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Theorem symgtgp 22806
 Description: The symmetric group is a topological group. (Contributed by Mario Carneiro, 2-Sep-2015.) (Proof shortened by AV, 30-Mar-2024.)
Hypothesis
Ref Expression
symgtgp.g 𝐺 = (SymGrp‘𝐴)
Assertion
Ref Expression
symgtgp (𝐴𝑉𝐺 ∈ TopGrp)

Proof of Theorem symgtgp
Dummy variables 𝑡 𝑓 𝑢 𝑣 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 symgtgp.g . . 3 𝐺 = (SymGrp‘𝐴)
21symggrp 18595 . 2 (𝐴𝑉𝐺 ∈ Grp)
3 eqid 2758 . . . 4 (EndoFMnd‘𝐴) = (EndoFMnd‘𝐴)
43efmndtmd 22801 . . 3 (𝐴𝑉 → (EndoFMnd‘𝐴) ∈ TopMnd)
5 eqid 2758 . . . 4 (Base‘𝐺) = (Base‘𝐺)
63, 1, 5symgsubmefmnd 18593 . . 3 (𝐴𝑉 → (Base‘𝐺) ∈ (SubMnd‘(EndoFMnd‘𝐴)))
71, 5, 3symgressbas 18577 . . . 4 𝐺 = ((EndoFMnd‘𝐴) ↾s (Base‘𝐺))
87submtmd 22804 . . 3 (((EndoFMnd‘𝐴) ∈ TopMnd ∧ (Base‘𝐺) ∈ (SubMnd‘(EndoFMnd‘𝐴))) → 𝐺 ∈ TopMnd)
94, 6, 8syl2anc 587 . 2 (𝐴𝑉𝐺 ∈ TopMnd)
10 eqid 2758 . . . . . 6 (∏t‘(𝐴 × {𝒫 𝐴})) = (∏t‘(𝐴 × {𝒫 𝐴}))
111, 5symgtopn 18601 . . . . . . 7 (𝐴𝑉 → ((∏t‘(𝐴 × {𝒫 𝐴})) ↾t (Base‘𝐺)) = (TopOpen‘𝐺))
12 distopon 21697 . . . . . . . . 9 (𝐴𝑉 → 𝒫 𝐴 ∈ (TopOn‘𝐴))
1310pttoponconst 22297 . . . . . . . . 9 ((𝐴𝑉 ∧ 𝒫 𝐴 ∈ (TopOn‘𝐴)) → (∏t‘(𝐴 × {𝒫 𝐴})) ∈ (TopOn‘(𝐴m 𝐴)))
1412, 13mpdan 686 . . . . . . . 8 (𝐴𝑉 → (∏t‘(𝐴 × {𝒫 𝐴})) ∈ (TopOn‘(𝐴m 𝐴)))
151, 5elsymgbas 18569 . . . . . . . . . 10 (𝐴𝑉 → (𝑥 ∈ (Base‘𝐺) ↔ 𝑥:𝐴1-1-onto𝐴))
16 f1of 6602 . . . . . . . . . . 11 (𝑥:𝐴1-1-onto𝐴𝑥:𝐴𝐴)
17 elmapg 8429 . . . . . . . . . . . 12 ((𝐴𝑉𝐴𝑉) → (𝑥 ∈ (𝐴m 𝐴) ↔ 𝑥:𝐴𝐴))
1817anidms 570 . . . . . . . . . . 11 (𝐴𝑉 → (𝑥 ∈ (𝐴m 𝐴) ↔ 𝑥:𝐴𝐴))
1916, 18syl5ibr 249 . . . . . . . . . 10 (𝐴𝑉 → (𝑥:𝐴1-1-onto𝐴𝑥 ∈ (𝐴m 𝐴)))
2015, 19sylbid 243 . . . . . . . . 9 (𝐴𝑉 → (𝑥 ∈ (Base‘𝐺) → 𝑥 ∈ (𝐴m 𝐴)))
2120ssrdv 3898 . . . . . . . 8 (𝐴𝑉 → (Base‘𝐺) ⊆ (𝐴m 𝐴))
22 resttopon 21861 . . . . . . . 8 (((∏t‘(𝐴 × {𝒫 𝐴})) ∈ (TopOn‘(𝐴m 𝐴)) ∧ (Base‘𝐺) ⊆ (𝐴m 𝐴)) → ((∏t‘(𝐴 × {𝒫 𝐴})) ↾t (Base‘𝐺)) ∈ (TopOn‘(Base‘𝐺)))
2314, 21, 22syl2anc 587 . . . . . . 7 (𝐴𝑉 → ((∏t‘(𝐴 × {𝒫 𝐴})) ↾t (Base‘𝐺)) ∈ (TopOn‘(Base‘𝐺)))
2411, 23eqeltrrd 2853 . . . . . 6 (𝐴𝑉 → (TopOpen‘𝐺) ∈ (TopOn‘(Base‘𝐺)))
25 id 22 . . . . . 6 (𝐴𝑉𝐴𝑉)
26 distop 21695 . . . . . . 7 (𝐴𝑉 → 𝒫 𝐴 ∈ Top)
27 fconst6g 6553 . . . . . . 7 (𝒫 𝐴 ∈ Top → (𝐴 × {𝒫 𝐴}):𝐴⟶Top)
2826, 27syl 17 . . . . . 6 (𝐴𝑉 → (𝐴 × {𝒫 𝐴}):𝐴⟶Top)
2915biimpa 480 . . . . . . . . . . . 12 ((𝐴𝑉𝑥 ∈ (Base‘𝐺)) → 𝑥:𝐴1-1-onto𝐴)
30 f1ocnv 6614 . . . . . . . . . . . 12 (𝑥:𝐴1-1-onto𝐴𝑥:𝐴1-1-onto𝐴)
31 f1of 6602 . . . . . . . . . . . 12 (𝑥:𝐴1-1-onto𝐴𝑥:𝐴𝐴)
3229, 30, 313syl 18 . . . . . . . . . . 11 ((𝐴𝑉𝑥 ∈ (Base‘𝐺)) → 𝑥:𝐴𝐴)
3332ffvelrnda 6842 . . . . . . . . . 10 (((𝐴𝑉𝑥 ∈ (Base‘𝐺)) ∧ 𝑦𝐴) → (𝑥𝑦) ∈ 𝐴)
3433an32s 651 . . . . . . . . 9 (((𝐴𝑉𝑦𝐴) ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑥𝑦) ∈ 𝐴)
3534fmpttd 6870 . . . . . . . 8 ((𝐴𝑉𝑦𝐴) → (𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦)):(Base‘𝐺)⟶𝐴)
3635adantr 484 . . . . . . . . . 10 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → (𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦)):(Base‘𝐺)⟶𝐴)
37 cnveq 5713 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑓𝑥 = 𝑓)
3837fveq1d 6660 . . . . . . . . . . . . . . 15 (𝑥 = 𝑓 → (𝑥𝑦) = (𝑓𝑦))
39 eqid 2758 . . . . . . . . . . . . . . 15 (𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦)) = (𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦))
