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Theorem tgpt0 22727
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 22726 . 2 (𝐺 ∈ TopGrp → (𝐽 ∈ Haus ↔ 𝐽 ∈ Fre))
3 t1t0 21956 . . 3 (𝐽 ∈ Fre → 𝐽 ∈ Kol2)
4 eleq2 2881 . . . . . . . . . . . 12 (𝑤 = 𝑧 → (𝑥𝑤𝑥𝑧))
5 eleq2 2881 . . . . . . . . . . . 12 (𝑤 = 𝑧 → (𝑦𝑤𝑦𝑧))
64, 5imbi12d 348 . . . . . . . . . . 11 (𝑤 = 𝑧 → ((𝑥𝑤𝑦𝑤) ↔ (𝑥𝑧𝑦𝑧)))
76rspccva 3573 . . . . . . . . . 10 ((∀𝑤𝐽 (𝑥𝑤𝑦𝑤) ∧ 𝑧𝐽) → (𝑥𝑧𝑦𝑧))
87adantll 713 . . . . . . . . 9 ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) ∧ 𝑧𝐽) → (𝑥𝑧𝑦𝑧))
9 tgpgrp 22686 . . . . . . . . . . . . . . 15 (𝐺 ∈ TopGrp → 𝐺 ∈ Grp)
109ad3antrrr 729 . . . . . . . . . . . . . 14 ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) ∧ (𝑧𝐽𝑦𝑧)) → 𝐺 ∈ Grp)
11 simpllr 775 . . . . . . . . . . . . . . 15 ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) ∧ (𝑧𝐽𝑦𝑧)) → (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)))
1211simprd 499 . . . . . . . . . . . . . 14 ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) ∧ (𝑧𝐽𝑦𝑧)) → 𝑦 ∈ (Base‘𝐺))
13 eqid 2801 . . . . . . . . . . . . . . 15 (Base‘𝐺) = (Base‘𝐺)
14 eqid 2801 . . . . . . . . . . . . . . 15 (0g𝐺) = (0g𝐺)
15 eqid 2801 . . . . . . . . . . . . . . 15 (-g𝐺) = (-g𝐺)
1613, 14, 15grpsubid 18178 . . . . . . . . . . . . . 14 ((𝐺 ∈ Grp ∧ 𝑦 ∈ (Base‘𝐺)) → (𝑦(-g𝐺)𝑦) = (0g𝐺))
1710, 12, 16syl2anc 587 . . . . . . . . . . . . 13 ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) ∧ (𝑧𝐽𝑦𝑧)) → (𝑦(-g𝐺)𝑦) = (0g𝐺))
1817oveq1d 7154 . . . . . . . . . . . 12 ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) ∧ (𝑧𝐽𝑦𝑧)) → ((𝑦(-g𝐺)𝑦)(+g𝐺)𝑥) = ((0g𝐺)(+g𝐺)𝑥))
1911simpld 498 . . . . . . . . . . . . 13 ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) ∧ (𝑧𝐽𝑦𝑧)) → 𝑥 ∈ (Base‘𝐺))
20 eqid 2801 . . . . . . . . . . . . . 14 (+g𝐺) = (+g𝐺)
2113, 20, 14grplid 18128 . . . . . . . . . . . . 13 ((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) → ((0g𝐺)(+g𝐺)𝑥) = 𝑥)
2210, 19, 21syl2anc 587 . . . . . . . . . . . 12 ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) ∧ (𝑧𝐽𝑦𝑧)) → ((0g𝐺)(+g𝐺)𝑥) = 𝑥)
2318, 22eqtrd 2836 . . . . . . . . . . 11 ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) ∧ (𝑧𝐽𝑦𝑧)) → ((𝑦(-g𝐺)𝑦)(+g𝐺)𝑥) = 𝑥)
2413, 20, 15grpnpcan 18186 . . . . . . . . . . . . . . . 16 ((𝐺 ∈ Grp ∧ 𝑦 ∈ (Base‘𝐺) ∧ 𝑥 ∈ (Base‘𝐺)) → ((𝑦(-g𝐺)𝑥)(+g𝐺)𝑥) = 𝑦)
2510, 12, 19, 24syl3anc 1368 . . . . . . . . . . . . . . 15 ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) ∧ (𝑧𝐽𝑦𝑧)) → ((𝑦(-g𝐺)𝑥)(+g𝐺)𝑥) = 𝑦)
26 simprr 772 . . . . . . . . . . . . . . 15 ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) ∧ (𝑧𝐽𝑦𝑧)) → 𝑦𝑧)
2725, 26eqeltrd 2893 . . . . . . . . . . . . . 14 ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) ∧ (𝑧𝐽𝑦𝑧)) → ((𝑦(-g𝐺)𝑥)(+g𝐺)𝑥) ∈ 𝑧)
