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Theorem symgtgp 24130
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 19433 . 2 (𝐴𝑉𝐺 ∈ Grp)
3 eqid 2735 . . . 4 (EndoFMnd‘𝐴) = (EndoFMnd‘𝐴)
43efmndtmd 24125 . . 3 (𝐴𝑉 → (EndoFMnd‘𝐴) ∈ TopMnd)
5 eqid 2735 . . . 4 (Base‘𝐺) = (Base‘𝐺)
63, 1, 5symgsubmefmnd 19431 . . 3 (𝐴𝑉 → (Base‘𝐺) ∈ (SubMnd‘(EndoFMnd‘𝐴)))
71, 5, 3symgressbas 19414 . . . 4 𝐺 = ((EndoFMnd‘𝐴) ↾s (Base‘𝐺))
87submtmd 24128 . . 3 (((EndoFMnd‘𝐴) ∈ TopMnd ∧ (Base‘𝐺) ∈ (SubMnd‘(EndoFMnd‘𝐴))) → 𝐺 ∈ TopMnd)
94, 6, 8syl2anc 584 . 2 (𝐴𝑉𝐺 ∈ TopMnd)
10 eqid 2735 . . . . . 6 (∏t‘(𝐴 × {𝒫 𝐴})) = (∏t‘(𝐴 × {𝒫 𝐴}))
111, 5symgtopn 19439 . . . . . . 7 (𝐴𝑉 → ((∏t‘(𝐴 × {𝒫 𝐴})) ↾t (Base‘𝐺)) = (TopOpen‘𝐺))
12 distopon 23020 . . . . . . . . 9 (𝐴𝑉 → 𝒫 𝐴 ∈ (TopOn‘𝐴))
1310pttoponconst 23621 . . . . . . . . 9 ((𝐴𝑉 ∧ 𝒫 𝐴 ∈ (TopOn‘𝐴)) → (∏t‘(𝐴 × {𝒫 𝐴})) ∈ (TopOn‘(𝐴m 𝐴)))
1412, 13mpdan 687 . . . . . . . 8 (𝐴𝑉 → (∏t‘(𝐴 × {𝒫 𝐴})) ∈ (TopOn‘(𝐴m 𝐴)))
151, 5elsymgbas 19406 . . . . . . . . . 10 (𝐴𝑉 → (𝑥 ∈ (Base‘𝐺) ↔ 𝑥:𝐴1-1-onto𝐴))
16 f1of 6849 . . . . . . . . . . 11 (𝑥:𝐴1-1-onto𝐴𝑥:𝐴𝐴)
17 elmapg 8878 . . . . . . . . . . . 12 ((𝐴𝑉𝐴𝑉) → (𝑥 ∈ (𝐴m 𝐴) ↔ 𝑥:𝐴𝐴))
1817anidms 566 . . . . . . . . . . 11 (𝐴𝑉 → (𝑥 ∈ (𝐴m 𝐴) ↔ 𝑥:𝐴𝐴))
1916, 18imbitrrid 246 . . . . . . . . . 10 (𝐴𝑉 → (𝑥:𝐴1-1-onto𝐴𝑥 ∈ (𝐴m 𝐴)))
2015, 19sylbid 240 . . . . . . . . 9 (𝐴𝑉 → (𝑥 ∈ (Base‘𝐺) → 𝑥 ∈ (𝐴m 𝐴)))
2120ssrdv 4001 . . . . . . . 8 (𝐴𝑉 → (Base‘𝐺) ⊆ (𝐴m 𝐴))
22 resttopon 23185 . . . . . . . 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 2840 . . . . . 6 (𝐴𝑉 → (TopOpen‘𝐺) ∈ (TopOn‘(Base‘𝐺)))
25 id 22 . . . . . 6 (𝐴𝑉𝐴𝑉)
26 distop 23018 . . . . . . 7 (𝐴𝑉 → 𝒫 𝐴 ∈ Top)
27 fconst6g 6798 . . . . . . 7 (𝒫 𝐴 ∈ Top → (𝐴 × {𝒫 𝐴}):𝐴⟶Top)
2826, 27syl 17 . . . . . 6 (𝐴𝑉 → (𝐴 × {𝒫 𝐴}):𝐴⟶Top)
2915biimpa 476 . . . . . . . . . . . 12 ((𝐴𝑉𝑥 ∈ (Base‘𝐺)) → 𝑥:𝐴1-1-onto𝐴)
30 f1ocnv 6861 . . . . . . . . . . . 12 (𝑥:𝐴1-1-onto𝐴𝑥:𝐴1-1-onto𝐴)
31 f1of 6849 . . . . . . . . . . . 12 (𝑥:𝐴1-1-onto𝐴𝑥:𝐴𝐴)
3229, 30, 313syl 18 . . . . . . . . . . 11 ((𝐴𝑉𝑥 ∈ (Base‘𝐺)) → 𝑥:𝐴𝐴)
3332ffvelcdmda 7104 . . . . . . . . . 10 (((𝐴𝑉𝑥 ∈ (Base‘𝐺)) ∧ 𝑦𝐴) → (𝑥𝑦) ∈ 𝐴)
3433an32s 652 . . . . . . . . 9 (((𝐴𝑉𝑦𝐴) ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑥𝑦) ∈ 𝐴)
3534fmpttd 7135 . . . . . . . 8 ((𝐴𝑉𝑦𝐴) → (𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦)):(Base‘𝐺)⟶𝐴)