40 fvex 6671 . . . . . . . . . . . . . . 15 (𝑓𝑦) ∈ V
4138, 39, 40fvmpt 6759 . . . . . . . . . . . . . 14 (𝑓 ∈ (Base‘𝐺) → ((𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦))‘𝑓) = (𝑓𝑦))
4241ad2antlr 726 . . . . . . . . . . . . 13 ((((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ 𝑡 ∈ 𝒫 𝐴) → ((𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦))‘𝑓) = (𝑓𝑦))
4342eleq1d 2836 . . . . . . . . . . . 12 ((((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ 𝑡 ∈ 𝒫 𝐴) → (((𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦))‘𝑓) ∈ 𝑡 ↔ (𝑓𝑦) ∈ 𝑡))
44 eqid 2758 . . . . . . . . . . . . . . . . . 18 (𝑢 ∈ (Base‘𝐺) ↦ (𝑢‘(𝑓𝑦))) = (𝑢 ∈ (Base‘𝐺) ↦ (𝑢‘(𝑓𝑦)))
4544mptiniseg 6068 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ V → ((𝑢 ∈ (Base‘𝐺) ↦ (𝑢‘(𝑓𝑦))) “ {𝑦}) = {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(𝑓𝑦)) = 𝑦})
4645elv 3415 . . . . . . . . . . . . . . . 16 ((𝑢 ∈ (Base‘𝐺) ↦ (𝑢‘(𝑓𝑦))) “ {𝑦}) = {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(𝑓𝑦)) = 𝑦}
47 eqid 2758 . . . . . . . . . . . . . . . . . . 19 ((∏t‘(𝐴 × {𝒫 𝐴})) ↾t (Base‘𝐺)) = ((∏t‘(𝐴 × {𝒫 𝐴})) ↾t (Base‘𝐺))
4814ad2antrr 725 . . . . . . . . . . . . . . . . . . 19 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → (∏t‘(𝐴 × {𝒫 𝐴})) ∈ (TopOn‘(𝐴m 𝐴)))
4921ad2antrr 725 . . . . . . . . . . . . . . . . . . 19 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → (Base‘𝐺) ⊆ (𝐴m 𝐴))
50 toponuni 21614 . . . . . . . . . . . . . . . . . . . . 21 ((∏t‘(𝐴 × {𝒫 𝐴})) ∈ (TopOn‘(𝐴m 𝐴)) → (𝐴m 𝐴) = (∏t‘(𝐴 × {𝒫 𝐴})))
51 mpteq1 5120 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴m 𝐴) = (∏t‘(𝐴 × {𝒫 𝐴})) → (𝑢 ∈ (𝐴m 𝐴) ↦ (𝑢‘(𝑓𝑦))) = (𝑢 (∏t‘(𝐴 × {𝒫 𝐴})) ↦ (𝑢‘(𝑓𝑦))))
5248, 50, 513syl 18 . . . . . . . . . . . . . . . . . . . 20 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → (𝑢 ∈ (𝐴m 𝐴) ↦ (𝑢‘(𝑓𝑦))) = (𝑢 (∏t‘(𝐴 × {𝒫 𝐴})) ↦ (𝑢‘(𝑓𝑦))))
53 simpll 766 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → 𝐴𝑉)
5428ad2antrr 725 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → (𝐴 × {𝒫 𝐴}):𝐴⟶Top)
551, 5elsymgbas 18569 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝐴𝑉 → (𝑓 ∈ (Base‘𝐺) ↔ 𝑓:𝐴1-1-onto𝐴))
5655adantr 484 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝐴𝑉𝑦𝐴) → (𝑓 ∈ (Base‘𝐺) ↔ 𝑓:𝐴1-1-onto𝐴))
5756biimpa 480 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → 𝑓:𝐴1-1-onto𝐴)
58 f1ocnv 6614 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑓:𝐴1-1-onto𝐴𝑓:𝐴1-1-onto𝐴)
59 f1of 6602 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑓:𝐴1-1-onto𝐴𝑓:𝐴𝐴)
6057, 58, 593syl 18 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → 𝑓:𝐴𝐴)
61 simplr 768 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → 𝑦𝐴)
6260, 61ffvelrnd 6843 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → (𝑓𝑦) ∈ 𝐴)
63 eqid 2758 . . . . . . . . . . . . . . . . . . . . . . 23 (∏t‘(𝐴 × {𝒫 𝐴})) = (∏t‘(𝐴 × {𝒫 𝐴}))
6463, 10ptpjcn 22311 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴𝑉 ∧ (𝐴 × {𝒫 𝐴}):𝐴⟶Top ∧ (𝑓𝑦) ∈ 𝐴) → (𝑢 (∏t‘(𝐴 × {𝒫 𝐴})) ↦ (𝑢‘(𝑓𝑦))) ∈ ((∏t‘(𝐴 × {𝒫 𝐴})) Cn ((𝐴 × {𝒫 𝐴})‘(𝑓𝑦))))
6553, 54, 62, 64syl3anc 1368 . . . . . . . . . . . . . . . . . . . . 21 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → (𝑢 (∏t‘(𝐴 × {𝒫 𝐴})) ↦ (𝑢‘(𝑓𝑦))) ∈ ((∏t‘(𝐴 × {𝒫 𝐴})) Cn ((𝐴 × {𝒫 𝐴})‘(𝑓𝑦))))
6626ad2antrr 725 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → 𝒫 𝐴 ∈ Top)