28 oveq2 7147 . . . . . . . . . . . . . . . . 17 (𝑎 = 𝑥 → (𝑦(-g𝐺)𝑎) = (𝑦(-g𝐺)𝑥))
2928oveq1d 7154 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑥 → ((𝑦(-g𝐺)𝑎)(+g𝐺)𝑥) = ((𝑦(-g𝐺)𝑥)(+g𝐺)𝑥))
3029eleq1d 2877 . . . . . . . . . . . . . . 15 (𝑎 = 𝑥 → (((𝑦(-g𝐺)𝑎)(+g𝐺)𝑥) ∈ 𝑧 ↔ ((𝑦(-g𝐺)𝑥)(+g𝐺)𝑥) ∈ 𝑧))
31 eqid 2801 . . . . . . . . . . . . . . . 16 (𝑎 ∈ (Base‘𝐺) ↦ ((𝑦(-g𝐺)𝑎)(+g𝐺)𝑥)) = (𝑎 ∈ (Base‘𝐺) ↦ ((𝑦(-g𝐺)𝑎)(+g𝐺)𝑥))
3231mptpreima 6063 . . . . . . . . . . . . . . 15 ((𝑎 ∈ (Base‘𝐺) ↦ ((𝑦(-g𝐺)𝑎)(+g𝐺)𝑥)) “ 𝑧) = {𝑎 ∈ (Base‘𝐺) ∣ ((𝑦(-g𝐺)𝑎)(+g𝐺)𝑥) ∈ 𝑧}
3330, 32elrab2 3634 . . . . . . . . . . . . . 14 (𝑥 ∈ ((𝑎 ∈ (Base‘𝐺) ↦ ((𝑦(-g𝐺)𝑎)(+g𝐺)𝑥)) “ 𝑧) ↔ (𝑥 ∈ (Base‘𝐺) ∧ ((𝑦(-g𝐺)𝑥)(+g𝐺)𝑥) ∈ 𝑧))
3419, 27, 33sylanbrc 586 . . . . . . . . . . . . 13 ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) ∧ (𝑧𝐽𝑦𝑧)) → 𝑥 ∈ ((𝑎 ∈ (Base‘𝐺) ↦ ((𝑦(-g𝐺)𝑎)(+g𝐺)𝑥)) “ 𝑧))
35 eleq2 2881 . . . . . . . . . . . . . . 15 (𝑤 = ((𝑎 ∈ (Base‘𝐺) ↦ ((𝑦(-g𝐺)𝑎)(+g𝐺)𝑥)) “ 𝑧) → (𝑥𝑤𝑥 ∈ ((𝑎 ∈ (Base‘𝐺) ↦ ((𝑦(-g𝐺)𝑎)(+g𝐺)𝑥)) “ 𝑧)))
36 eleq2 2881 . . . . . . . . . . . . . . 15 (𝑤 = ((𝑎 ∈ (Base‘𝐺) ↦ ((𝑦(-g𝐺)𝑎)(+g𝐺)𝑥)) “ 𝑧) → (𝑦𝑤𝑦 ∈ ((𝑎 ∈ (Base‘𝐺) ↦ ((𝑦(-g𝐺)𝑎)(+g𝐺)𝑥)) “ 𝑧)))
3735, 36imbi12d 348 . . . . . . . . . . . . . 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 768 . . . . . . . . . . . . . 14 ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) ∧ (𝑧𝐽𝑦𝑧)) → ∀𝑤𝐽 (𝑥𝑤𝑦𝑤))
39 tgptmd 22687 . . . . . . . . . . . . . . . . 17 (𝐺 ∈ TopGrp → 𝐺 ∈ TopMnd)
4039ad3antrrr 729 . . . . . . . . . . . . . . . 16 ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) ∧ (𝑧𝐽𝑦𝑧)) → 𝐺 ∈ TopMnd)
411, 13tgptopon 22690 . . . . . . . . . . . . . . . . 17 (𝐺 ∈ TopGrp → 𝐽 ∈ (TopOn‘(Base‘𝐺)))
4241ad3antrrr 729 . . . . . . . . . . . . . . . 16 ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) ∧ (𝑧𝐽𝑦𝑧)) → 𝐽 ∈ (TopOn‘(Base‘𝐺)))
4342, 42, 12cnmptc 22270 . . . . . . . . . . . . . . . . 17 ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) ∧ (𝑧𝐽𝑦𝑧)) → (𝑎 ∈ (Base‘𝐺) ↦ 𝑦) ∈ (𝐽 Cn 𝐽))
4442cnmptid 22269 . . . . . . . . . . . . . . . . 17 ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) ∧ (𝑧𝐽𝑦𝑧)) → (𝑎 ∈ (Base‘𝐺) ↦ 𝑎) ∈ (𝐽 Cn 𝐽))
451, 15tgpsubcn 22698 . . . . . . . . . . . . . . . . . 18 (𝐺 ∈ TopGrp → (-g𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
4645ad3antrrr 729 . . . . . . . . . . . . . . . . 17 ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) ∧ (𝑧𝐽𝑦𝑧)) → (-g𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
4742, 43, 44, 46cnmpt12f 22274 . . . . . . . . . . . . . . . 16 ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) ∧ (𝑧𝐽𝑦𝑧)) → (𝑎 ∈ (Base‘𝐺) ↦ (𝑦(-g𝐺)𝑎)) ∈ (𝐽 Cn 𝐽))
