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Theorem symgtgp 23403
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 19135 . 2 (𝐴𝑉𝐺 ∈ Grp)
3 eqid 2736 . . . 4 (EndoFMnd‘𝐴) = (EndoFMnd‘𝐴)
43efmndtmd 23398 . . 3 (𝐴𝑉 → (EndoFMnd‘𝐴) ∈ TopMnd)
5 eqid 2736 . . . 4 (Base‘𝐺) = (Base‘𝐺)
63, 1, 5symgsubmefmnd 19133 . . 3 (𝐴𝑉 → (Base‘𝐺) ∈ (SubMnd‘(EndoFMnd‘𝐴)))
71, 5, 3symgressbas 19116 . . . 4 𝐺 = ((EndoFMnd‘𝐴) ↾s (Base‘𝐺))
87submtmd 23401 . . 3 (((EndoFMnd‘𝐴) ∈ TopMnd ∧ (Base‘𝐺) ∈ (SubMnd‘(EndoFMnd‘𝐴))) → 𝐺 ∈ TopMnd)
94, 6, 8syl2anc 584 . 2 (𝐴𝑉𝐺 ∈ TopMnd)
10 eqid 2736 . . . . . 6 (∏t‘(𝐴 × {𝒫 𝐴})) = (∏t‘(𝐴 × {𝒫 𝐴}))
111, 5symgtopn 19141 . . . . . . 7 (𝐴𝑉 → ((∏t‘(𝐴 × {𝒫 𝐴})) ↾t (Base‘𝐺)) = (TopOpen‘𝐺))
12 distopon 22293 . . . . . . . . 9 (𝐴𝑉 → 𝒫 𝐴 ∈ (TopOn‘𝐴))
1310pttoponconst 22894 . . . . . . . . 9 ((𝐴𝑉 ∧ 𝒫 𝐴 ∈ (TopOn‘𝐴)) → (∏t‘(𝐴 × {𝒫 𝐴})) ∈ (TopOn‘(𝐴m 𝐴)))
1412, 13mpdan 685 . . . . . . . 8 (𝐴𝑉 → (∏t‘(𝐴 × {𝒫 𝐴})) ∈ (TopOn‘(𝐴m 𝐴)))
151, 5elsymgbas 19108 . . . . . . . . . 10 (𝐴𝑉 → (𝑥 ∈ (Base‘𝐺) ↔ 𝑥:𝐴1-1-onto𝐴))
16 f1of 6781 . . . . . . . . . . 11 (𝑥:𝐴1-1-onto𝐴𝑥:𝐴𝐴)
17 elmapg 8736 . . . . . . . . . . . 12 ((𝐴𝑉𝐴𝑉) → (𝑥 ∈ (𝐴m 𝐴) ↔ 𝑥:𝐴𝐴))
1817anidms 567 . . . . . . . . . . 11 (𝐴𝑉 → (𝑥 ∈ (𝐴m 𝐴) ↔ 𝑥:𝐴𝐴))
1916, 18syl5ibr 245 . . . . . . . . . 10 (𝐴𝑉 → (𝑥:𝐴1-1-onto𝐴𝑥 ∈ (𝐴m 𝐴)))
2015, 19sylbid 239 . . . . . . . . 9 (𝐴𝑉 → (𝑥 ∈ (Base‘𝐺) → 𝑥 ∈ (𝐴m 𝐴)))
2120ssrdv 3948 . . . . . . . 8 (𝐴𝑉 → (Base‘𝐺) ⊆ (𝐴m 𝐴))
22 resttopon 22458 . . . . . . . 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 584 . . . . . . 7 (𝐴𝑉 → ((∏t‘(𝐴 × {𝒫 𝐴})) ↾t (Base‘𝐺)) ∈ (TopOn‘(Base‘𝐺)))
2411, 23eqeltrrd 2839 . . . . . 6 (𝐴𝑉 → (TopOpen‘𝐺) ∈ (TopOn‘(Base‘𝐺)))
25 id 22 . . . . . 6 (𝐴𝑉𝐴𝑉)
26 distop 22291 . . . . . . 7 (𝐴𝑉 → 𝒫 𝐴 ∈ Top)
27 fconst6g 6728 . . . . . . 7 (𝒫 𝐴 ∈ Top → (𝐴 × {𝒫 𝐴}):𝐴⟶Top)
2826, 27syl 17 . . . . . 6 (𝐴𝑉 → (𝐴 × {𝒫 𝐴}):𝐴⟶Top)
2915biimpa 477 . . . . . . . . . . . 12 ((𝐴𝑉𝑥 ∈ (Base‘𝐺)) → 𝑥:𝐴1-1-onto𝐴)
30 f1ocnv 6793 . . . . . . . . . . . 12 (𝑥:𝐴1-1-onto𝐴𝑥:𝐴1-1-onto𝐴)
31 f1of 6781 . . . . . . . . . . . 12 (𝑥:𝐴1-1-onto𝐴𝑥:𝐴𝐴)
3229, 30, 313syl 18 . . . . . . . . . . 11 ((𝐴𝑉𝑥 ∈ (Base‘𝐺)) → 𝑥:𝐴𝐴)
3332ffvelcdmda 7031 . . . . . . . . . 10 (((𝐴𝑉𝑥 ∈ (Base‘𝐺)) ∧ 𝑦𝐴) → (𝑥𝑦) ∈ 𝐴)
3433an32s 650 . . . . . . . . 9 (((𝐴𝑉𝑦𝐴) ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑥𝑦) ∈ 𝐴)
3534fmpttd 7059 . . . . . . . 8 ((𝐴𝑉𝑦𝐴) → (𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦)):(Base‘𝐺)⟶𝐴)
3635adantr 481 . . . . . . . . . 10 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → (𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦)):(Base‘𝐺)⟶𝐴)
37 cnveq 5827 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑓𝑥 = 𝑓)