67 fvconst2g 6955 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝒫 𝐴 ∈ Top ∧ (𝑓𝑦) ∈ 𝐴) → ((𝐴 × {𝒫 𝐴})‘(𝑓𝑦)) = 𝒫 𝐴)
6866, 62, 67syl2anc 587 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → ((𝐴 × {𝒫 𝐴})‘(𝑓𝑦)) = 𝒫 𝐴)
6968oveq2d 7166 . . . . . . . . . . . . . . . . . . . . 21 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → ((∏t‘(𝐴 × {𝒫 𝐴})) Cn ((𝐴 × {𝒫 𝐴})‘(𝑓𝑦))) = ((∏t‘(𝐴 × {𝒫 𝐴})) Cn 𝒫 𝐴))
7065, 69eleqtrd 2854 . . . . . . . . . . . . . . . . . . . 20 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → (𝑢 (∏t‘(𝐴 × {𝒫 𝐴})) ↦ (𝑢‘(𝑓𝑦))) ∈ ((∏t‘(𝐴 × {𝒫 𝐴})) Cn 𝒫 𝐴))
7152, 70eqeltrd 2852 . . . . . . . . . . . . . . . . . . 19 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → (𝑢 ∈ (𝐴m 𝐴) ↦ (𝑢‘(𝑓𝑦))) ∈ ((∏t‘(𝐴 × {𝒫 𝐴})) Cn 𝒫 𝐴))
7247, 48, 49, 71cnmpt1res 22376 . . . . . . . . . . . . . . . . . 18 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → (𝑢 ∈ (Base‘𝐺) ↦ (𝑢‘(𝑓𝑦))) ∈ (((∏t‘(𝐴 × {𝒫 𝐴})) ↾t (Base‘𝐺)) Cn 𝒫 𝐴))
7311oveq1d 7165 . . . . . . . . . . . . . . . . . . 19 (𝐴𝑉 → (((∏t‘(𝐴 × {𝒫 𝐴})) ↾t (Base‘𝐺)) Cn 𝒫 𝐴) = ((TopOpen‘𝐺) Cn 𝒫 𝐴))
7473ad2antrr 725 . . . . . . . . . . . . . . . . . 18 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → (((∏t‘(𝐴 × {𝒫 𝐴})) ↾t (Base‘𝐺)) Cn 𝒫 𝐴) = ((TopOpen‘𝐺) Cn 𝒫 𝐴))
7572, 74eleqtrd 2854 . . . . . . . . . . . . . . . . 17 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → (𝑢 ∈ (Base‘𝐺) ↦ (𝑢‘(𝑓𝑦))) ∈ ((TopOpen‘𝐺) Cn 𝒫 𝐴))
76 snelpwi 5305 . . . . . . . . . . . . . . . . . 18 (𝑦𝐴 → {𝑦} ∈ 𝒫 𝐴)
7776ad2antlr 726 . . . . . . . . . . . . . . . . 17 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → {𝑦} ∈ 𝒫 𝐴)
78 cnima 21965 . . . . . . . . . . . . . . . . 17 (((𝑢 ∈ (Base‘𝐺) ↦ (𝑢‘(𝑓𝑦))) ∈ ((TopOpen‘𝐺) Cn 𝒫 𝐴) ∧ {𝑦} ∈ 𝒫 𝐴) → ((𝑢 ∈ (Base‘𝐺) ↦ (𝑢‘(𝑓𝑦))) “ {𝑦}) ∈ (TopOpen‘𝐺))
7975, 77, 78syl2anc 587 . . . . . . . . . . . . . . . 16 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → ((𝑢 ∈ (Base‘𝐺) ↦ (𝑢‘(𝑓𝑦))) “ {𝑦}) ∈ (TopOpen‘𝐺))
8046, 79eqeltrrid 2857 . . . . . . . . . . . . . . 15 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(𝑓𝑦)) = 𝑦} ∈ (TopOpen‘𝐺))
8180adantr 484 . . . . . . . . . . . . . 14 ((((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ (𝑡 ∈ 𝒫 𝐴 ∧ (𝑓𝑦) ∈ 𝑡)) → {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(𝑓𝑦)) = 𝑦} ∈ (TopOpen‘𝐺))
82 fveq1 6657 . . . . . . . . . . . . . . . 16 (𝑢 = 𝑓 → (𝑢‘(𝑓𝑦)) = (𝑓‘(𝑓𝑦)))
8382eqeq1d 2760 . . . . . . . . . . . . . . 15 (𝑢 = 𝑓 → ((𝑢‘(𝑓𝑦)) = 𝑦 ↔ (𝑓‘(𝑓𝑦)) = 𝑦))
84 simplr 768 . . . . . . . . . . . . . . 15 ((((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ (𝑡 ∈ 𝒫 𝐴 ∧ (𝑓𝑦) ∈ 𝑡)) → 𝑓 ∈ (Base‘𝐺))
8557adantr 484 . . . . . . . . . . . . . . . 16 ((((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ (𝑡 ∈ 𝒫 𝐴 ∧ (𝑓𝑦) ∈ 𝑡)) → 𝑓:𝐴1-1-onto𝐴)
86 simpllr 775 . . . . . . . . . . . . . . . 16 ((((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ (𝑡 ∈ 𝒫 𝐴 ∧ (𝑓𝑦) ∈ 𝑡)) → 𝑦𝐴)
87 f1ocnvfv2 7026 . . . . . . . . . . . . . . . 16 ((𝑓:𝐴1-1-onto𝐴𝑦𝐴) → (𝑓‘(𝑓𝑦)) = 𝑦)
8885, 86, 87syl2anc 587 . . . . . . . . . . . . . . 15 ((((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ (𝑡 ∈ 𝒫 𝐴 ∧ (𝑓𝑦) ∈ 𝑡)) → (𝑓‘(𝑓𝑦)) = 𝑦)
8983, 84, 88elrabd 3604 . . . . . . . . . . . . . 14 ((((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ (𝑡 ∈ 𝒫 𝐴 ∧ (𝑓𝑦) ∈ 𝑡)) → 𝑓 ∈ {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(𝑓𝑦)) = 𝑦})
