦(-g𝐺)𝑥)(+g𝐺)𝑥) = 𝑦)
41 simprr 795 . . . . . . . . . . . . . . 15 ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) ∧ (𝑧𝐽𝑦𝑧)) → 𝑦𝑧)
4240, 41eqeltrd 2698 . . . . . . . . . . . . . 14 ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) ∧ (𝑧𝐽𝑦𝑧)) → ((𝑦(-g𝐺)𝑥)(+g𝐺)𝑥) ∈ 𝑧)
43 oveq2 6612 . . . . . . . . . . . . . . . . 17 (𝑎 = 𝑥 → (𝑦(-g𝐺)𝑎) = (𝑦(-g𝐺)𝑥))
4443oveq1d 6619 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑥 → ((𝑦(-g𝐺)𝑎)(+g𝐺)𝑥) = ((𝑦(-g𝐺)𝑥)(+g𝐺)𝑥))
4544eleq1d 2683 . . . . . . . . . . . . . . 15 (𝑎 = 𝑥 → (((𝑦(-g𝐺)𝑎)(+g𝐺)𝑥) ∈ 𝑧 ↔ ((𝑦(-g𝐺)𝑥)(+g𝐺)𝑥) ∈ 𝑧))
46 eqid 2621 . . . . . . . . . . . . . . . 16 (𝑎 ∈ (Base‘𝐺) ↦ ((𝑦(-g𝐺)𝑎)(+g𝐺)𝑥)) = (𝑎 ∈ (Base‘𝐺) ↦ ((𝑦(-g𝐺)𝑎)(+g𝐺)𝑥))
4746mptpreima 5587 . . . . . . . . . . . . . . 15 ((𝑎 ∈ (Base‘𝐺) ↦ ((𝑦(-g𝐺)𝑎)(+g𝐺)𝑥)) “ 𝑧) = {𝑎 ∈ (Base‘𝐺) ∣ ((𝑦(-g𝐺)𝑎)(+g𝐺)𝑥) ∈ 𝑧}
4845, 47elrab2 3348 . . . . . . . . . . . . . 14 (𝑥 ∈ ((𝑎 ∈ (Base‘𝐺) ↦ ((𝑦(-g𝐺)𝑎)(+g𝐺)𝑥)) “ 𝑧) ↔ (𝑥 ∈ (Base‘𝐺) ∧ ((𝑦(-g𝐺)𝑥)(+g𝐺)𝑥) ∈ 𝑧))
4919, 42, 48sylanbrc 697 . . . . . . . . . . . . 13 ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) ∧ (𝑧𝐽𝑦𝑧)) → 𝑥 ∈ ((𝑎 ∈ (Base‘𝐺) ↦ ((𝑦(-g𝐺)𝑎)(+g𝐺)𝑥)) “ 𝑧))
50 eleq2 2687 . . . . . . . . . . . . . . 15 (𝑤 = ((𝑎 ∈ (Base‘𝐺) ↦ ((𝑦(-g𝐺)𝑎)(+g𝐺)𝑥)) “ 𝑧) → (𝑥𝑤𝑥 ∈ ((𝑎 ∈ (Base‘𝐺) ↦ ((𝑦(-g𝐺)𝑎)(+g𝐺)𝑥)) “ 𝑧)))
51 eleq2 2687 . . . . . . . . . . . . . . 15 (𝑤 = ((𝑎 ∈ (Base‘𝐺) ↦ ((𝑦(-g𝐺)𝑎)(+g𝐺)𝑥)) “ 𝑧) → (𝑦𝑤𝑦 ∈ ((𝑎 ∈ (Base‘𝐺) ↦ ((𝑦(-g𝐺)𝑎)(+g𝐺)𝑥)) “ 𝑧)))
5250, 51imbi12d 334 . . . . . . . . . . . . . 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))))
5352rspcv 3291 . . . . . . . . . . . . 13 (((𝑎 ∈ (Base‘𝐺) ↦ ((𝑦(-g𝐺)𝑎)(+g𝐺)𝑥)) “ 𝑧) ∈ 𝐽 → (∀𝑤𝐽 (𝑥𝑤𝑦𝑤) → (𝑥 ∈ ((𝑎 ∈ (Base‘𝐺) ↦ ((𝑦(-g𝐺)𝑎)(+g𝐺)𝑥)) “ 𝑧) → 𝑦 ∈ ((𝑎 ∈ (Base‘𝐺) ↦ ((𝑦(-g𝐺)𝑎)(+g𝐺)𝑥)) “ 𝑧))))
5437, 38, 49, 53syl3c 66 . . . . . . . . . . . 12 ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) ∧ (𝑧𝐽𝑦𝑧)) → 𝑦 ∈ ((𝑎 ∈ (Base‘𝐺) ↦ ((𝑦(-g𝐺)𝑎)(+g𝐺)𝑥)) “ 𝑧))
55 oveq2 6612 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑦 → (𝑦(-g𝐺)𝑎) = (𝑦(-g𝐺)𝑦))
5655oveq1d 6619 . . . . . . . . . . . . . . 15 (𝑎 = 𝑦 → ((𝑦(-g𝐺)𝑎)(+g𝐺)𝑥) = ((𝑦(-g𝐺)𝑦)(+g𝐺)𝑥))
