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Theorem symgtgp 22637
Description: The symmetric group is a topological group. (Contributed by Mario Carneiro, 2-Sep-2015.)
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 18458 . 2 (𝐴𝑉𝐺 ∈ Grp)
3 grpmnd 18048 . . . 4 (𝐺 ∈ Grp → 𝐺 ∈ Mnd)
42, 3syl 17 . . 3 (𝐴𝑉𝐺 ∈ Mnd)
5 eqid 2818 . . . . . 6 (Base‘𝐺) = (Base‘𝐺)
61, 5symgtopn 18463 . . . . 5 (𝐴𝑉 → ((∏t‘(𝐴 × {𝒫 𝐴})) ↾t (Base‘𝐺)) = (TopOpen‘𝐺))
7 distopon 21533 . . . . . . 7 (𝐴𝑉 → 𝒫 𝐴 ∈ (TopOn‘𝐴))
8 eqid 2818 . . . . . . . 8 (∏t‘(𝐴 × {𝒫 𝐴})) = (∏t‘(𝐴 × {𝒫 𝐴}))
98pttoponconst 22133 . . . . . . 7 ((𝐴𝑉 ∧ 𝒫 𝐴 ∈ (TopOn‘𝐴)) → (∏t‘(𝐴 × {𝒫 𝐴})) ∈ (TopOn‘(𝐴m 𝐴)))
107, 9mpdan 683 . . . . . 6 (𝐴𝑉 → (∏t‘(𝐴 × {𝒫 𝐴})) ∈ (TopOn‘(𝐴m 𝐴)))
111, 5elsymgbas 18438 . . . . . . . 8 (𝐴𝑉 → (𝑥 ∈ (Base‘𝐺) ↔ 𝑥:𝐴1-1-onto𝐴))
12 f1of 6608 . . . . . . . . 9 (𝑥:𝐴1-1-onto𝐴𝑥:𝐴𝐴)
13 elmapg 8408 . . . . . . . . . 10 ((𝐴𝑉𝐴𝑉) → (𝑥 ∈ (𝐴m 𝐴) ↔ 𝑥:𝐴𝐴))
1413anidms 567 . . . . . . . . 9 (𝐴𝑉 → (𝑥 ∈ (𝐴m 𝐴) ↔ 𝑥:𝐴𝐴))
1512, 14syl5ibr 247 . . . . . . . 8 (𝐴𝑉 → (𝑥:𝐴1-1-onto𝐴𝑥 ∈ (𝐴m 𝐴)))
1611, 15sylbid 241 . . . . . . 7 (𝐴𝑉 → (𝑥 ∈ (Base‘𝐺) → 𝑥 ∈ (𝐴m 𝐴)))
1716ssrdv 3970 . . . . . 6 (𝐴𝑉 → (Base‘𝐺) ⊆ (𝐴m 𝐴))
18 resttopon 21697 . . . . . 6 (((∏t‘(𝐴 × {𝒫 𝐴})) ∈ (TopOn‘(𝐴m 𝐴)) ∧ (Base‘𝐺) ⊆ (𝐴m 𝐴)) → ((∏t‘(𝐴 × {𝒫 𝐴})) ↾t (Base‘𝐺)) ∈ (TopOn‘(Base‘𝐺)))
1910, 17, 18syl2anc 584 . . . . 5 (𝐴𝑉 → ((∏t‘(𝐴 × {𝒫 𝐴})) ↾t (Base‘𝐺)) ∈ (TopOn‘(Base‘𝐺)))
206, 19eqeltrrd 2911 . . . 4 (𝐴𝑉 → (TopOpen‘𝐺) ∈ (TopOn‘(Base‘𝐺)))
21 eqid 2818 . . . . 5 (TopOpen‘𝐺) = (TopOpen‘𝐺)
225, 21istps 21470 . . . 4 (𝐺 ∈ TopSp ↔ (TopOpen‘𝐺) ∈ (TopOn‘(Base‘𝐺)))
2320, 22sylibr 235 . . 3 (𝐴𝑉𝐺 ∈ TopSp)
24 eqid 2818 . . . . . . . 8 (+g𝐺) = (+g𝐺)
251, 5, 24symgplusg 18445 . . . . . . 7 (+g𝐺) = (𝑥 ∈ (Base‘𝐺), 𝑦 ∈ (Base‘𝐺) ↦ (𝑥𝑦))
26 eqid 2818 . . . . . . . 8 ((𝒫 𝐴ko 𝒫 𝐴) ↾t (Base‘𝐺)) = ((𝒫 𝐴ko 𝒫 𝐴) ↾t (Base‘𝐺))
27 distop 21531 . . . . . . . . 9 (𝐴𝑉 → 𝒫 𝐴 ∈ Top)
28 eqid 2818 . . . . . . . . . 10 (𝒫 𝐴ko 𝒫 𝐴) = (𝒫 𝐴ko 𝒫 𝐴)
2928xkotopon 22136 . . . . . . . . 9 ((𝒫 𝐴 ∈ Top ∧ 𝒫 𝐴 ∈ Top) → (𝒫 𝐴ko 𝒫 𝐴) ∈ (TopOn‘(𝒫 𝐴 Cn 𝒫 𝐴)))
3027, 27, 29syl2anc 584 . . . . . . . 8 (𝐴𝑉 → (𝒫 𝐴ko 𝒫 𝐴) ∈ (TopOn‘(𝒫 𝐴 Cn 𝒫 𝐴)))
31 cndis 21827 . . . . . . . . . 10 ((𝐴𝑉 ∧ 𝒫 𝐴 ∈ (TopOn‘𝐴)) → (𝒫 𝐴 Cn 𝒫 𝐴) = (𝐴m 𝐴))
327, 31mpdan 683 . . . . . . . . 9 (𝐴𝑉 → (𝒫 𝐴 Cn 𝒫 𝐴) = (𝐴m 𝐴))
3317, 32sseqtrrd 4005 . . . . . . . 8 (𝐴𝑉 → (Base‘𝐺) ⊆ (𝒫 𝐴 Cn 𝒫 𝐴))
34 disllycmp 22034 . . . . . . . . . 10 (𝐴𝑉 → 𝒫 𝐴 ∈ Locally Comp)
35 llynlly 22013 . . . . . . . . . 10 (𝒫 𝐴 ∈ Locally Comp → 𝒫 𝐴 ∈ 𝑛-Locally Comp)
3634, 35syl 17 . . . . . . . . 9 (𝐴𝑉 → 𝒫 𝐴 ∈ 𝑛-Locally Comp)
37 eqid 2818 . . . . . . . . . 10 (𝑥 ∈ (𝒫 𝐴 Cn 𝒫 𝐴), 𝑦 ∈ (𝒫 𝐴 Cn 𝒫 𝐴) ↦ (𝑥𝑦)) = (𝑥 ∈ (𝒫 𝐴 Cn 𝒫 𝐴), 𝑦 ∈ (𝒫 𝐴 Cn 𝒫 𝐴) ↦ (𝑥𝑦))