3837fveq1d 6841 . . . . . . . . . . . . . . 15 (𝑥 = 𝑓 → (𝑥𝑦) = (𝑓𝑦))
39 eqid 2736 . . . . . . . . . . . . . . 15 (𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦)) = (𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦))
40 fvex 6852 . . . . . . . . . . . . . . 15 (𝑓𝑦) ∈ V
4138, 39, 40fvmpt 6945 . . . . . . . . . . . . . 14 (𝑓 ∈ (Base‘𝐺) → ((𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦))‘𝑓) = (𝑓𝑦))
4241ad2antlr 725 . . . . . . . . . . . . 13 ((((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ 𝑡 ∈ 𝒫 𝐴) → ((𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦))‘𝑓) = (𝑓𝑦))
4342eleq1d 2822 . . . . . . . . . . . 12 ((((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ 𝑡 ∈ 𝒫 𝐴) → (((𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦))‘𝑓) ∈ 𝑡 ↔ (𝑓𝑦) ∈ 𝑡))
44 eqid 2736 . . . . . . . . . . . . . . . . . 18 (𝑢 ∈ (Base‘𝐺) ↦ (𝑢‘(𝑓𝑦))) = (𝑢 ∈ (Base‘𝐺) ↦ (𝑢‘(𝑓𝑦)))
4544mptiniseg 6189 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ V → ((𝑢 ∈ (Base‘𝐺) ↦ (𝑢‘(𝑓𝑦))) “ {𝑦}) = {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(𝑓𝑦)) = 𝑦})
4645elv 3449 . . . . . . . . . . . . . . . 16 ((𝑢 ∈ (Base‘𝐺) ↦ (𝑢‘(𝑓𝑦))) “ {𝑦}) = {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(𝑓𝑦)) = 𝑦}
47 eqid 2736 . . . . . . . . . . . . . . . . . . 19 ((∏t‘(𝐴 × {𝒫 𝐴})) ↾t (Base‘𝐺)) = ((∏t‘(𝐴 × {𝒫 𝐴})) ↾t (Base‘𝐺))
4814ad2antrr 724 . . . . . . . . . . . . . . . . . . 19 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → (∏t‘(𝐴 × {𝒫 𝐴})) ∈ (TopOn‘(𝐴m 𝐴)))
4921ad2antrr 724 . . . . . . . . . . . . . . . . . . 19 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → (Base‘𝐺) ⊆ (𝐴m 𝐴))
50 toponuni 22209 . . . . . . . . . . . . . . . . . . . . 21 ((∏t‘(𝐴 × {𝒫 𝐴})) ∈ (TopOn‘(𝐴m 𝐴)) → (𝐴m 𝐴) = (∏t‘(𝐴 × {𝒫 𝐴})))
51 mpteq1 5196 . . . . . . . . . . . . . . . . . . . . 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 765 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → 𝐴𝑉)
5428ad2antrr 724 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → (𝐴 × {𝒫 𝐴}):𝐴⟶Top)
551, 5elsymgbas 19108 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝐴𝑉 → (𝑓 ∈ (Base‘𝐺) ↔ 𝑓:𝐴1-1-onto𝐴))
5655adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝐴𝑉𝑦𝐴) → (𝑓 ∈ (Base‘𝐺) ↔ 𝑓:𝐴1-1-onto𝐴))
5756biimpa 477 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → 𝑓:𝐴1-1-onto𝐴)
58 f1ocnv 6793 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑓:𝐴1-1-onto𝐴𝑓:𝐴1-1-onto𝐴)
59 f1of 6781 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑓:𝐴1-1-onto𝐴𝑓:𝐴𝐴)
6057, 58, 593syl 18 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → 𝑓:𝐴𝐴)