4842, 42, 19cnmptc 22270 . . . . . . . . . . . . . . . 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 22695 . . . . . . . . . . . . . . 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 770 . . . . . . . . . . . . . . 15 ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) ∧ (𝑧𝐽𝑦𝑧)) → 𝑧𝐽)
51 cnima 21873 . . . . . . . . . . . . . . 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 587 . . . . . . . . . . . . . 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 3576 . . . . . . . . . . . . 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 7147 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑦 → (𝑦(-g𝐺)𝑎) = (𝑦(-g𝐺)𝑦))
5655oveq1d 7154 . . . . . . . . . . . . . . 15 (𝑎 = 𝑦 → ((𝑦(-g𝐺)𝑎)(+g𝐺)𝑥) = ((𝑦(-g𝐺)𝑦)(+g𝐺)𝑥))
5756eleq1d 2877 . . . . . . . . . . . . . 14 (𝑎 = 𝑦 → (((𝑦(-g𝐺)𝑎)(+g𝐺)𝑥) ∈ 𝑧 ↔ ((𝑦(-g𝐺)𝑦)(+g𝐺)𝑥) ∈ 𝑧))
5857, 32elrab2 3634 . . . . . . . . . . . . 13 (𝑦 ∈ ((𝑎 ∈ (Base‘𝐺) ↦ ((𝑦(-g𝐺)𝑎)(+g𝐺)𝑥)) “ 𝑧) ↔ (𝑦 ∈ (Base‘𝐺) ∧ ((𝑦(-g𝐺)𝑦)(+g𝐺)𝑥) ∈ 𝑧))
5958simprbi 500 . . . . . . . . . . . 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 2894 . . . . . . . . . 10 ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) ∧ (𝑧𝐽𝑦𝑧)) → 𝑥𝑧)
6261expr 460 . . . . . . . . 9 ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) ∧ 𝑧𝐽) → (𝑦𝑧𝑥𝑧))
638, 62impbid 215 . . . . . . . 8 ((((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) ∧ 𝑧𝐽) → (𝑥𝑧𝑦𝑧))
6463ralrimiva 3152 . . . . . . 7 (((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ ∀𝑤𝐽 (𝑥𝑤𝑦𝑤)) → ∀𝑧𝐽 (𝑥𝑧𝑦𝑧))
6564ex 416 . . . . . 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 3149 . . . 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 21952 . . . . 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 21955 . . . . 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 297 . . 3 (𝐺 ∈ TopGrp → (𝐽 ∈ Kol2 → 𝐽 ∈ Fre))
733, 72impbid2 229 . 2 (𝐺 ∈ TopGrp → (𝐽 ∈ Fre ↔ 𝐽 ∈ Kol2))
742, 73bitrd 282 1 (𝐺 ∈ TopGrp → (𝐽 ∈ Haus ↔ 𝐽 ∈ Kol2))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2112  wral 3109  cmpt 5113  ccnv 5522  cima 5526  cfv 6328  (class class class)co 7139  Basecbs 16478  +gcplusg 16560  TopOpenctopn 16690  0gc0g 16708  Grpcgrp 18098  -gcsg 18100  TopOnctopon 21518   Cn ccn 21832  Kol2ct0 21914  Frect1 21915  Hauscha 21916   ×t ctx 22168  TopMndctmd 22678  TopGrpctgp 22679
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rmo 3117  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-1st 7675  df-2nd 7676  df-map 8395  df-0g 16710  df-topgen 16712  df-plusf 17846  df-mgm 17847  df-sgrp 17896  df-mnd 17907  df-grp 18101  df-minusg 18102  df-sbg 18103  df-top 21502  df-topon 21519  df-topsp 21541  df-bases 21554  df-cld 21627  df-cn 21835  df-cnp 21836  df-t0 21921  df-t1 21922  df-haus 21923  df-tx 22170  df-tmd 22680  df-tgp 22681
This theorem is referenced by: (None)
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