3837xkococn 22196 . . . . . . . . 9 ((𝒫 𝐴 ∈ Top ∧ 𝒫 𝐴 ∈ 𝑛-Locally Comp ∧ 𝒫 𝐴 ∈ Top) → (𝑥 ∈ (𝒫 𝐴 Cn 𝒫 𝐴), 𝑦 ∈ (𝒫 𝐴 Cn 𝒫 𝐴) ↦ (𝑥𝑦)) ∈ (((𝒫 𝐴ko 𝒫 𝐴) ×t (𝒫 𝐴ko 𝒫 𝐴)) Cn (𝒫 𝐴ko 𝒫 𝐴)))
3927, 36, 27, 38syl3anc 1363 . . . . . . . 8 (𝐴𝑉 → (𝑥 ∈ (𝒫 𝐴 Cn 𝒫 𝐴), 𝑦 ∈ (𝒫 𝐴 Cn 𝒫 𝐴) ↦ (𝑥𝑦)) ∈ (((𝒫 𝐴ko 𝒫 𝐴) ×t (𝒫 𝐴ko 𝒫 𝐴)) Cn (𝒫 𝐴ko 𝒫 𝐴)))
4026, 30, 33, 26, 30, 33, 39cnmpt2res 22213 . . . . . . 7 (𝐴𝑉 → (𝑥 ∈ (Base‘𝐺), 𝑦 ∈ (Base‘𝐺) ↦ (𝑥𝑦)) ∈ ((((𝒫 𝐴ko 𝒫 𝐴) ↾t (Base‘𝐺)) ×t ((𝒫 𝐴ko 𝒫 𝐴) ↾t (Base‘𝐺))) Cn (𝒫 𝐴ko 𝒫 𝐴)))
4125, 40eqeltrid 2914 . . . . . 6 (𝐴𝑉 → (+g𝐺) ∈ ((((𝒫 𝐴ko 𝒫 𝐴) ↾t (Base‘𝐺)) ×t ((𝒫 𝐴ko 𝒫 𝐴) ↾t (Base‘𝐺))) Cn (𝒫 𝐴ko 𝒫 𝐴)))
42 xkopt 22191 . . . . . . . . . . 11 ((𝒫 𝐴 ∈ Top ∧ 𝐴𝑉) → (𝒫 𝐴ko 𝒫 𝐴) = (∏t‘(𝐴 × {𝒫 𝐴})))
4327, 42mpancom 684 . . . . . . . . . 10 (𝐴𝑉 → (𝒫 𝐴ko 𝒫 𝐴) = (∏t‘(𝐴 × {𝒫 𝐴})))
4443oveq1d 7160 . . . . . . . . 9 (𝐴𝑉 → ((𝒫 𝐴ko 𝒫 𝐴) ↾t (Base‘𝐺)) = ((∏t‘(𝐴 × {𝒫 𝐴})) ↾t (Base‘𝐺)))
4544, 6eqtrd 2853 . . . . . . . 8 (𝐴𝑉 → ((𝒫 𝐴ko 𝒫 𝐴) ↾t (Base‘𝐺)) = (TopOpen‘𝐺))
4645, 45oveq12d 7163 . . . . . . 7 (𝐴𝑉 → (((𝒫 𝐴ko 𝒫 𝐴) ↾t (Base‘𝐺)) ×t ((𝒫 𝐴ko 𝒫 𝐴) ↾t (Base‘𝐺))) = ((TopOpen‘𝐺) ×t (TopOpen‘𝐺)))
4746oveq1d 7160 . . . . . 6 (𝐴𝑉 → ((((𝒫 𝐴ko 𝒫 𝐴) ↾t (Base‘𝐺)) ×t ((𝒫 𝐴ko 𝒫 𝐴) ↾t (Base‘𝐺))) Cn (𝒫 𝐴ko 𝒫 𝐴)) = (((TopOpen‘𝐺) ×t (TopOpen‘𝐺)) Cn (𝒫 𝐴ko 𝒫 𝐴)))
4841, 47eleqtrd 2912 . . . . 5 (𝐴𝑉 → (+g𝐺) ∈ (((TopOpen‘𝐺) ×t (TopOpen‘𝐺)) Cn (𝒫 𝐴ko 𝒫 𝐴)))
49 vex 3495 . . . . . . . . . . . 12 𝑥 ∈ V
50 vex 3495 . . . . . . . . . . . 12 𝑦 ∈ V
5149, 50coex 7624 . . . . . . . . . . 11 (𝑥𝑦) ∈ V
5225, 51fnmpoi 7757 . . . . . . . . . 10 (+g𝐺) Fn ((Base‘𝐺) × (Base‘𝐺))
53 eqid 2818 . . . . . . . . . . 11 (+𝑓𝐺) = (+𝑓𝐺)
545, 24, 53plusfeq 17848 . . . . . . . . . 10 ((+g𝐺) Fn ((Base‘𝐺) × (Base‘𝐺)) → (+𝑓𝐺) = (+g𝐺))
5552, 54ax-mp 5 . . . . . . . . 9 (+𝑓𝐺) = (+g𝐺)
5655eqcomi 2827 . . . . . . . 8 (+g𝐺) = (+𝑓𝐺)
575, 56grpplusf 18053 . . . . . . 7 (𝐺 ∈ Grp → (+g𝐺):((Base‘𝐺) × (Base‘𝐺))⟶(Base‘𝐺))
58 frn 6513 . . . . . . 7 ((+g𝐺):((Base‘𝐺) × (Base‘𝐺))⟶(Base‘𝐺) → ran (+g𝐺) ⊆ (Base‘𝐺))
592, 57, 583syl 18 . . . . . 6 (𝐴𝑉 → ran (+g𝐺) ⊆ (Base‘𝐺))
60 cnrest2 21822 . . . . . 6 (((𝒫 𝐴ko 𝒫 𝐴) ∈ (TopOn‘(𝒫 𝐴 Cn 𝒫 𝐴)) ∧ ran (+g𝐺) ⊆ (Base‘𝐺) ∧ (Base‘𝐺) ⊆ (𝒫 𝐴 Cn 𝒫 𝐴)) → ((+g𝐺) ∈ (((TopOpen‘𝐺) ×t (TopOpen‘𝐺)) Cn (𝒫 𝐴ko 𝒫 𝐴)) ↔ (+g𝐺) ∈ (((TopOpen‘𝐺) ×t (TopOpen‘𝐺)) Cn ((𝒫 𝐴ko 𝒫 𝐴) ↾t (Base‘𝐺)))))
6130, 59, 33, 60syl3anc 1363 . . . . 5 (𝐴𝑉 → ((+g𝐺) ∈ (((TopOpen‘𝐺) ×t (TopOpen‘𝐺)) Cn (𝒫 𝐴ko 𝒫 𝐴)) ↔ (+g𝐺) ∈ (((TopOpen‘𝐺) ×t (TopOpen‘𝐺)) Cn ((𝒫 𝐴ko 𝒫 𝐴) ↾t (Base‘𝐺)))))
6248, 61mpbid 233 . . . 4 (𝐴𝑉 → (+g𝐺) ∈ (((TopOpen‘𝐺) ×t (TopOpen‘𝐺)) Cn ((𝒫 𝐴ko 𝒫 𝐴) ↾t (Base‘𝐺))))
6345oveq2d 7161 . . . 4 (𝐴𝑉 → (((TopOpen‘𝐺) ×t (TopOpen‘𝐺)) Cn ((𝒫 𝐴ko 𝒫 𝐴) ↾t (Base‘𝐺))) = (((TopOpen‘𝐺) ×t (TopOpen‘𝐺)) Cn (TopOpen‘𝐺)))
6462, 63eleqtrd 2912 . . 3 (𝐴𝑉 → (+g𝐺) ∈ (((TopOpen‘𝐺) ×t (TopOpen‘𝐺)) Cn (TopOpen‘𝐺)))