61 simplr 767 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → 𝑦𝐴)
6260, 61ffvelcdmd 7032 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → (𝑓𝑦) ∈ 𝐴)
63 eqid 2736 . . . . . . . . . . . . . . . . . . . . . . 23 (∏t‘(𝐴 × {𝒫 𝐴})) = (∏t‘(𝐴 × {𝒫 𝐴}))
6463, 10ptpjcn 22908 . . . . . . . . . . . . . . . . . . . . . 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 1371 . . . . . . . . . . . . . . . . . . . . 21 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → (𝑢 (∏t‘(𝐴 × {𝒫 𝐴})) ↦ (𝑢‘(𝑓𝑦))) ∈ ((∏t‘(𝐴 × {𝒫 𝐴})) Cn ((𝐴 × {𝒫 𝐴})‘(𝑓𝑦))))
6626ad2antrr 724 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → 𝒫 𝐴 ∈ Top)
67 fvconst2g 7147 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝒫 𝐴 ∈ Top ∧ (𝑓𝑦) ∈ 𝐴) → ((𝐴 × {𝒫 𝐴})‘(𝑓𝑦)) = 𝒫 𝐴)
6866, 62, 67syl2anc 584 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → ((𝐴 × {𝒫 𝐴})‘(𝑓𝑦)) = 𝒫 𝐴)
6968oveq2d 7367 . . . . . . . . . . . . . . . . . . . . 21 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → ((∏t‘(𝐴 × {𝒫 𝐴})) Cn ((𝐴 × {𝒫 𝐴})‘(𝑓𝑦))) = ((∏t‘(𝐴 × {𝒫 𝐴})) Cn 𝒫 𝐴))
7065, 69eleqtrd 2840 . . . . . . . . . . . . . . . . . . . 20 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → (𝑢 (∏t‘(𝐴 × {𝒫 𝐴})) ↦ (𝑢‘(𝑓𝑦))) ∈ ((∏t‘(𝐴 × {𝒫 𝐴})) Cn 𝒫 𝐴))
7152, 70eqeltrd 2838 . . . . . . . . . . . . . . . . . . 19 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → (𝑢 ∈ (𝐴m 𝐴) ↦ (𝑢‘(𝑓𝑦))) ∈ ((∏t‘(𝐴 × {𝒫 𝐴})) Cn 𝒫 𝐴))
7247, 48, 49, 71cnmpt1res 22973 . . . . . . . . . . . . . . . . . 18 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → (𝑢 ∈ (Base‘𝐺) ↦ (𝑢‘(𝑓𝑦))) ∈ (((∏t‘(𝐴 × {𝒫 𝐴})) ↾t (Base‘𝐺)) Cn 𝒫 𝐴))
7311oveq1d 7366 . . . . . . . . . . . . . . . . . . 19 (𝐴𝑉 → (((∏t‘(𝐴 × {𝒫 𝐴})) ↾t (Base‘𝐺)) Cn 𝒫 𝐴) = ((TopOpen‘𝐺) Cn 𝒫 𝐴))
7473ad2antrr 724 . . . . . . . . . . . . . . . . . 18 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → (((∏t‘(𝐴 × {𝒫 𝐴})) ↾t (Base‘𝐺)) Cn 𝒫 𝐴) = ((TopOpen‘𝐺) Cn 𝒫 𝐴))
7572, 74eleqtrd 2840 . . . . . . . . . . . . . . . . 17 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → (𝑢 ∈ (Base‘𝐺) ↦ (𝑢‘(𝑓𝑦))) ∈ ((TopOpen‘𝐺) Cn 𝒫 𝐴))
76 snelpwi 5398 . . . . . . . . . . . . . . . . . 18 (𝑦𝐴 → {𝑦} ∈ 𝒫 𝐴)
7776ad2antlr 725 . . . . . . . . . . . . . . . . 17 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → {𝑦} ∈ 𝒫 𝐴)
78 cnima 22562 . . . . . . . . . . . . . . . . 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 584 . . . . . . . . . . . . . . . 16 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → ((𝑢 ∈ (Base‘𝐺) ↦ (𝑢‘(𝑓𝑦))) “ {𝑦}) ∈ (TopOpen‘𝐺))
8046, 79eqeltrrid 2843 . . . . . . . . . . . . . . 15 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(𝑓𝑦)) = 𝑦} ∈ (TopOpen‘𝐺))