3635adantr 480 . . . . . . . . . 10 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → (𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦)):(Base‘𝐺)⟶𝐴)
37 cnveq 5887 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑓𝑥 = 𝑓)
3837fveq1d 6909 . . . . . . . . . . . . . . 15 (𝑥 = 𝑓 → (𝑥𝑦) = (𝑓𝑦))
39 eqid 2735 . . . . . . . . . . . . . . 15 (𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦)) = (𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦))
40 fvex 6920 . . . . . . . . . . . . . . 15 (𝑓𝑦) ∈ V
4138, 39, 40fvmpt 7016 . . . . . . . . . . . . . 14 (𝑓 ∈ (Base‘𝐺) → ((𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦))‘𝑓) = (𝑓𝑦))
4241ad2antlr 727 . . . . . . . . . . . . 13 ((((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ 𝑡 ∈ 𝒫 𝐴) → ((𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦))‘𝑓) = (𝑓𝑦))
4342eleq1d 2824 . . . . . . . . . . . 12 ((((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ 𝑡 ∈ 𝒫 𝐴) → (((𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦))‘𝑓) ∈ 𝑡 ↔ (𝑓𝑦) ∈ 𝑡))
44 eqid 2735 . . . . . . . . . . . . . . . . . 18 (𝑢 ∈ (Base‘𝐺) ↦ (𝑢‘(𝑓𝑦))) = (𝑢 ∈ (Base‘𝐺) ↦ (𝑢‘(𝑓𝑦)))
4544mptiniseg 6261 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ V → ((𝑢 ∈ (Base‘𝐺) ↦ (𝑢‘(𝑓𝑦))) “ {𝑦}) = {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(𝑓𝑦)) = 𝑦})
4645elv 3483 . . . . . . . . . . . . . . . 16 ((𝑢 ∈ (Base‘𝐺) ↦ (𝑢‘(𝑓𝑦))) “ {𝑦}) = {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(𝑓𝑦)) = 𝑦}
47 eqid 2735 . . . . . . . . . . . . . . . . . . 19 ((∏t‘(𝐴 × {𝒫 𝐴})) ↾t (Base‘𝐺)) = ((∏t‘(𝐴 × {𝒫 𝐴})) ↾t (Base‘𝐺))
4814ad2antrr 726 . . . . . . . . . . . . . . . . . . 19 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → (∏t‘(𝐴 × {𝒫 𝐴})) ∈ (TopOn‘(𝐴m 𝐴)))
4921ad2antrr 726 . . . . . . . . . . . . . . . . . . 19 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → (Base‘𝐺) ⊆ (𝐴m 𝐴))
50 toponuni 22936 . . . . . . . . . . . . . . . . . . . . 21 ((∏t‘(𝐴 × {𝒫 𝐴})) ∈ (TopOn‘(𝐴m 𝐴)) → (𝐴m 𝐴) = (∏t‘(𝐴 × {𝒫 𝐴})))
51 mpteq1 5241 . . . . . . . . . . . . . . . . . . . . 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 767 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → 𝐴𝑉)
5428ad2antrr 726 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → (𝐴 × {𝒫 𝐴}):𝐴⟶Top)
551, 5elsymgbas 19406 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝐴𝑉 → (𝑓 ∈ (Base‘𝐺) ↔ 𝑓:𝐴1-1-onto𝐴))
5655adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝐴𝑉𝑦𝐴) → (𝑓 ∈ (Base‘𝐺) ↔ 𝑓:𝐴1-1-onto𝐴))