5756eleq1d 2683 . . . . . . . . . . . . . 14 (𝑎 = 𝑦 → (((𝑦(-g𝐺)𝑎)(+g𝐺)𝑥) ∈ 𝑧 ↔ ((𝑦(-g𝐺)𝑦)(+g𝐺)𝑥) ∈ 𝑧))
5857, 47elrab2 3348 . . . . . . . . . . . . 13 (𝑦 ∈ ((𝑎 ∈ (Base‘𝐺) ↦ ((𝑦(-g𝐺)𝑎)(+g𝐺)𝑥)) “ 𝑧) ↔ (𝑦 ∈ (Base‘𝐺) ∧ ((𝑦(-g𝐺)𝑦)(+g𝐺)𝑥) ∈ 𝑧))
5958simprbi 480 . . . . . . . . . . . 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 2699 . . . . . . . . . 10 ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) ∧ (𝑧𝐽𝑦𝑧)) → 𝑥𝑧)
6261expr 642 . . . . . . . . 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 2960 . . . . . . 7 (((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) → ∀𝑧𝐽 (𝑥𝑧𝑦𝑧))
6564ex 450 . . . . . 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 2958 . . . 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 21058 . . . . 5 (𝐽 ∈ (TopOn‘(Base‘𝐺)) → (𝐽 ∈ Kol2 ↔ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(∀𝑧𝐽 (𝑥𝑧𝑦𝑧) → 𝑥 = 𝑦)))
6926, 68syl 17 . . . 4 (𝐺 ∈ TopGrp → (𝐽 ∈ Kol2 ↔ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(∀𝑧𝐽 (𝑥𝑧𝑦𝑧) → 𝑥 = 𝑦)))
70 ist1-2 21061 . . . . 5 (𝐽 ∈ (TopOn‘(Base‘𝐺)) → (𝐽 ∈ Fre ↔ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(∀𝑤𝐽 (𝑥𝑤𝑦𝑤) → 𝑥 = 𝑦)))
7126, 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 384   = wceq 1480  wcel 1987  wral 2907  cmpt 4673  ccnv 5073  cima 5077  cfv 5847  (class class class)co 6604  Basecbs 15781  +gcplusg 15862  TopOpenctopn 16003  0gc0g 16021  Grpcgrp 17343  -gcsg 17345  TopOnctopon 20618   Cn ccn 20938  Kol2ct0 21020  Frect1 21021  Hauscha 21022   ×t ctx 21273  TopMndctmd 21784  TopGrpctgp 21785
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-1st 7113  df-2nd 7114  df-map 7804  df-0g 16023  df-topgen 16025  df-plusf 17162  df-mgm 17163  df-sgrp 17205  df-mnd 17216  df-grp 17346  df-minusg 17347  df-sbg 17348  df-top 20621  df-bases 20622  df-topon 20623  df-topsp 20624  df-cld 20733  df-cn 20941  df-cnp 20942  df-t0 21027  df-t1 21028  df-haus 21029  df-tx 21275  df-tmd 21786  df-tgp 21787
This theorem is referenced by: (None)
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