6556, 21istmd 22610 . . 3 (𝐺 ∈ TopMnd ↔ (𝐺 ∈ Mnd ∧ 𝐺 ∈ TopSp ∧ (+g𝐺) ∈ (((TopOpen‘𝐺) ×t (TopOpen‘𝐺)) Cn (TopOpen‘𝐺))))
664, 23, 64, 65syl3anbrc 1335 . 2 (𝐴𝑉𝐺 ∈ TopMnd)
67 id 22 . . . . . 6 (𝐴𝑉𝐴𝑉)
68 fconst6g 6561 . . . . . . 7 (𝒫 𝐴 ∈ Top → (𝐴 × {𝒫 𝐴}):𝐴⟶Top)
6927, 68syl 17 . . . . . 6 (𝐴𝑉 → (𝐴 × {𝒫 𝐴}):𝐴⟶Top)
7011biimpa 477 . . . . . . . . . . . 12 ((𝐴𝑉𝑥 ∈ (Base‘𝐺)) → 𝑥:𝐴1-1-onto𝐴)
71 f1ocnv 6620 . . . . . . . . . . . 12 (𝑥:𝐴1-1-onto𝐴𝑥:𝐴1-1-onto𝐴)
72 f1of 6608 . . . . . . . . . . . 12 (𝑥:𝐴1-1-onto𝐴𝑥:𝐴𝐴)
7370, 71, 723syl 18 . . . . . . . . . . 11 ((𝐴𝑉𝑥 ∈ (Base‘𝐺)) → 𝑥:𝐴𝐴)
7473ffvelrnda 6843 . . . . . . . . . 10 (((𝐴𝑉𝑥 ∈ (Base‘𝐺)) ∧ 𝑦𝐴) → (𝑥𝑦) ∈ 𝐴)
7574an32s 648 . . . . . . . . 9 (((𝐴𝑉𝑦𝐴) ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑥𝑦) ∈ 𝐴)
7675fmpttd 6871 . . . . . . . 8 ((𝐴𝑉𝑦𝐴) → (𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦)):(Base‘𝐺)⟶𝐴)
7776adantr 481 . . . . . . . . . 10 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → (𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦)):(Base‘𝐺)⟶𝐴)
78 cnveq 5737 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑓𝑥 = 𝑓)
7978fveq1d 6665 . . . . . . . . . . . . . . 15 (𝑥 = 𝑓 → (𝑥𝑦) = (𝑓𝑦))
80 eqid 2818 . . . . . . . . . . . . . . 15 (𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦)) = (𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦))
81 fvex 6676 . . . . . . . . . . . . . . 15 (𝑓𝑦) ∈ V
8279, 80, 81fvmpt 6761 . . . . . . . . . . . . . 14 (𝑓 ∈ (Base‘𝐺) → ((𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦))‘𝑓) = (𝑓𝑦))
8382ad2antlr 723 . . . . . . . . . . . . 13 ((((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ 𝑡 ∈ 𝒫 𝐴) → ((𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦))‘𝑓) = (𝑓𝑦))
8483eleq1d 2894 . . . . . . . . . . . 12 ((((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ 𝑡 ∈ 𝒫 𝐴) → (((𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦))‘𝑓) ∈ 𝑡 ↔ (𝑓𝑦) ∈ 𝑡))
85 eqid 2818 . . . . . . . . . . . . . . . . . 18 (𝑢 ∈ (Base‘𝐺) ↦ (𝑢‘(𝑓𝑦))) = (𝑢 ∈ (Base‘𝐺) ↦ (𝑢‘(𝑓𝑦)))
8685mptiniseg 6086 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ V → ((𝑢 ∈ (Base‘𝐺) ↦ (𝑢‘(𝑓𝑦))) “ {𝑦}) = {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(𝑓𝑦)) = 𝑦})
8786elv 3497 . . . . . . . . . . . . . . . 16 ((𝑢 ∈ (Base‘𝐺) ↦ (𝑢‘(𝑓𝑦))) “ {𝑦}) = {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(𝑓𝑦)) = 𝑦}
88 eqid 2818 . . . . . . . . . . . . . . . . . . 19 ((∏t‘(𝐴 × {𝒫 𝐴})) ↾t (Base‘𝐺)) = ((∏t‘(𝐴 × {𝒫 𝐴})) ↾t (Base‘𝐺))
8910ad2antrr 722 . . . . . . . . . . . . . . . . . . 19 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → (∏t‘(𝐴 × {𝒫 𝐴})) ∈ (TopOn‘(𝐴m 𝐴)))