5756biimpa 476 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → 𝑓:𝐴1-1-onto𝐴)
58 f1ocnv 6861 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑓:𝐴1-1-onto𝐴𝑓:𝐴1-1-onto𝐴)
59 f1of 6849 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑓:𝐴1-1-onto𝐴𝑓:𝐴𝐴)
6057, 58, 593syl 18 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → 𝑓:𝐴𝐴)
61 simplr 769 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → 𝑦𝐴)
6260, 61ffvelcdmd 7105 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → (𝑓𝑦) ∈ 𝐴)
63 eqid 2735 . . . . . . . . . . . . . . . . . . . . . . 23 (∏t‘(𝐴 × {𝒫 𝐴})) = (∏t‘(𝐴 × {𝒫 𝐴}))
6463, 10ptpjcn 23635 . . . . . . . . . . . . . . . . . . . . . 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 1370 . . . . . . . . . . . . . . . . . . . . 21 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → (𝑢 (∏t‘(𝐴 × {𝒫 𝐴})) ↦ (𝑢‘(𝑓𝑦))) ∈ ((∏t‘(𝐴 × {𝒫 𝐴})) Cn ((𝐴 × {𝒫 𝐴})‘(𝑓𝑦))))
6626ad2antrr 726 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → 𝒫 𝐴 ∈ Top)
67 fvconst2g 7222 . . . . . . . . . . . . . . . . . . . . . . 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 7447 . . . . . . . . . . . . . . . . . . . . 21 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → ((∏t‘(𝐴 × {𝒫 𝐴})) Cn ((𝐴 × {𝒫 𝐴})‘(𝑓𝑦))) = ((∏t‘(𝐴 × {𝒫 𝐴})) Cn 𝒫 𝐴))
7065, 69eleqtrd 2841 . . . . . . . . . . . . . . . . . . . 20 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → (𝑢 (∏t‘(𝐴 × {𝒫 𝐴})) ↦ (𝑢‘(𝑓𝑦))) ∈ ((∏t‘(𝐴 × {𝒫 𝐴})) Cn 𝒫 𝐴))
7152, 70eqeltrd 2839 . . . . . . . . . . . . . . . . . . 19 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → (𝑢 ∈ (𝐴m 𝐴) ↦ (𝑢‘(𝑓𝑦))) ∈ ((∏t‘(𝐴 × {𝒫 𝐴})) Cn 𝒫 𝐴))
7247, 48, 49, 71cnmpt1res 23700 . . . . . . . . . . . . . . . . . 18 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → (𝑢 ∈ (Base‘𝐺) ↦ (𝑢‘(𝑓𝑦))) ∈ (((∏t‘(𝐴 × {𝒫 𝐴})) ↾t (Base‘𝐺)) Cn 𝒫 𝐴))
7311oveq1d 7446 . . . . . . . . . . . . . . . . . . 19 (𝐴𝑉 → (((∏t‘(𝐴 × {𝒫 𝐴})) ↾t (Base‘𝐺)) Cn 𝒫 𝐴) = ((TopOpen‘𝐺) Cn 𝒫 𝐴))
7473ad2antrr 726 . . . . . . . . . . . . . . . . . 18 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → (((∏t‘(𝐴 × {𝒫 𝐴})) ↾t (Base‘𝐺)) Cn 𝒫 𝐴) = ((TopOpen‘𝐺) Cn 𝒫 𝐴))
7572, 74eleqtrd 2841 . . . . . . . . . . . . . . . . 17 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → (𝑢 ∈ (Base‘𝐺) ↦ (𝑢‘(𝑓𝑦))) ∈ ((TopOpen‘𝐺) Cn 𝒫 𝐴))
76 snelpwi 5454 . . . . . . . . . . . . . . . . . 18 (𝑦𝐴 → {𝑦} ∈ 𝒫 𝐴)