90 ssrab2 3984 . . . . . . . . . . . . . . . . . 18 {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(𝑓𝑦)) = 𝑦} ⊆ (Base‘𝐺)
9190a1i 11 . . . . . . . . . . . . . . . . 17 ((((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ (𝑡 ∈ 𝒫 𝐴 ∧ (𝑓𝑦) ∈ 𝑡)) → {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(𝑓𝑦)) = 𝑦} ⊆ (Base‘𝐺))
9215ad3antrrr 729 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ (𝑡 ∈ 𝒫 𝐴 ∧ (𝑓𝑦) ∈ 𝑡)) → (𝑥 ∈ (Base‘𝐺) ↔ 𝑥:𝐴1-1-onto𝐴))
9392biimpa 480 . . . . . . . . . . . . . . . . . . . . 21 (((((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ (𝑡 ∈ 𝒫 𝐴 ∧ (𝑓𝑦) ∈ 𝑡)) ∧ 𝑥 ∈ (Base‘𝐺)) → 𝑥:𝐴1-1-onto𝐴)
9462ad2antrr 725 . . . . . . . . . . . . . . . . . . . . 21 (((((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ (𝑡 ∈ 𝒫 𝐴 ∧ (𝑓𝑦) ∈ 𝑡)) ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑓𝑦) ∈ 𝐴)
95 f1ocnvfv 7027 . . . . . . . . . . . . . . . . . . . . 21 ((𝑥:𝐴1-1-onto𝐴 ∧ (𝑓𝑦) ∈ 𝐴) → ((𝑥‘(𝑓𝑦)) = 𝑦 → (𝑥𝑦) = (𝑓𝑦)))
9693, 94, 95syl2anc 587 . . . . . . . . . . . . . . . . . . . 20 (((((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ (𝑡 ∈ 𝒫 𝐴 ∧ (𝑓𝑦) ∈ 𝑡)) ∧ 𝑥 ∈ (Base‘𝐺)) → ((𝑥‘(𝑓𝑦)) = 𝑦 → (𝑥𝑦) = (𝑓𝑦)))
97 simplrr 777 . . . . . . . . . . . . . . . . . . . . 21 (((((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ (𝑡 ∈ 𝒫 𝐴 ∧ (𝑓𝑦) ∈ 𝑡)) ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑓𝑦) ∈ 𝑡)
98 eleq1 2839 . . . . . . . . . . . . . . . . . . . . 21 ((𝑥𝑦) = (𝑓𝑦) → ((𝑥𝑦) ∈ 𝑡 ↔ (𝑓𝑦) ∈ 𝑡))
9997, 98syl5ibrcom 250 . . . . . . . . . . . . . . . . . . . 20 (((((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ (𝑡 ∈ 𝒫 𝐴 ∧ (𝑓𝑦) ∈ 𝑡)) ∧ 𝑥 ∈ (Base‘𝐺)) → ((𝑥𝑦) = (𝑓𝑦) → (𝑥𝑦) ∈ 𝑡))
10096, 99syld 47 . . . . . . . . . . . . . . . . . . 19 (((((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ (𝑡 ∈ 𝒫 𝐴 ∧ (𝑓𝑦) ∈ 𝑡)) ∧ 𝑥 ∈ (Base‘𝐺)) → ((𝑥‘(𝑓𝑦)) = 𝑦 → (𝑥𝑦) ∈ 𝑡))
101100ralrimiva 3113 . . . . . . . . . . . . . . . . . 18 ((((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ (𝑡 ∈ 𝒫 𝐴 ∧ (𝑓𝑦) ∈ 𝑡)) → ∀𝑥 ∈ (Base‘𝐺)((𝑥‘(𝑓𝑦)) = 𝑦 → (𝑥𝑦) ∈ 𝑡))
102 fveq1 6657 . . . . . . . . . . . . . . . . . . . 20 (𝑢 = 𝑥 → (𝑢‘(𝑓𝑦)) = (𝑥‘(𝑓𝑦)))
103102eqeq1d 2760 . . . . . . . . . . . . . . . . . . 19 (𝑢 = 𝑥 → ((𝑢‘(𝑓𝑦)) = 𝑦 ↔ (𝑥‘(𝑓𝑦)) = 𝑦))
104103ralrab 3608 . . . . . . . . . . . . . . . . . 18 (∀𝑥 ∈ {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(𝑓𝑦)) = 𝑦} (𝑥𝑦) ∈ 𝑡 ↔ ∀𝑥 ∈ (Base‘𝐺)((𝑥‘(𝑓𝑦)) = 𝑦 → (𝑥𝑦) ∈ 𝑡))
105101, 104sylibr 237 . . . . . . . . . . . . . . . . 17 ((((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ (𝑡 ∈ 𝒫 𝐴 ∧ (𝑓𝑦) ∈ 𝑡)) → ∀𝑥 ∈ {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(𝑓𝑦)) = 𝑦} (𝑥𝑦) ∈ 𝑡)
106 ssrab 3977 . . . . . . . . . . . . . . . . 17 ({𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(𝑓𝑦)) = 𝑦} ⊆ {𝑥 ∈ (Base‘𝐺) ∣ (𝑥𝑦) ∈ 𝑡} ↔ ({𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(𝑓𝑦)) = 𝑦} ⊆ (Base‘𝐺) ∧ ∀𝑥 ∈ {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(𝑓𝑦)) = 𝑦} (𝑥𝑦) ∈ 𝑡))
10791, 105, 106sylanbrc 586 . . . . . . . . . . . . . . . 16 ((((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ (𝑡 ∈ 𝒫 𝐴 ∧ (𝑓𝑦) ∈ 𝑡)) → {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(𝑓𝑦)) = 𝑦} ⊆ {𝑥 ∈ (Base‘𝐺) ∣ (𝑥𝑦) ∈ 𝑡})
10839mptpreima 6067 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦)) “ 𝑡) = {𝑥 ∈ (Base‘𝐺) ∣ (𝑥𝑦) ∈ 𝑡}