9017ad2antrr 722 . . . . . . . . . . . . . . . . . . 19 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → (Base‘𝐺) ⊆ (𝐴m 𝐴))
91 toponuni 21450 . . . . . . . . . . . . . . . . . . . . 21 ((∏t‘(𝐴 × {𝒫 𝐴})) ∈ (TopOn‘(𝐴m 𝐴)) → (𝐴m 𝐴) = (∏t‘(𝐴 × {𝒫 𝐴})))
92 mpteq1 5145 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴m 𝐴) = (∏t‘(𝐴 × {𝒫 𝐴})) → (𝑢 ∈ (𝐴m 𝐴) ↦ (𝑢‘(𝑓𝑦))) = (𝑢 (∏t‘(𝐴 × {𝒫 𝐴})) ↦ (𝑢‘(𝑓𝑦))))
9389, 91, 923syl 18 . . . . . . . . . . . . . . . . . . . 20 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → (𝑢 ∈ (𝐴m 𝐴) ↦ (𝑢‘(𝑓𝑦))) = (𝑢 (∏t‘(𝐴 × {𝒫 𝐴})) ↦ (𝑢‘(𝑓𝑦))))
94 simpll 763 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → 𝐴𝑉)
9569ad2antrr 722 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → (𝐴 × {𝒫 𝐴}):𝐴⟶Top)
961, 5elsymgbas 18438 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝐴𝑉 → (𝑓 ∈ (Base‘𝐺) ↔ 𝑓:𝐴1-1-onto𝐴))
9796adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝐴𝑉𝑦𝐴) → (𝑓 ∈ (Base‘𝐺) ↔ 𝑓:𝐴1-1-onto𝐴))
9897biimpa 477 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → 𝑓:𝐴1-1-onto𝐴)
99 f1ocnv 6620 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑓:𝐴1-1-onto𝐴𝑓:𝐴1-1-onto𝐴)
100 f1of 6608 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑓:𝐴1-1-onto𝐴𝑓:𝐴𝐴)
10198, 99, 1003syl 18 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → 𝑓:𝐴𝐴)
102 simplr 765 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → 𝑦𝐴)
103101, 102ffvelrnd 6844 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → (𝑓𝑦) ∈ 𝐴)
104 eqid 2818 . . . . . . . . . . . . . . . . . . . . . . 23 (∏t‘(𝐴 × {𝒫 𝐴})) = (∏t‘(𝐴 × {𝒫 𝐴}))
105104, 8ptpjcn 22147 . . . . . . . . . . . . . . . . . . . . . 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))))
10694, 95, 103, 105syl3anc 1363 . . . . . . . . . . . . . . . . . . . . 21 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → (𝑢 (∏t‘(𝐴 × {𝒫 𝐴})) ↦ (𝑢‘(𝑓𝑦))) ∈ ((∏t‘(𝐴 × {𝒫 𝐴})) Cn ((𝐴 × {𝒫 𝐴})‘(𝑓𝑦))))
10727ad2antrr 722 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → 𝒫 𝐴 ∈ Top)
108 fvconst2g 6956 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝒫 𝐴 ∈ Top ∧ (𝑓𝑦) ∈ 𝐴) → ((𝐴 × {𝒫 𝐴})‘(𝑓𝑦)) = 𝒫 𝐴)
109107, 103, 108syl2anc 584 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → ((𝐴 × {𝒫 𝐴})‘(𝑓𝑦)) = 𝒫 𝐴)
110109oveq2d 7161 . . . . . . . . . . . . . . . . . . . . 21 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → ((∏t‘(𝐴 × {𝒫 𝐴})) Cn ((𝐴 × {𝒫 𝐴})‘(𝑓𝑦))) = ((∏t‘(𝐴 × {𝒫 𝐴})) Cn 𝒫 𝐴))
111106, 110eleqtrd 2912 . . . . . . . . . . . . . . . . . . . 20 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → (𝑢 (∏t‘(𝐴 × {𝒫 𝐴})) ↦ (𝑢‘(𝑓𝑦))) ∈ ((∏t‘(𝐴 × {𝒫 𝐴})) Cn 𝒫 𝐴))
11293, 111eqeltrd 2910 . . . . . . . . . . . . . . . . . . 19 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → (𝑢 ∈ (𝐴m 𝐴) ↦ (𝑢‘(𝑓𝑦))) ∈ ((∏t‘(𝐴 × {𝒫 𝐴})) Cn 𝒫 𝐴))