7776ad2antlr 727 . . . . . . . . . . . . . . . . 17 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → {𝑦} ∈ 𝒫 𝐴)
78 cnima 23289 . . . . . . . . . . . . . . . . 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 2844 . . . . . . . . . . . . . . 15 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(𝑓𝑦)) = 𝑦} ∈ (TopOpen‘𝐺))
8180adantr 480 . . . . . . . . . . . . . 14 ((((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ (𝑡 ∈ 𝒫 𝐴 ∧ (𝑓𝑦) ∈ 𝑡)) → {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(𝑓𝑦)) = 𝑦} ∈ (TopOpen‘𝐺))
82 fveq1 6906 . . . . . . . . . . . . . . . 16 (𝑢 = 𝑓 → (𝑢‘(𝑓𝑦)) = (𝑓‘(𝑓𝑦)))
8382eqeq1d 2737 . . . . . . . . . . . . . . 15 (𝑢 = 𝑓 → ((𝑢‘(𝑓𝑦)) = 𝑦 ↔ (𝑓‘(𝑓𝑦)) = 𝑦))
84 simplr 769 . . . . . . . . . . . . . . 15 ((((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ (𝑡 ∈ 𝒫 𝐴 ∧ (𝑓𝑦) ∈ 𝑡)) → 𝑓 ∈ (Base‘𝐺))
8557adantr 480 . . . . . . . . . . . . . . . 16 ((((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ (𝑡 ∈ 𝒫 𝐴 ∧ (𝑓𝑦) ∈ 𝑡)) → 𝑓:𝐴1-1-onto𝐴)
86 simpllr 776 . . . . . . . . . . . . . . . 16 ((((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ (𝑡 ∈ 𝒫 𝐴 ∧ (𝑓𝑦) ∈ 𝑡)) → 𝑦𝐴)
87 f1ocnvfv2 7297 . . . . . . . . . . . . . . . 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 3697 . . . . . . . . . . . . . 14 ((((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ (𝑡 ∈ 𝒫 𝐴 ∧ (𝑓𝑦) ∈ 𝑡)) → 𝑓 ∈ {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(𝑓𝑦)) = 𝑦})
90 ssrab2 4090 . . . . . . . . . . . . . . . . . 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 730 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ (𝑡 ∈ 𝒫 𝐴 ∧ (𝑓𝑦) ∈ 𝑡)) → (𝑥 ∈ (Base‘𝐺) ↔ 𝑥:𝐴1-1-onto𝐴))
9392biimpa 476 . . . . . . . . . . . . . . . . . . . . 21 (((((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ (𝑡 ∈ 𝒫 𝐴 ∧ (𝑓𝑦) ∈ 𝑡)) ∧ 𝑥 ∈ (Base‘𝐺)) → 𝑥:𝐴1-1-onto𝐴)
9462ad2antrr 726 . . . . . . . . . . . . . . . . . . . . 21 (((((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ (𝑡 ∈ 𝒫 𝐴 ∧ (𝑓𝑦) ∈ 𝑡)) ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑓𝑦) ∈ 𝐴)
95 f1ocnvfv 7298 . . . . . . . . . . . . . . . . . . . . 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 778 . . . . . . . . . . . . . . . . . . . . 21 (((((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ (𝑡 ∈ 𝒫 𝐴 ∧ (𝑓𝑦) ∈ 𝑡)) ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑓𝑦) ∈ 𝑡)
98 eleq1 2827 . . . . . . . . . . . . . . . . . . . . 21 ((𝑥𝑦) = (𝑓𝑦) → ((𝑥𝑦) ∈ 𝑡 ↔ (𝑓𝑦) ∈ 𝑡))