109107, 108sseqtrrdi 3943 . . . . . . . . . . . . . . 15 ((((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ (𝑡 ∈ 𝒫 𝐴 ∧ (𝑓𝑦) ∈ 𝑡)) → {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(𝑓𝑦)) = 𝑦} ⊆ ((𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦)) “ 𝑡))
110 funmpt 6373 . . . . . . . . . . . . . . . 16 Fun (𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦))
111 fvex 6671 . . . . . . . . . . . . . . . . . 18 (𝑥𝑦) ∈ V
112111, 39dmmpti 6475 . . . . . . . . . . . . . . . . 17 dom (𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦)) = (Base‘𝐺)
11391, 112sseqtrrdi 3943 . . . . . . . . . . . . . . . 16 ((((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ (𝑡 ∈ 𝒫 𝐴 ∧ (𝑓𝑦) ∈ 𝑡)) → {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(𝑓𝑦)) = 𝑦} ⊆ dom (𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦)))
114 funimass3 6815 . . . . . . . . . . . . . . . 16 ((Fun (𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦)) ∧ {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(𝑓𝑦)) = 𝑦} ⊆ dom (𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦))) → (((𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦)) “ {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(𝑓𝑦)) = 𝑦}) ⊆ 𝑡 ↔ {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(𝑓𝑦)) = 𝑦} ⊆ ((𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦)) “ 𝑡)))
115110, 113, 114sylancr 590 . . . . . . . . . . . . . . 15 ((((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ (𝑡 ∈ 𝒫 𝐴 ∧ (𝑓𝑦) ∈ 𝑡)) → (((𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦)) “ {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(𝑓𝑦)) = 𝑦}) ⊆ 𝑡 ↔ {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(𝑓𝑦)) = 𝑦} ⊆ ((𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦)) “ 𝑡)))
116109, 115mpbird 260 . . . . . . . . . . . . . 14 ((((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ (𝑡 ∈ 𝒫 𝐴 ∧ (𝑓𝑦) ∈ 𝑡)) → ((𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦)) “ {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(𝑓𝑦)) = 𝑦}) ⊆ 𝑡)
117 eleq2 2840 . . . . . . . . . . . . . . . 16 (𝑣 = {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(𝑓𝑦)) = 𝑦} → (𝑓𝑣𝑓 ∈ {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(𝑓𝑦)) = 𝑦}))
118 imaeq2 5897 . . . . . . . . . . . . . . . . 17 (𝑣 = {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(𝑓𝑦)) = 𝑦} → ((𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦)) “ 𝑣) = ((𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦)) “ {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(𝑓𝑦)) = 𝑦}))
119118sseq1d 3923 . . . . . . . . . . . . . . . 16 (𝑣 = {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(𝑓𝑦)) = 𝑦} → (((𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦)) “ 𝑣) ⊆ 𝑡 ↔ ((𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦)) “ {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(𝑓𝑦)) = 𝑦}) ⊆ 𝑡))
120117, 119anbi12d 633 . . . . . . . . . . . . . . 15 (𝑣 = {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(𝑓𝑦)) = 𝑦} → ((𝑓𝑣 ∧ ((𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦)) “ 𝑣) ⊆ 𝑡) ↔ (𝑓 ∈ {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(𝑓𝑦)) = 𝑦} ∧ ((𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦)) “ {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(𝑓𝑦)) = 𝑦}) ⊆ 𝑡)))