11388, 89, 90, 112cnmpt1res 22212 . . . . . . . . . . . . . . . . . 18 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → (𝑢 ∈ (Base‘𝐺) ↦ (𝑢‘(𝑓𝑦))) ∈ (((∏t‘(𝐴 × {𝒫 𝐴})) ↾t (Base‘𝐺)) Cn 𝒫 𝐴))
1146oveq1d 7160 . . . . . . . . . . . . . . . . . . 19 (𝐴𝑉 → (((∏t‘(𝐴 × {𝒫 𝐴})) ↾t (Base‘𝐺)) Cn 𝒫 𝐴) = ((TopOpen‘𝐺) Cn 𝒫 𝐴))
115114ad2antrr 722 . . . . . . . . . . . . . . . . . 18 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → (((∏t‘(𝐴 × {𝒫 𝐴})) ↾t (Base‘𝐺)) Cn 𝒫 𝐴) = ((TopOpen‘𝐺) Cn 𝒫 𝐴))
116113, 115eleqtrd 2912 . . . . . . . . . . . . . . . . 17 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → (𝑢 ∈ (Base‘𝐺) ↦ (𝑢‘(𝑓𝑦))) ∈ ((TopOpen‘𝐺) Cn 𝒫 𝐴))
117 snelpwi 5327 . . . . . . . . . . . . . . . . . 18 (𝑦𝐴 → {𝑦} ∈ 𝒫 𝐴)
118117ad2antlr 723 . . . . . . . . . . . . . . . . 17 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → {𝑦} ∈ 𝒫 𝐴)
119 cnima 21801 . . . . . . . . . . . . . . . . 17 (((𝑢 ∈ (Base‘𝐺) ↦ (𝑢‘(𝑓𝑦))) ∈ ((TopOpen‘𝐺) Cn 𝒫 𝐴) ∧ {𝑦} ∈ 𝒫 𝐴) → ((𝑢 ∈ (Base‘𝐺) ↦ (𝑢‘(𝑓𝑦))) “ {𝑦}) ∈ (TopOpen‘𝐺))
120116, 118, 119syl2anc 584 . . . . . . . . . . . . . . . 16 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → ((𝑢 ∈ (Base‘𝐺) ↦ (𝑢‘(𝑓𝑦))) “ {𝑦}) ∈ (TopOpen‘𝐺))
12187, 120eqeltrrid 2915 . . . . . . . . . . . . . . 15 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(𝑓𝑦)) = 𝑦} ∈ (TopOpen‘𝐺))
122121adantr 481 . . . . . . . . . . . . . 14 ((((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ (𝑡 ∈ 𝒫 𝐴 ∧ (𝑓𝑦) ∈ 𝑡)) → {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(𝑓𝑦)) = 𝑦} ∈ (TopOpen‘𝐺))
123 fveq1 6662 . . . . . . . . . . . . . . . 16 (𝑢 = 𝑓 → (𝑢‘(𝑓𝑦)) = (𝑓‘(𝑓𝑦)))
124123eqeq1d 2820 . . . . . . . . . . . . . . 15 (𝑢 = 𝑓 → ((𝑢‘(𝑓𝑦)) = 𝑦 ↔ (𝑓‘(𝑓𝑦)) = 𝑦))
125 simplr 765 . . . . . . . . . . . . . . 15 ((((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ (𝑡 ∈ 𝒫 𝐴 ∧ (𝑓𝑦) ∈ 𝑡)) → 𝑓 ∈ (Base‘𝐺))
12698adantr 481 . . . . . . . . . . . . . . . 16 ((((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ (𝑡 ∈ 𝒫 𝐴 ∧ (𝑓𝑦) ∈ 𝑡)) → 𝑓:𝐴1-1-onto𝐴)
127 simpllr 772 . . . . . . . . . . . . . . . 16 ((((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ (𝑡 ∈ 𝒫 𝐴 ∧ (𝑓𝑦) ∈ 𝑡)) → 𝑦𝐴)
128 f1ocnvfv2 7025 . . . . . . . . . . . . . . . 16 ((𝑓:𝐴1-1-onto𝐴𝑦𝐴) → (𝑓‘(𝑓𝑦)) = 𝑦)
129126, 127, 128syl2anc 584 . . . . . . . . . . . . . . 15 ((((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ (𝑡 ∈ 𝒫 𝐴 ∧ (𝑓𝑦) ∈ 𝑡)) → (𝑓‘(𝑓𝑦)) = 𝑦)
130124, 125, 129elrabd 3679 . . . . . . . . . . . . . 14 ((((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ (𝑡 ∈ 𝒫 𝐴 ∧ (𝑓𝑦) ∈ 𝑡)) → 𝑓 ∈ {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(𝑓𝑦)) = 𝑦})
131 ssrab2 4053 . . . . . . . . . . . . . . . . . 18 {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(𝑓𝑦)) = 𝑦} ⊆ (Base‘𝐺)