8180adantr 481 . . . . . . . . . . . . . 14 ((((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ (𝑡 ∈ 𝒫 𝐴 ∧ (𝑓𝑦) ∈ 𝑡)) → {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(𝑓𝑦)) = 𝑦} ∈ (TopOpen‘𝐺))
82 fveq1 6838 . . . . . . . . . . . . . . . 16 (𝑢 = 𝑓 → (𝑢‘(𝑓𝑦)) = (𝑓‘(𝑓𝑦)))
8382eqeq1d 2738 . . . . . . . . . . . . . . 15 (𝑢 = 𝑓 → ((𝑢‘(𝑓𝑦)) = 𝑦 ↔ (𝑓‘(𝑓𝑦)) = 𝑦))
84 simplr 767 . . . . . . . . . . . . . . 15 ((((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ (𝑡 ∈ 𝒫 𝐴 ∧ (𝑓𝑦) ∈ 𝑡)) → 𝑓 ∈ (Base‘𝐺))
8557adantr 481 . . . . . . . . . . . . . . . 16 ((((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ (𝑡 ∈ 𝒫 𝐴 ∧ (𝑓𝑦) ∈ 𝑡)) → 𝑓:𝐴1-1-onto𝐴)
86 simpllr 774 . . . . . . . . . . . . . . . 16 ((((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ (𝑡 ∈ 𝒫 𝐴 ∧ (𝑓𝑦) ∈ 𝑡)) → 𝑦𝐴)
87 f1ocnvfv2 7219 . . . . . . . . . . . . . . . 16 ((𝑓:𝐴1-1-onto𝐴𝑦𝐴) → (𝑓‘(𝑓𝑦)) = 𝑦)
8885, 86, 87syl2anc 584 . . . . . . . . . . . . . . 15 ((((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ (𝑡 ∈ 𝒫 𝐴 ∧ (𝑓𝑦) ∈ 𝑡)) → (𝑓‘(𝑓𝑦)) = 𝑦)
8983, 84, 88elrabd 3645 . . . . . . . . . . . . . 14 ((((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ (𝑡 ∈ 𝒫 𝐴 ∧ (𝑓𝑦) ∈ 𝑡)) → 𝑓 ∈ {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(𝑓𝑦)) = 𝑦})
90 ssrab2 4035 . . . . . . . . . . . . . . . . . 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 728 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ (𝑡 ∈ 𝒫 𝐴 ∧ (𝑓𝑦) ∈ 𝑡)) → (𝑥 ∈ (Base‘𝐺) ↔ 𝑥:𝐴1-1-onto𝐴))
9392biimpa 477 . . . . . . . . . . . . . . . . . . . . 21 (((((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ (𝑡 ∈ 𝒫 𝐴 ∧ (𝑓𝑦) ∈ 𝑡)) ∧ 𝑥 ∈ (Base‘𝐺)) → 𝑥:𝐴1-1-onto𝐴)
9462ad2antrr 724 . . . . . . . . . . . . . . . . . . . . 21 (((((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ (𝑡 ∈ 𝒫 𝐴 ∧ (𝑓𝑦) ∈ 𝑡)) ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑓𝑦) ∈ 𝐴)
95 f1ocnvfv 7220 . . . . . . . . . . . . . . . . . . . . 21 ((𝑥:𝐴1-1-onto𝐴 ∧ (𝑓𝑦) ∈ 𝐴) → ((𝑥‘(𝑓𝑦)) = 𝑦 → (𝑥𝑦) = (𝑓𝑦)))
9693, 94, 95syl2anc 584 . . . . . . . . . . . . . . . . . . . 20 (((((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ (𝑡 ∈ 𝒫 𝐴 ∧ (𝑓𝑦) ∈ 𝑡)) ∧ 𝑥 ∈ (Base‘𝐺)) → ((𝑥‘(𝑓𝑦)) = 𝑦 → (𝑥𝑦) = (𝑓𝑦)))
97 simplrr 776 . . . . . . . . . . . . . . . . . . . . 21 (((((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ (𝑡 ∈ 𝒫 𝐴 ∧ (𝑓𝑦) ∈ 𝑡)) ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑓𝑦) ∈ 𝑡)
98 eleq1 2825 . . . . . . . . . . . . . . . . . . . . 21 ((𝑥𝑦) = (𝑓𝑦) → ((𝑥𝑦) ∈ 𝑡 ↔ (𝑓𝑦) ∈ 𝑡))
9997, 98syl5ibrcom 246 . . . . . . . . . . . . . . . . . . . 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 3141 . . . . . . . . . . . . . . . . . 18 ((((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ (𝑡 ∈ 𝒫 𝐴 ∧ (𝑓𝑦) ∈ 𝑡)) → ∀𝑥 ∈ (Base‘𝐺)((𝑥‘(𝑓𝑦)) = 𝑦 → (𝑥𝑦) ∈ 𝑡))