121120rspcev 3541 . . . . . . . . . . . . . 14 (({𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(𝑓𝑦)) = 𝑦} ∈ (TopOpen‘𝐺) ∧ (𝑓 ∈ {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(𝑓𝑦)) = 𝑦} ∧ ((𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦)) “ {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(𝑓𝑦)) = 𝑦}) ⊆ 𝑡)) → ∃𝑣 ∈ (TopOpen‘𝐺)(𝑓𝑣 ∧ ((𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦)) “ 𝑣) ⊆ 𝑡))
12281, 89, 116, 121syl12anc 835 . . . . . . . . . . . . 13 ((((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ (𝑡 ∈ 𝒫 𝐴 ∧ (𝑓𝑦) ∈ 𝑡)) → ∃𝑣 ∈ (TopOpen‘𝐺)(𝑓𝑣 ∧ ((𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦)) “ 𝑣) ⊆ 𝑡))
123122expr 460 . . . . . . . . . . . 12 ((((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ 𝑡 ∈ 𝒫 𝐴) → ((𝑓𝑦) ∈ 𝑡 → ∃𝑣 ∈ (TopOpen‘𝐺)(𝑓𝑣 ∧ ((𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦)) “ 𝑣) ⊆ 𝑡)))
12443, 123sylbid 243 . . . . . . . . . . 11 ((((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ 𝑡 ∈ 𝒫 𝐴) → (((𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦))‘𝑓) ∈ 𝑡 → ∃𝑣 ∈ (TopOpen‘𝐺)(𝑓𝑣 ∧ ((𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦)) “ 𝑣) ⊆ 𝑡)))
125124ralrimiva 3113 . . . . . . . . . 10 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → ∀𝑡 ∈ 𝒫 𝐴(((𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦))‘𝑓) ∈ 𝑡 → ∃𝑣 ∈ (TopOpen‘𝐺)(𝑓𝑣 ∧ ((𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦)) “ 𝑣) ⊆ 𝑡)))
12624ad2antrr 725 . . . . . . . . . . 11 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → (TopOpen‘𝐺) ∈ (TopOn‘(Base‘𝐺)))
12712ad2antrr 725 . . . . . . . . . . 11 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → 𝒫 𝐴 ∈ (TopOn‘𝐴))
128 simpr 488 . . . . . . . . . . 11 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → 𝑓 ∈ (Base‘𝐺))
129 iscnp 21937 . . . . . . . . . . 11 (((TopOpen‘𝐺) ∈ (TopOn‘(Base‘𝐺)) ∧ 𝒫 𝐴 ∈ (TopOn‘𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → ((𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦)) ∈ (((TopOpen‘𝐺) CnP 𝒫 𝐴)‘𝑓) ↔ ((𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦)):(Base‘𝐺)⟶𝐴 ∧ ∀𝑡 ∈ 𝒫 𝐴(((𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦))‘𝑓) ∈ 𝑡 → ∃𝑣 ∈ (TopOpen‘𝐺)(𝑓𝑣 ∧ ((𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦)) “ 𝑣) ⊆ 𝑡)))))
130126, 127, 128, 129syl3anc 1368 . . . . . . . . . 10 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → ((𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦)) ∈ (((TopOpen‘𝐺) CnP 𝒫 𝐴)‘𝑓) ↔ ((𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦)):(Base‘𝐺)⟶𝐴 ∧ ∀𝑡 ∈ 𝒫 𝐴(((𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦))‘𝑓) ∈ 𝑡 → ∃𝑣 ∈ (TopOpen‘𝐺)(𝑓𝑣 ∧ ((𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦)) “ 𝑣) ⊆ 𝑡)))))
13136, 125, 130mpbir2and 712 . . . . . . . . 9 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → (𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦)) ∈ (((TopOpen‘𝐺) CnP 𝒫 𝐴)‘𝑓))
132131ralrimiva 3113 . . . . . . . 8 ((𝐴𝑉𝑦𝐴) → ∀𝑓 ∈ (Base‘𝐺)(𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦)) ∈ (((TopOpen‘𝐺) CnP 𝒫 𝐴)‘𝑓))
133 cncnp 21980 . . . . . . . . . 10 (((TopOpen‘𝐺) ∈ (TopOn‘(Base‘𝐺)) ∧ 𝒫 𝐴 ∈ (TopOn‘𝐴)) → ((𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦)) ∈ ((TopOpen‘𝐺) Cn 𝒫 𝐴) ↔ ((𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦)):(Base‘𝐺)⟶𝐴 ∧ ∀𝑓 ∈ (Base‘𝐺)(𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦)) ∈ (((TopOpen‘𝐺) CnP 𝒫 𝐴)‘𝑓))))