132131a1i 11 . . . . . . . . . . . . . . . . 17 ((((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ (𝑡 ∈ 𝒫 𝐴 ∧ (𝑓𝑦) ∈ 𝑡)) → {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(𝑓𝑦)) = 𝑦} ⊆ (Base‘𝐺))
13311ad3antrrr 726 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ (𝑡 ∈ 𝒫 𝐴 ∧ (𝑓𝑦) ∈ 𝑡)) → (𝑥 ∈ (Base‘𝐺) ↔ 𝑥:𝐴1-1-onto𝐴))
134133biimpa 477 . . . . . . . . . . . . . . . . . . . . 21 (((((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ (𝑡 ∈ 𝒫 𝐴 ∧ (𝑓𝑦) ∈ 𝑡)) ∧ 𝑥 ∈ (Base‘𝐺)) → 𝑥:𝐴1-1-onto𝐴)
135103ad2antrr 722 . . . . . . . . . . . . . . . . . . . . 21 (((((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ (𝑡 ∈ 𝒫 𝐴 ∧ (𝑓𝑦) ∈ 𝑡)) ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑓𝑦) ∈ 𝐴)
136 f1ocnvfv 7026 . . . . . . . . . . . . . . . . . . . . 21 ((𝑥:𝐴1-1-onto𝐴 ∧ (𝑓𝑦) ∈ 𝐴) → ((𝑥‘(𝑓𝑦)) = 𝑦 → (𝑥𝑦) = (𝑓𝑦)))
137134, 135, 136syl2anc 584 . . . . . . . . . . . . . . . . . . . 20 (((((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ (𝑡 ∈ 𝒫 𝐴 ∧ (𝑓𝑦) ∈ 𝑡)) ∧ 𝑥 ∈ (Base‘𝐺)) → ((𝑥‘(𝑓𝑦)) = 𝑦 → (𝑥𝑦) = (𝑓𝑦)))
138 simplrr 774 . . . . . . . . . . . . . . . . . . . . 21 (((((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ (𝑡 ∈ 𝒫 𝐴 ∧ (𝑓𝑦) ∈ 𝑡)) ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑓𝑦) ∈ 𝑡)
139 eleq1 2897 . . . . . . . . . . . . . . . . . . . . 21 ((𝑥𝑦) = (𝑓𝑦) → ((𝑥𝑦) ∈ 𝑡 ↔ (𝑓𝑦) ∈ 𝑡))
140138, 139syl5ibrcom 248 . . . . . . . . . . . . . . . . . . . 20 (((((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ (𝑡 ∈ 𝒫 𝐴 ∧ (𝑓𝑦) ∈ 𝑡)) ∧ 𝑥 ∈ (Base‘𝐺)) → ((𝑥𝑦) = (𝑓𝑦) → (𝑥𝑦) ∈ 𝑡))
141137, 140syld 47 . . . . . . . . . . . . . . . . . . 19 (((((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ (𝑡 ∈ 𝒫 𝐴 ∧ (𝑓𝑦) ∈ 𝑡)) ∧ 𝑥 ∈ (Base‘𝐺)) → ((𝑥‘(𝑓𝑦)) = 𝑦 → (𝑥𝑦) ∈ 𝑡))
142141ralrimiva 3179 . . . . . . . . . . . . . . . . . 18 ((((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ (𝑡 ∈ 𝒫 𝐴 ∧ (𝑓𝑦) ∈ 𝑡)) → ∀𝑥 ∈ (Base‘𝐺)((𝑥‘(𝑓𝑦)) = 𝑦 → (𝑥𝑦) ∈ 𝑡))
143 fveq1 6662 . . . . . . . . . . . . . . . . . . . 20 (𝑢 = 𝑥 → (𝑢‘(𝑓𝑦)) = (𝑥‘(𝑓𝑦)))
144143eqeq1d 2820 . . . . . . . . . . . . . . . . . . 19 (𝑢 = 𝑥 → ((𝑢‘(𝑓𝑦)) = 𝑦 ↔ (𝑥‘(𝑓𝑦)) = 𝑦))
145144ralrab 3682 . . . . . . . . . . . . . . . . . 18 (∀𝑥 ∈ {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(𝑓𝑦)) = 𝑦} (𝑥𝑦) ∈ 𝑡 ↔ ∀𝑥 ∈ (Base‘𝐺)((𝑥‘(𝑓𝑦)) = 𝑦 → (𝑥𝑦) ∈ 𝑡))
146142, 145sylibr 235 . . . . . . . . . . . . . . . . 17 ((((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ (𝑡 ∈ 𝒫 𝐴 ∧ (𝑓𝑦) ∈ 𝑡)) → ∀𝑥 ∈ {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(𝑓𝑦)) = 𝑦} (𝑥𝑦) ∈ 𝑡)
147 ssrab 4046 . . . . . . . . . . . . . . . . 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))
148132, 146, 147sylanbrc 583 . . . . . . . . . . . . . . . 16 ((((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ (𝑡 ∈ 𝒫 𝐴 ∧ (𝑓𝑦) ∈ 𝑡)) → {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(𝑓𝑦)) = 𝑦} ⊆ {𝑥 ∈ (Base‘𝐺) ∣ (𝑥𝑦) ∈ 𝑡})