9997, 98syl5ibrcom 247 . . . . . . . . . . . . . . . . . . . 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 3144 . . . . . . . . . . . . . . . . . 18 ((((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ (𝑡 ∈ 𝒫 𝐴 ∧ (𝑓𝑦) ∈ 𝑡)) → ∀𝑥 ∈ (Base‘𝐺)((𝑥‘(𝑓𝑦)) = 𝑦 → (𝑥𝑦) ∈ 𝑡))
102 fveq1 6906 . . . . . . . . . . . . . . . . . . . 20 (𝑢 = 𝑥 → (𝑢‘(𝑓𝑦)) = (𝑥‘(𝑓𝑦)))
103102eqeq1d 2737 . . . . . . . . . . . . . . . . . . 19 (𝑢 = 𝑥 → ((𝑢‘(𝑓𝑦)) = 𝑦 ↔ (𝑥‘(𝑓𝑦)) = 𝑦))
104103ralrab 3702 . . . . . . . . . . . . . . . . . 18 (∀𝑥 ∈ {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(𝑓𝑦)) = 𝑦} (𝑥𝑦) ∈ 𝑡 ↔ ∀𝑥 ∈ (Base‘𝐺)((𝑥‘(𝑓𝑦)) = 𝑦 → (𝑥𝑦) ∈ 𝑡))
105101, 104sylibr 234 . . . . . . . . . . . . . . . . 17 ((((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ (𝑡 ∈ 𝒫 𝐴 ∧ (𝑓𝑦) ∈ 𝑡)) → ∀𝑥 ∈ {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(𝑓𝑦)) = 𝑦} (𝑥𝑦) ∈ 𝑡)
106 ssrab 4083 . . . . . . . . . . . . . . . . 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 6260 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦)) “ 𝑡) = {𝑥 ∈ (Base‘𝐺) ∣ (𝑥𝑦) ∈ 𝑡}
109107, 108sseqtrrdi 4047 . . . . . . . . . . . . . . 15 ((((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ (𝑡 ∈ 𝒫 𝐴 ∧ (𝑓𝑦) ∈ 𝑡)) → {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(𝑓𝑦)) = 𝑦} ⊆ ((𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦)) “ 𝑡))
110 funmpt 6606 . . . . . . . . . . . . . . . 16 Fun (𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦))
111 fvex 6920 . . . . . . . . . . . . . . . . . 18 (𝑥𝑦) ∈ V
112111, 39dmmpti 6713 . . . . . . . . . . . . . . . . 17 dom (𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦)) = (Base‘𝐺)
11391, 112sseqtrrdi 4047 . . . . . . . . . . . . . . . 16 ((((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ (𝑡 ∈ 𝒫 𝐴 ∧ (𝑓𝑦) ∈ 𝑡)) → {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(𝑓𝑦)) = 𝑦} ⊆ dom (𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦)))
114 funimass3 7074 . . . . . . . . . . . . . . . 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 257 . . . . . . . . . . . . . 14 ((((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ (𝑡 ∈ 𝒫 𝐴 ∧ (𝑓𝑦) ∈ 𝑡)) → ((𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦)) “ {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(𝑓𝑦)) = 𝑦}) ⊆ 𝑡)
117 eleq2 2828 . . . . . . . . . . . . . . . 16 (𝑣 = {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(𝑓𝑦)) = 𝑦} → (𝑓𝑣𝑓 ∈ {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(𝑓𝑦)) = 𝑦}))