13424, 12, 133syl2anc 587 . . . . . . . . 9 (𝐴𝑉 → ((𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦)) ∈ ((TopOpen‘𝐺) Cn 𝒫 𝐴) ↔ ((𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦)):(Base‘𝐺)⟶𝐴 ∧ ∀𝑓 ∈ (Base‘𝐺)(𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦)) ∈ (((TopOpen‘𝐺) CnP 𝒫 𝐴)‘𝑓))))
135134adantr 484 . . . . . . . 8 ((𝐴𝑉𝑦𝐴) → ((𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦)) ∈ ((TopOpen‘𝐺) Cn 𝒫 𝐴) ↔ ((𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦)):(Base‘𝐺)⟶𝐴 ∧ ∀𝑓 ∈ (Base‘𝐺)(𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦)) ∈ (((TopOpen‘𝐺) CnP 𝒫 𝐴)‘𝑓))))
13635, 132, 135mpbir2and 712 . . . . . . 7 ((𝐴𝑉𝑦𝐴) → (𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦)) ∈ ((TopOpen‘𝐺) Cn 𝒫 𝐴))
137 fvconst2g 6955 . . . . . . . . 9 ((𝒫 𝐴 ∈ Top ∧ 𝑦𝐴) → ((𝐴 × {𝒫 𝐴})‘𝑦) = 𝒫 𝐴)
13826, 137sylan 583 . . . . . . . 8 ((𝐴𝑉𝑦𝐴) → ((𝐴 × {𝒫 𝐴})‘𝑦) = 𝒫 𝐴)
139138oveq2d 7166 . . . . . . 7 ((𝐴𝑉𝑦𝐴) → ((TopOpen‘𝐺) Cn ((𝐴 × {𝒫 𝐴})‘𝑦)) = ((TopOpen‘𝐺) Cn 𝒫 𝐴))
140136, 139eleqtrrd 2855 . . . . . 6 ((𝐴𝑉𝑦𝐴) → (𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦)) ∈ ((TopOpen‘𝐺) Cn ((𝐴 × {𝒫 𝐴})‘𝑦)))
14110, 24, 25, 28, 140ptcn 22327 . . . . 5 (𝐴𝑉 → (𝑥 ∈ (Base‘𝐺) ↦ (𝑦𝐴 ↦ (𝑥𝑦))) ∈ ((TopOpen‘𝐺) Cn (∏t‘(𝐴 × {𝒫 𝐴}))))
142 eqid 2758 . . . . . . . . 9 (invg𝐺) = (invg𝐺)
1435, 142grpinvf 18217 . . . . . . . 8 (𝐺 ∈ Grp → (invg𝐺):(Base‘𝐺)⟶(Base‘𝐺))
1442, 143syl 17 . . . . . . 7 (𝐴𝑉 → (invg𝐺):(Base‘𝐺)⟶(Base‘𝐺))
145144feqmptd 6721 . . . . . 6 (𝐴𝑉 → (invg𝐺) = (𝑥 ∈ (Base‘𝐺) ↦ ((invg𝐺)‘𝑥)))
1461, 5, 142symginv 18597 . . . . . . . . 9 (𝑥 ∈ (Base‘𝐺) → ((invg𝐺)‘𝑥) = 𝑥)
147146adantl 485 . . . . . . . 8 ((𝐴𝑉𝑥 ∈ (Base‘𝐺)) → ((invg𝐺)‘𝑥) = 𝑥)
14832feqmptd 6721 . . . . . . . 8 ((𝐴𝑉𝑥 ∈ (Base‘𝐺)) → 𝑥 = (𝑦𝐴 ↦ (𝑥𝑦)))
149147, 148eqtrd 2793 . . . . . . 7 ((𝐴𝑉𝑥 ∈ (Base‘𝐺)) → ((invg𝐺)‘𝑥) = (𝑦𝐴 ↦ (𝑥𝑦)))
150149mpteq2dva 5127 . . . . . 6 (𝐴𝑉 → (𝑥 ∈ (Base‘𝐺) ↦ ((invg𝐺)‘𝑥)) = (𝑥 ∈ (Base‘𝐺) ↦ (𝑦𝐴 ↦ (𝑥𝑦))))
151145, 150eqtrd 2793 . . . . 5 (𝐴𝑉 → (invg𝐺) = (𝑥 ∈ (Base‘𝐺) ↦ (𝑦𝐴 ↦ (𝑥𝑦))))
152 xkopt 22355 . . . . . . 7 ((𝒫 𝐴 ∈ Top ∧ 𝐴𝑉) → (𝒫 𝐴ko 𝒫 𝐴) = (∏t‘(𝐴 × {𝒫 𝐴})))
15326, 152mpancom 687 . . . . . 6 (𝐴𝑉 → (𝒫 𝐴ko 𝒫 𝐴) = (∏t‘(𝐴 × {𝒫 𝐴})))
154153oveq2d 7166 . . . . 5 (𝐴𝑉 → ((TopOpen‘𝐺) Cn (𝒫 𝐴ko 𝒫 𝐴)) = ((TopOpen‘𝐺) Cn (∏t‘(𝐴 × {𝒫 𝐴}))))
155141, 151, 1543eltr4d 2867 . . . 4 (𝐴𝑉 → (invg𝐺) ∈ ((TopOpen‘𝐺) Cn (𝒫 𝐴ko 𝒫 𝐴)))
156 eqid 2758 . . . . . . 7 (𝒫 𝐴ko 𝒫 𝐴) = (𝒫 𝐴ko 𝒫 𝐴)
157156xkotopon 22300 . . . . . 6 ((𝒫 𝐴 ∈ Top ∧ 𝒫 𝐴 ∈ Top) → (𝒫 𝐴ko 𝒫 𝐴) ∈ (TopOn‘(𝒫 𝐴 Cn 𝒫 𝐴)))
15826, 26, 157syl2anc 587 . . . . 5 (𝐴𝑉 → (𝒫 𝐴ko 𝒫 𝐴) ∈ (TopOn‘(𝒫 𝐴 Cn 𝒫 𝐴)))
159 frn 6504 . . . . . 6 ((invg𝐺):(Base‘𝐺)⟶(Base‘𝐺) → ran (invg𝐺) ⊆ (Base‘𝐺))
1602, 143, 1593syl 18 . . . . 5 (𝐴𝑉 → ran (invg𝐺) ⊆ (Base‘𝐺))
161 cndis 21991 . . . . . . 7 ((𝐴𝑉 ∧ 𝒫 𝐴 ∈ (TopOn‘𝐴)) → (𝒫 𝐴 Cn 𝒫 𝐴) = (𝐴m 𝐴))
16212, 161mpdan 686 . . . . . 6 (𝐴𝑉 → (𝒫 𝐴 Cn 𝒫 𝐴) = (𝐴m 𝐴))
16321, 162sseqtrrd 3933 . . . . 5 (𝐴𝑉 → (Base‘𝐺) ⊆ (𝒫 𝐴 Cn 𝒫 𝐴))
164 cnrest2 21986 . . . . 5 (((𝒫 𝐴ko 𝒫 𝐴) ∈ (TopOn‘(𝒫 𝐴 Cn 𝒫 𝐴)) ∧ ran (invg𝐺) ⊆ (Base‘𝐺) ∧ (Base‘𝐺) ⊆ (𝒫 𝐴 Cn 𝒫 𝐴)) → ((invg𝐺) ∈ ((TopOpen‘𝐺) Cn (𝒫 𝐴ko 𝒫 𝐴)) ↔ (invg𝐺) ∈ ((TopOpen‘𝐺) Cn ((𝒫 𝐴ko 𝒫 𝐴) ↾t (Base‘𝐺)))))
165158, 160, 163, 164syl3anc 1368 . . . 4 (𝐴𝑉 → ((invg𝐺) ∈ ((TopOpen‘𝐺) Cn (𝒫 𝐴ko 𝒫 𝐴)) ↔ (invg𝐺) ∈ ((TopOpen‘𝐺) Cn ((𝒫 𝐴ko 𝒫 𝐴) ↾t (Base‘𝐺)))))
166155, 165mpbid 235 . . 3 (𝐴𝑉 → (invg𝐺) ∈ ((TopOpen‘𝐺) Cn ((𝒫 𝐴ko 𝒫 𝐴) ↾t (Base‘𝐺))))