14980mptpreima 6085 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦)) “ 𝑡) = {𝑥 ∈ (Base‘𝐺) ∣ (𝑥𝑦) ∈ 𝑡}
150148, 149sseqtrrdi 4015 . . . . . . . . . . . . . . 15 ((((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ (𝑡 ∈ 𝒫 𝐴 ∧ (𝑓𝑦) ∈ 𝑡)) → {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(𝑓𝑦)) = 𝑦} ⊆ ((𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦)) “ 𝑡))
151 funmpt 6386 . . . . . . . . . . . . . . . 16 Fun (𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦))
152 fvex 6676 . . . . . . . . . . . . . . . . . 18 (𝑥𝑦) ∈ V
153152, 80dmmpti 6485 . . . . . . . . . . . . . . . . 17 dom (𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦)) = (Base‘𝐺)
154132, 153sseqtrrdi 4015 . . . . . . . . . . . . . . . 16 ((((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ (𝑡 ∈ 𝒫 𝐴 ∧ (𝑓𝑦) ∈ 𝑡)) → {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(𝑓𝑦)) = 𝑦} ⊆ dom (𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦)))
155 funimass3 6816 . . . . . . . . . . . . . . . 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)))
156151, 154, 155sylancr 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)))
157150, 156mpbird 258 . . . . . . . . . . . . . 14 ((((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ (𝑡 ∈ 𝒫 𝐴 ∧ (𝑓𝑦) ∈ 𝑡)) → ((𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦)) “ {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(𝑓𝑦)) = 𝑦}) ⊆ 𝑡)
158 eleq2 2898 . . . . . . . . . . . . . . . 16 (𝑣 = {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(𝑓𝑦)) = 𝑦} → (𝑓𝑣𝑓 ∈ {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(𝑓𝑦)) = 𝑦}))
159 imaeq2 5918 . . . . . . . . . . . . . . . . 17 (𝑣 = {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(𝑓𝑦)) = 𝑦} → ((𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦)) “ 𝑣) = ((𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦)) “ {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(𝑓𝑦)) = 𝑦}))
160159sseq1d 3995 . . . . . . . . . . . . . . . 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))
161158, 160anbi12d 630 . . . . . . . . . . . . . . 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)))
162161rspcev 3620 . . . . . . . . . . . . . 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))
163122, 130, 157, 162syl12anc 832 . . . . . . . . . . . . 13 ((((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ (𝑡 ∈ 𝒫 𝐴 ∧ (𝑓𝑦) ∈ 𝑡)) → ∃𝑣 ∈ (TopOpen‘𝐺)(𝑓𝑣 ∧ ((𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦)) “ 𝑣) ⊆ 𝑡))
164163expr 457 . . . . . . . . . . . 12 ((((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ 𝑡 ∈ 𝒫 𝐴) → ((𝑓𝑦) ∈ 𝑡 → ∃𝑣 ∈ (TopOpen‘𝐺)(𝑓𝑣 ∧ ((𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦)) “ 𝑣) ⊆ 𝑡)))