102 fveq1 6838 . . . . . . . . . . . . . . . . . . . 20 (𝑢 = 𝑥 → (𝑢‘(𝑓𝑦)) = (𝑥‘(𝑓𝑦)))
103102eqeq1d 2738 . . . . . . . . . . . . . . . . . . 19 (𝑢 = 𝑥 → ((𝑢‘(𝑓𝑦)) = 𝑦 ↔ (𝑥‘(𝑓𝑦)) = 𝑦))
104103ralrab 3649 . . . . . . . . . . . . . . . . . 18 (∀𝑥 ∈ {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(𝑓𝑦)) = 𝑦} (𝑥𝑦) ∈ 𝑡 ↔ ∀𝑥 ∈ (Base‘𝐺)((𝑥‘(𝑓𝑦)) = 𝑦 → (𝑥𝑦) ∈ 𝑡))
105101, 104sylibr 233 . . . . . . . . . . . . . . . . 17 ((((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ (𝑡 ∈ 𝒫 𝐴 ∧ (𝑓𝑦) ∈ 𝑡)) → ∀𝑥 ∈ {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(𝑓𝑦)) = 𝑦} (𝑥𝑦) ∈ 𝑡)
106 ssrab 4028 . . . . . . . . . . . . . . . . 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 583 . . . . . . . . . . . . . . . 16 ((((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ (𝑡 ∈ 𝒫 𝐴 ∧ (𝑓𝑦) ∈ 𝑡)) → {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(𝑓𝑦)) = 𝑦} ⊆ {𝑥 ∈ (Base‘𝐺) ∣ (𝑥𝑦) ∈ 𝑡})
10839mptpreima 6188 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦)) “ 𝑡) = {𝑥 ∈ (Base‘𝐺) ∣ (𝑥𝑦) ∈ 𝑡}
109107, 108sseqtrrdi 3993 . . . . . . . . . . . . . . 15 ((((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ (𝑡 ∈ 𝒫 𝐴 ∧ (𝑓𝑦) ∈ 𝑡)) → {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(𝑓𝑦)) = 𝑦} ⊆ ((𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦)) “ 𝑡))
110 funmpt 6536 . . . . . . . . . . . . . . . 16 Fun (𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦))
111 fvex 6852 . . . . . . . . . . . . . . . . . 18 (𝑥𝑦) ∈ V
112111, 39dmmpti 6642 . . . . . . . . . . . . . . . . 17 dom (𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦)) = (Base‘𝐺)
11391, 112sseqtrrdi 3993 . . . . . . . . . . . . . . . 16 ((((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ (𝑡 ∈ 𝒫 𝐴 ∧ (𝑓𝑦) ∈ 𝑡)) → {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(𝑓𝑦)) = 𝑦} ⊆ dom (𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦)))
114 funimass3 7001 . . . . . . . . . . . . . . . 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 587 . . . . . . . . . . . . . . 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 256 . . . . . . . . . . . . . 14 ((((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ (𝑡 ∈ 𝒫 𝐴 ∧ (𝑓𝑦) ∈ 𝑡)) → ((𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦)) “ {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(𝑓𝑦)) = 𝑦}) ⊆ 𝑡)
117 eleq2 2826 . . . . . . . . . . . . . . . 16 (𝑣 = {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(𝑓𝑦)) = 𝑦} → (𝑓𝑣𝑓 ∈ {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(𝑓𝑦)) = 𝑦}))