118 imaeq2 6076 . . . . . . . . . . . . . . . . 17 (𝑣 = {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(𝑓𝑦)) = 𝑦} → ((𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦)) “ 𝑣) = ((𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦)) “ {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(𝑓𝑦)) = 𝑦}))
119118sseq1d 4027 . . . . . . . . . . . . . . . 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 632 . . . . . . . . . . . . . . 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 3622 . . . . . . . . . . . . . 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 837 . . . . . . . . . . . . 13 ((((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ (𝑡 ∈ 𝒫 𝐴 ∧ (𝑓𝑦) ∈ 𝑡)) → ∃𝑣 ∈ (TopOpen‘𝐺)(𝑓𝑣 ∧ ((𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦)) “ 𝑣) ⊆ 𝑡))
123122expr 456 . . . . . . . . . . . 12 ((((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ 𝑡 ∈ 𝒫 𝐴) → ((𝑓𝑦) ∈ 𝑡 → ∃𝑣 ∈ (TopOpen‘𝐺)(𝑓𝑣 ∧ ((𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦)) “ 𝑣) ⊆ 𝑡)))
12443, 123sylbid 240 . . . . . . . . . . 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 3144 . . . . . . . . . 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 726 . . . . . . . . . . 11 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → (TopOpen‘𝐺) ∈ (TopOn‘(Base‘𝐺)))
12712ad2antrr 726 . . . . . . . . . . 11 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → 𝒫 𝐴 ∈ (TopOn‘𝐴))
128 simpr 484 . . . . . . . . . . 11 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → 𝑓 ∈ (Base‘𝐺))
129 iscnp 23261 . . . . . . . . . . 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 1370 . . . . . . . . . 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 713 . . . . . . . . 9 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → (𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦)) ∈ (((TopOpen‘𝐺) CnP 𝒫 𝐴)‘𝑓))
132131ralrimiva 3144 . . . . . . . 8 ((𝐴𝑉𝑦𝐴) → ∀𝑓 ∈ (Base‘𝐺)(𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦)) ∈ (((TopOpen‘𝐺) CnP 𝒫 𝐴)‘𝑓))
133 cncnp 23304 . . . . . . . . . 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 480 . . . . . . . 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 713 . . . . . . 7 ((𝐴𝑉𝑦𝐴) → (𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦)) ∈ ((TopOpen‘𝐺) Cn 𝒫 𝐴))
137 fvconst2g 7222 . . . . . . . . 9 ((𝒫 𝐴 ∈ Top ∧ 𝑦𝐴) → ((𝐴 × {𝒫 𝐴})‘𝑦) = 𝒫 𝐴)