167153oveq1d 7165 . . . . 5 (𝐴𝑉 → ((𝒫 𝐴ko 𝒫 𝐴) ↾t (Base‘𝐺)) = ((∏t‘(𝐴 × {𝒫 𝐴})) ↾t (Base‘𝐺)))
168167, 11eqtrd 2793 . . . 4 (𝐴𝑉 → ((𝒫 𝐴ko 𝒫 𝐴) ↾t (Base‘𝐺)) = (TopOpen‘𝐺))
169168oveq2d 7166 . . 3 (𝐴𝑉 → ((TopOpen‘𝐺) Cn ((𝒫 𝐴ko 𝒫 𝐴) ↾t (Base‘𝐺))) = ((TopOpen‘𝐺) Cn (TopOpen‘𝐺)))
170166, 169eleqtrd 2854 . 2 (𝐴𝑉 → (invg𝐺) ∈ ((TopOpen‘𝐺) Cn (TopOpen‘𝐺)))
171 eqid 2758 . . 3 (TopOpen‘𝐺) = (TopOpen‘𝐺)
172171, 142istgp 22777 . 2 (𝐺 ∈ TopGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ TopMnd ∧ (invg𝐺) ∈ ((TopOpen‘𝐺) Cn (TopOpen‘𝐺))))
1732, 9, 170, 172syl3anbrc 1340 1 (𝐴𝑉𝐺 ∈ TopGrp)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3070  ∃wrex 3071  {crab 3074  Vcvv 3409   ⊆ wss 3858  𝒫 cpw 4494  {csn 4522  ∪ cuni 4798   ↦ cmpt 5112   × cxp 5522  ◡ccnv 5523  dom cdm 5524  ran crn 5525   “ cima 5527  Fun wfun 6329  ⟶wf 6331  –1-1-onto→wf1o 6334  ‘cfv 6335  (class class class)co 7150   ↑m cmap 8416  Basecbs 16541   ↾t crest 16752  TopOpenctopn 16753  ∏tcpt 16770  SubMndcsubmnd 18021  EndoFMndcefmnd 18099  Grpcgrp 18169  invgcminusg 18170  SymGrpcsymg 18562  Topctop 21593  TopOnctopon 21610   Cn ccn 21924   CnP ccnp 21925   ↑ko cxko 22261  TopMndctmd 22770  TopGrpctgp 22771 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 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5156  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298  ax-un 7459  ax-cnex 10631  ax-resscn 10632  ax-1cn 10633  ax-icn 10634  ax-addcl 10635  ax-addrcl 10636  ax-mulcl 10637  ax-mulrcl 10638  ax-mulcom 10639  ax-addass 10640  ax-mulass 10641  ax-distr 10642  ax-i2m1 10643  ax-1ne0 10644  ax-1rid 10645  ax-rnegex 10646  ax-rrecex 10647  ax-cnre 10648  ax-pre-lttri 10649  ax-pre-lttrn 10650  ax-pre-ltadd 10651  ax-pre-mulgt0 10652 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-nel 3056  df-ral 3075  df-rex 3076  df-reu 3077  df-rmo 3078  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-pss 3877  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-tp 4527  df-op 4529  df-uni 4799  df-int 4839  df-iun 4885  df-iin 4886  df-br 5033  df-opab 5095  df-mpt 5113  df-tr 5139  df-id 5430  df-eprel 5435  df-po 5443  df-so 5444  df-fr 5483  df-we 5485  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7580  df-1st 7693  df-2nd 7694  df-wrecs 7957  df-recs 8018  df-rdg 8056  df-1o 8112  df-er 8299  df-map 8418  df-ixp 8480  df-en 8528  df-dom 8529  df-sdom 8530  df-fin 8531  df-fi 8908  df-pnf 10715  df-mnf 10716  df-xr 10717  df-ltxr 10718  df-le 10719  df-sub 10910  df-neg 10911  df-nn 11675  df-2 11737  df-3 11738  df-4 11739  df-5 11740  df-6 11741  df-7 11742  df-8 11743  df-9 11744  df-n0 11935  df-z 12021  df-uz 12283  df-fz 12940  df-struct 16543  df-ndx 16544  df-slot 16545  df-base 16547  df-sets 16548  df-ress 16549  df-plusg 16636  df-tset 16642  df-rest 16754  df-topn 16755  df-0g 16773  df-topgen 16775  df-pt 16776  df-plusf 17917  df-mgm 17918  df-sgrp 17967  df-mnd 17978  df-submnd 18023  df-efmnd 18100  df-grp 18172  df-minusg 18173  df-symg 18563  df-top 21594  df-topon 21611  df-topsp 21633  df-bases 21646  df-ntr 21720  df-nei 21798  df-cn 21927  df-cnp 21928  df-cmp 22087  df-lly 22166  df-nlly 22167  df-tx 22262  df-xko 22263  df-tmd 22772  df-tgp 22773 This theorem is referenced by: (None)
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