118 imaeq2 6007 . . . . . . . . . . . . . . . . 17 (𝑣 = {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(𝑓𝑦)) = 𝑦} → ((𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦)) “ 𝑣) = ((𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦)) “ {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(𝑓𝑦)) = 𝑦}))
119118sseq1d 3973 . . . . . . . . . . . . . . . 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 631 . . . . . . . . . . . . . . 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 3579 . . . . . . . . . . . . . 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 457 . . . . . . . . . . . 12 ((((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ 𝑡 ∈ 𝒫 𝐴) → ((𝑓𝑦) ∈ 𝑡 → ∃𝑣 ∈ (TopOpen‘𝐺)(𝑓𝑣 ∧ ((𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦)) “ 𝑣) ⊆ 𝑡)))
12443, 123sylbid 239 . . . . . . . . . . 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 3141 . . . . . . . . . 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 724 . . . . . . . . . . 11 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → (TopOpen‘𝐺) ∈ (TopOn‘(Base‘𝐺)))
12712ad2antrr 724 . . . . . . . . . . 11 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → 𝒫 𝐴 ∈ (TopOn‘𝐴))
128 simpr 485 . . . . . . . . . . 11 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → 𝑓 ∈ (Base‘𝐺))
129 iscnp 22534 . . . . . . . . . . 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 1371 . . . . . . . . . 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 711 . . . . . . . . 9 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → (𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦)) ∈ (((TopOpen‘𝐺) CnP 𝒫 𝐴)‘𝑓))
132131ralrimiva 3141 . . . . . . . 8 ((𝐴𝑉𝑦𝐴) → ∀𝑓 ∈ (Base‘𝐺)(𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦)) ∈ (((TopOpen‘𝐺) CnP 𝒫 𝐴)‘𝑓))
133 cncnp 22577 . . . . . . . . . 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 584 . . . . . . . . 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 481 . . . . . . . 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 711 . . . . . . 7 ((𝐴𝑉𝑦𝐴) → (𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦)) ∈ ((TopOpen‘𝐺) Cn 𝒫 𝐴))
137 fvconst2g 7147 . . . . . . . . 9 ((𝒫 𝐴 ∈ Top ∧ 𝑦𝐴) → ((𝐴 × {𝒫 𝐴})‘𝑦) = 𝒫 𝐴)
13826, 137sylan 580 . . . . . . . 8 ((𝐴𝑉𝑦𝐴) → ((𝐴 × {𝒫 𝐴})‘𝑦) = 𝒫 𝐴)
139138oveq2d 7367 . . . . . . 7 ((𝐴𝑉𝑦𝐴) → ((TopOpen‘𝐺) Cn ((𝐴 × {𝒫 𝐴})‘𝑦)) = ((TopOpen‘𝐺) Cn 𝒫 𝐴))
140136, 139eleqtrrd 2841 . . . . . 6 ((𝐴𝑉𝑦𝐴) → (𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦)) ∈ ((TopOpen‘𝐺) Cn ((𝐴 × {𝒫 𝐴})‘𝑦)))
14110, 24, 25, 28, 140ptcn 22924 . . . . 5 (𝐴𝑉 → (𝑥 ∈ (Base‘𝐺) ↦ (𝑦𝐴 ↦ (𝑥𝑦))) ∈ ((TopOpen‘𝐺) Cn (∏t‘(𝐴 × {𝒫 𝐴}))))