13826, 137sylan 580 . . . . . . . 8 ((𝐴𝑉𝑦𝐴) → ((𝐴 × {𝒫 𝐴})‘𝑦) = 𝒫 𝐴)
139138oveq2d 7447 . . . . . . 7 ((𝐴𝑉𝑦𝐴) → ((TopOpen‘𝐺) Cn ((𝐴 × {𝒫 𝐴})‘𝑦)) = ((TopOpen‘𝐺) Cn 𝒫 𝐴))
140136, 139eleqtrrd 2842 . . . . . 6 ((𝐴𝑉𝑦𝐴) → (𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦)) ∈ ((TopOpen‘𝐺) Cn ((𝐴 × {𝒫 𝐴})‘𝑦)))
14110, 24, 25, 28, 140ptcn 23651 . . . . 5 (𝐴𝑉 → (𝑥 ∈ (Base‘𝐺) ↦ (𝑦𝐴 ↦ (𝑥𝑦))) ∈ ((TopOpen‘𝐺) Cn (∏t‘(𝐴 × {𝒫 𝐴}))))
142 eqid 2735 . . . . . . . . 9 (invg𝐺) = (invg𝐺)
1435, 142grpinvf 19017 . . . . . . . 8 (𝐺 ∈ Grp → (invg𝐺):(Base‘𝐺)⟶(Base‘𝐺))
1442, 143syl 17 . . . . . . 7 (𝐴𝑉 → (invg𝐺):(Base‘𝐺)⟶(Base‘𝐺))
145144feqmptd 6977 . . . . . 6 (𝐴𝑉 → (invg𝐺) = (𝑥 ∈ (Base‘𝐺) ↦ ((invg𝐺)‘𝑥)))
1461, 5, 142symginv 19435 . . . . . . . . 9 (𝑥 ∈ (Base‘𝐺) → ((invg𝐺)‘𝑥) = 𝑥)
147146adantl 481 . . . . . . . 8 ((𝐴𝑉𝑥 ∈ (Base‘𝐺)) → ((invg𝐺)‘𝑥) = 𝑥)
14832feqmptd 6977 . . . . . . . 8 ((𝐴𝑉𝑥 ∈ (Base‘𝐺)) → 𝑥 = (𝑦𝐴 ↦ (𝑥𝑦)))
149147, 148eqtrd 2775 . . . . . . 7 ((𝐴𝑉𝑥 ∈ (Base‘𝐺)) → ((invg𝐺)‘𝑥) = (𝑦𝐴 ↦ (𝑥𝑦)))
150149mpteq2dva 5248 . . . . . 6 (𝐴𝑉 → (𝑥 ∈ (Base‘𝐺) ↦ ((invg𝐺)‘𝑥)) = (𝑥 ∈ (Base‘𝐺) ↦ (𝑦𝐴 ↦ (𝑥𝑦))))
151145, 150eqtrd 2775 . . . . 5 (𝐴𝑉 → (invg𝐺) = (𝑥 ∈ (Base‘𝐺) ↦ (𝑦𝐴 ↦ (𝑥𝑦))))
152 xkopt 23679 . . . . . . 7 ((𝒫 𝐴 ∈ Top ∧ 𝐴𝑉) → (𝒫 𝐴ko 𝒫 𝐴) = (∏t‘(𝐴 × {𝒫 𝐴})))
15326, 152mpancom 688 . . . . . 6 (𝐴𝑉 → (𝒫 𝐴ko 𝒫 𝐴) = (∏t‘(𝐴 × {𝒫 𝐴})))
154153oveq2d 7447 . . . . 5 (𝐴𝑉 → ((TopOpen‘𝐺) Cn (𝒫 𝐴ko 𝒫 𝐴)) = ((TopOpen‘𝐺) Cn (∏t‘(𝐴 × {𝒫 𝐴}))))
155141, 151, 1543eltr4d 2854 . . . 4 (𝐴𝑉 → (invg𝐺) ∈ ((TopOpen‘𝐺) Cn (𝒫 𝐴ko 𝒫 𝐴)))
156 eqid 2735 . . . . . . 7 (𝒫 𝐴ko 𝒫 𝐴) = (𝒫 𝐴ko 𝒫 𝐴)
157156xkotopon 23624 . . . . . 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 6744 . . . . . 6 ((invg𝐺):(Base‘𝐺)⟶(Base‘𝐺) → ran (invg𝐺) ⊆ (Base‘𝐺))
1602, 143, 1593syl 18 . . . . 5 (𝐴𝑉 → ran (invg𝐺) ⊆ (Base‘𝐺))
161 cndis 23315 . . . . . . 7 ((𝐴𝑉 ∧ 𝒫 𝐴 ∈ (TopOn‘𝐴)) → (𝒫 𝐴 Cn 𝒫 𝐴) = (𝐴m 𝐴))
16212, 161mpdan 687 . . . . . 6 (𝐴𝑉 → (𝒫 𝐴 Cn 𝒫 𝐴) = (𝐴m 𝐴))
16321, 162sseqtrrd 4037 . . . . 5 (𝐴𝑉 → (Base‘𝐺) ⊆ (𝒫 𝐴 Cn 𝒫 𝐴))
164 cnrest2 23310 . . . . 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 1370 . . . 4 (𝐴𝑉 → ((invg𝐺) ∈ ((TopOpen‘𝐺) Cn (𝒫 𝐴ko 𝒫 𝐴)) ↔ (invg𝐺) ∈ ((TopOpen‘𝐺) Cn ((𝒫 𝐴ko 𝒫 𝐴) ↾t (Base‘𝐺)))))
166155, 165mpbid 232 . . 3 (𝐴𝑉 → (invg𝐺) ∈ ((TopOpen‘𝐺) Cn ((𝒫 𝐴ko 𝒫 𝐴) ↾t (Base‘𝐺))))
167153oveq1d 7446 . . . . 5 (𝐴𝑉 → ((𝒫 𝐴ko 𝒫 𝐴) ↾t (Base‘𝐺)) = ((∏t‘(𝐴 × {𝒫 𝐴})) ↾t (Base‘𝐺)))