142 eqid 2736 . . . . . . . . 9 (invg𝐺) = (invg𝐺)
1435, 142grpinvf 18751 . . . . . . . 8 (𝐺 ∈ Grp → (invg𝐺):(Base‘𝐺)⟶(Base‘𝐺))
1442, 143syl 17 . . . . . . 7 (𝐴𝑉 → (invg𝐺):(Base‘𝐺)⟶(Base‘𝐺))
145144feqmptd 6907 . . . . . 6 (𝐴𝑉 → (invg𝐺) = (𝑥 ∈ (Base‘𝐺) ↦ ((invg𝐺)‘𝑥)))
1461, 5, 142symginv 19137 . . . . . . . . 9 (𝑥 ∈ (Base‘𝐺) → ((invg𝐺)‘𝑥) = 𝑥)
147146adantl 482 . . . . . . . 8 ((𝐴𝑉𝑥 ∈ (Base‘𝐺)) → ((invg𝐺)‘𝑥) = 𝑥)
14832feqmptd 6907 . . . . . . . 8 ((𝐴𝑉𝑥 ∈ (Base‘𝐺)) → 𝑥 = (𝑦𝐴 ↦ (𝑥𝑦)))
149147, 148eqtrd 2776 . . . . . . 7 ((𝐴𝑉𝑥 ∈ (Base‘𝐺)) → ((invg𝐺)‘𝑥) = (𝑦𝐴 ↦ (𝑥𝑦)))
150149mpteq2dva 5203 . . . . . 6 (𝐴𝑉 → (𝑥 ∈ (Base‘𝐺) ↦ ((invg𝐺)‘𝑥)) = (𝑥 ∈ (Base‘𝐺) ↦ (𝑦𝐴 ↦ (𝑥𝑦))))
151145, 150eqtrd 2776 . . . . 5 (𝐴𝑉 → (invg𝐺) = (𝑥 ∈ (Base‘𝐺) ↦ (𝑦𝐴 ↦ (𝑥𝑦))))
152 xkopt 22952 . . . . . . 7 ((𝒫 𝐴 ∈ Top ∧ 𝐴𝑉) → (𝒫 𝐴ko 𝒫 𝐴) = (∏t‘(𝐴 × {𝒫 𝐴})))
15326, 152mpancom 686 . . . . . 6 (𝐴𝑉 → (𝒫 𝐴ko 𝒫 𝐴) = (∏t‘(𝐴 × {𝒫 𝐴})))
154153oveq2d 7367 . . . . 5 (𝐴𝑉 → ((TopOpen‘𝐺) Cn (𝒫 𝐴ko 𝒫 𝐴)) = ((TopOpen‘𝐺) Cn (∏t‘(𝐴 × {𝒫 𝐴}))))
155141, 151, 1543eltr4d 2853 . . . 4 (𝐴𝑉 → (invg𝐺) ∈ ((TopOpen‘𝐺) Cn (𝒫 𝐴ko 𝒫 𝐴)))
156 eqid 2736 . . . . . . 7 (𝒫 𝐴ko 𝒫 𝐴) = (𝒫 𝐴ko 𝒫 𝐴)
157156xkotopon 22897 . . . . . 6 ((𝒫 𝐴 ∈ Top ∧ 𝒫 𝐴 ∈ Top) → (𝒫 𝐴ko 𝒫 𝐴) ∈ (TopOn‘(𝒫 𝐴 Cn 𝒫 𝐴)))
15826, 26, 157syl2anc 584 . . . . 5 (𝐴𝑉 → (𝒫 𝐴ko 𝒫 𝐴) ∈ (TopOn‘(𝒫 𝐴 Cn 𝒫 𝐴)))
159 frn 6672 . . . . . 6 ((invg𝐺):(Base‘𝐺)⟶(Base‘𝐺) → ran (invg𝐺) ⊆ (Base‘𝐺))
1602, 143, 1593syl 18 . . . . 5 (𝐴𝑉 → ran (invg𝐺) ⊆ (Base‘𝐺))
161 cndis 22588 . . . . . . 7 ((𝐴𝑉 ∧ 𝒫 𝐴 ∈ (TopOn‘𝐴)) → (𝒫 𝐴 Cn 𝒫 𝐴) = (𝐴m 𝐴))
16212, 161mpdan 685 . . . . . 6 (𝐴𝑉 → (𝒫 𝐴 Cn 𝒫 𝐴) = (𝐴m 𝐴))
16321, 162sseqtrrd 3983 . . . . 5 (𝐴𝑉 → (Base‘𝐺) ⊆ (𝒫 𝐴 Cn 𝒫 𝐴))
164 cnrest2 22583 . . . . 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 1371 . . . 4 (𝐴𝑉 → ((invg𝐺) ∈ ((TopOpen‘𝐺) Cn (𝒫 𝐴ko 𝒫 𝐴)) ↔ (invg𝐺) ∈ ((TopOpen‘𝐺) Cn ((𝒫 𝐴ko 𝒫 𝐴) ↾t (Base‘𝐺)))))
166155, 165mpbid 231 . . 3 (𝐴𝑉 → (invg𝐺) ∈ ((TopOpen‘𝐺) Cn ((𝒫 𝐴ko 𝒫 𝐴) ↾t (Base‘𝐺))))
167153oveq1d 7366 . . . . 5 (𝐴𝑉 → ((𝒫 𝐴ko 𝒫 𝐴) ↾t (Base‘𝐺)) = ((∏t‘(𝐴 × {𝒫 𝐴})) ↾t (Base‘𝐺)))
168167, 11eqtrd 2776 . . . 4 (𝐴𝑉 → ((𝒫 𝐴ko 𝒫 𝐴) ↾t (Base‘𝐺)) = (TopOpen‘𝐺))
169168oveq2d 7367 . . 3 (𝐴𝑉 → ((TopOpen‘𝐺) Cn ((𝒫 𝐴ko 𝒫 𝐴) ↾t (Base‘𝐺))) = ((TopOpen‘𝐺) Cn (TopOpen‘𝐺)))
170166, 169eleqtrd 2840 . 2 (𝐴𝑉 → (invg𝐺) ∈ ((TopOpen‘𝐺) Cn (TopOpen‘𝐺)))
171 eqid 2736 . . 3 (TopOpen‘𝐺) = (TopOpen‘𝐺)
172171, 142istgp 23374 . 2 (𝐺 ∈ TopGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ TopMnd ∧ (invg𝐺) ∈ ((TopOpen‘𝐺) Cn (TopOpen‘𝐺))))
1732, 9, 170, 172syl3anbrc 1343 1 (𝐴𝑉𝐺 ∈ TopGrp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  wral 3062  wrex 3071  {crab 3405  Vcvv 3443  wss 3908  𝒫 cpw 4558  {csn 4584   cuni 4863  cmpt 5186   × cxp 5629  ccnv 5630  dom cdm 5631  ran crn 5632  cima 5634  Fun wfun 6487  wf 6489  1-1-ontowf1o 6492  cfv 6493  (class class class)co 7351  m cmap 8723  Basecbs 17037  t crest 17256  TopOpenctopn 17257  tcpt 17274  SubMndcsubmnd 18554  EndoFMndcefmnd 18632  Grpcgrp 18702  invgcminusg 18703  SymGrpcsymg 19101  Topctop 22188  TopOnctopon 22205   Cn ccn 22521   CnP ccnp 22522  ko cxko 22858  TopMndctmd 23367  TopGrpctgp 23368
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5240  ax-sep 5254  ax-nul 5261  ax-pow 5318  ax-pr 5382  ax-un 7664  ax-cnex 11065  ax-resscn 11066  ax-1cn 11067  ax-icn 11068  ax-addcl 11069  ax-addrcl 11070  ax-mulcl 11071  ax-mulrcl 11072  ax-mulcom 11073  ax-addass 11074  ax-mulass 11075  ax-distr 11076  ax-i2m1 11077  ax-1ne0 11078  ax-1rid 11079  ax-rnegex 11080  ax-rrecex 11081  ax-cnre 11082  ax-pre-lttri 11083  ax-pre-lttrn 11084  ax-pre-ltadd 11085  ax-pre-mulgt0 11086
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3351  df-reu 3352  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4864  df-int 4906  df-iun 4954  df-iin 4955  df-br 5104  df-opab 5166  df-mpt 5187  df-tr 5221  df-id 5529  df-eprel 5535  df-po 5543  df-so 5544  df-fr 5586  df-we 5588  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6251  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6445  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-riota 7307  df-ov 7354  df-oprab 7355  df-mpo 7356  df-om 7795  df-1st 7913  df-2nd 7914  df-frecs 8204  df-wrecs 8235  df-recs 8309  df-rdg 8348  df-1o 8404  df-er 8606  df-map 8725  df-ixp 8794  df-en 8842  df-dom 8843  df-sdom 8844  df-fin 8845  df-fi 9305  df-pnf 11149  df-mnf 11150  df-xr 11151  df-ltxr 11152  df-le 11153  df-sub 11345  df-neg 11346  df-nn 12112  df-2 12174  df-3 12175  df-4 12176  df-5 12177  df-6 12178  df-7 12179  df-8 12180  df-9 12181  df-n0 12372  df-z 12458  df-uz 12722  df-fz 13379  df-struct 16973  df-sets 16990  df-slot 17008  df-ndx 17020  df-base 17038  df-ress 17067  df-plusg 17100  df-tset 17106  df-rest 17258  df-topn 17259  df-0g 17277  df-topgen 17279  df-pt 17280  df-plusf 18450  df-mgm 18451  df-sgrp 18500  df-mnd 18511  df-submnd 18556  df-efmnd 18633  df-grp 18705  df-minusg 18706  df-symg 19102  df-top 22189  df-topon 22206  df-topsp 22228  df-bases 22242  df-ntr 22317  df-nei 22395  df-cn 22524  df-cnp 22525  df-cmp 22684  df-lly 22763  df-nlly 22764  df-tx 22859  df-xko 22860  df-tmd 23369  df-tgp 23370
This theorem is referenced by: (None)
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