168167, 11eqtrd 2775 . . . 4 (𝐴𝑉 → ((𝒫 𝐴ko 𝒫 𝐴) ↾t (Base‘𝐺)) = (TopOpen‘𝐺))
169168oveq2d 7447 . . 3 (𝐴𝑉 → ((TopOpen‘𝐺) Cn ((𝒫 𝐴ko 𝒫 𝐴) ↾t (Base‘𝐺))) = ((TopOpen‘𝐺) Cn (TopOpen‘𝐺)))
170166, 169eleqtrd 2841 . 2 (𝐴𝑉 → (invg𝐺) ∈ ((TopOpen‘𝐺) Cn (TopOpen‘𝐺)))
171 eqid 2735 . . 3 (TopOpen‘𝐺) = (TopOpen‘𝐺)
172171, 142istgp 24101 . 2 (𝐺 ∈ TopGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ TopMnd ∧ (invg𝐺) ∈ ((TopOpen‘𝐺) Cn (TopOpen‘𝐺))))
1732, 9, 170, 172syl3anbrc 1342 1 (𝐴𝑉𝐺 ∈ TopGrp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  wral 3059  wrex 3068  {crab 3433  Vcvv 3478  wss 3963  𝒫 cpw 4605  {csn 4631   cuni 4912  cmpt 5231   × cxp 5687  ccnv 5688  dom cdm 5689  ran crn 5690  cima 5692  Fun wfun 6557  wf 6559  1-1-ontowf1o 6562  cfv 6563  (class class class)co 7431  m cmap 8865  Basecbs 17245  t crest 17467  TopOpenctopn 17468  tcpt 17485  SubMndcsubmnd 18808  EndoFMndcefmnd 18894  Grpcgrp 18964  invgcminusg 18965  SymGrpcsymg 19401  Topctop 22915  TopOnctopon 22932   Cn ccn 23248   CnP ccnp 23249  ko cxko 23585  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-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  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-nel 3045  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-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  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-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  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-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-er 8744  df-map 8867  df-ixp 8937  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-fi 9449  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12525  df-z 12612  df-uz 12877  df-fz 13545  df-struct 17181  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-ress 17275  df-plusg 17311  df-tset 17317  df-rest 17469  df-topn 17470  df-0g 17488  df-topgen 17490  df-pt 17491  df-plusf 18665  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-submnd 18810  df-efmnd 18895  df-grp 18967  df-minusg 18968  df-symg 19402  df-top 22916  df-topon 22933  df-topsp 22955  df-bases 22969  df-ntr 23044  df-nei 23122  df-cn 23251  df-cnp 23252  df-cmp 23411  df-lly 23490  df-nlly 23491  df-tx 23586  df-xko 23587  df-tmd 24096  df-tgp 24097
This theorem is referenced by: (None)
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