16584, 164sylbid 241 . . . . . . . . . . 11 ((((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ 𝑡 ∈ 𝒫 𝐴) → (((𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦))‘𝑓) ∈ 𝑡 → ∃𝑣 ∈ (TopOpen‘𝐺)(𝑓𝑣 ∧ ((𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦)) “ 𝑣) ⊆ 𝑡)))
166165ralrimiva 3179 . . . . . . . . . 10 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → ∀𝑡 ∈ 𝒫 𝐴(((𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦))‘𝑓) ∈ 𝑡 → ∃𝑣 ∈ (TopOpen‘𝐺)(𝑓𝑣 ∧ ((𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦)) “ 𝑣) ⊆ 𝑡)))
16720ad2antrr 722 . . . . . . . . . . 11 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → (TopOpen‘𝐺) ∈ (TopOn‘(Base‘𝐺)))
1687ad2antrr 722 . . . . . . . . . . 11 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → 𝒫 𝐴 ∈ (TopOn‘𝐴))
169 simpr 485 . . . . . . . . . . 11 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → 𝑓 ∈ (Base‘𝐺))
170 iscnp 21773 . . . . . . . . . . 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)))))
171167, 168, 169, 170syl3anc 1363 . . . . . . . . . 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)))))
17277, 166, 171mpbir2and 709 . . . . . . . . 9 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → (𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦)) ∈ (((TopOpen‘𝐺) CnP 𝒫 𝐴)‘𝑓))
173172ralrimiva 3179 . . . . . . . 8 ((𝐴𝑉𝑦𝐴) → ∀𝑓 ∈ (Base‘𝐺)(𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦)) ∈ (((TopOpen‘𝐺) CnP 𝒫 𝐴)‘𝑓))
174 cncnp 21816 . . . . . . . . . 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))))
17520, 7, 174syl2anc 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))))
176175adantr 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))))
17776, 173, 176mpbir2and 709 . . . . . . 7 ((𝐴𝑉𝑦𝐴) → (𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦)) ∈ ((TopOpen‘𝐺) Cn 𝒫 𝐴))
178 fvconst2g 6956 . . . . . . . . 9 ((𝒫 𝐴 ∈ Top ∧ 𝑦𝐴) → ((𝐴 × {𝒫 𝐴})‘𝑦) = 𝒫 𝐴)
17927, 178sylan 580 . . . . . . . 8 ((𝐴𝑉𝑦𝐴) → ((𝐴 × {𝒫 𝐴})‘𝑦) = 𝒫 𝐴)
180179oveq2d 7161 . . . . . . 7 ((𝐴𝑉𝑦𝐴) → ((TopOpen‘𝐺) Cn ((𝐴 × {𝒫 𝐴})‘𝑦)) = ((TopOpen‘𝐺) Cn 𝒫 𝐴))
181177, 180eleqtrrd 2913 . . . . . 6 ((𝐴𝑉𝑦𝐴) → (𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦)) ∈ ((TopOpen‘𝐺) Cn ((𝐴 × {𝒫 𝐴})‘𝑦)))
1828, 20, 67, 69, 181ptcn 22163 . . . . 5 (𝐴𝑉 → (𝑥 ∈ (Base‘𝐺) ↦ (𝑦𝐴 ↦ (𝑥𝑦))) ∈ ((TopOpen‘𝐺) Cn (∏t‘(𝐴 × {𝒫 𝐴}))))
183 eqid 2818 . . . . . . . . 9 (invg𝐺) = (invg𝐺)
1845, 183grpinvf 18088 . . . . . . . 8 (𝐺 ∈ Grp → (invg𝐺):(Base‘𝐺)⟶(Base‘𝐺))
1852, 184syl 17 . . . . . . 7 (𝐴𝑉 → (invg𝐺):(Base‘𝐺)⟶(Base‘𝐺))
186185feqmptd 6726 . . . . . 6 (𝐴𝑉 → (invg𝐺) = (𝑥 ∈ (Base‘𝐺) ↦ ((invg𝐺)‘𝑥)))
1871, 5, 183symginv 18460 . . . . . . . . 9 (𝑥 ∈ (Base‘𝐺) → ((invg𝐺)‘𝑥) = 𝑥)
188187adantl 482 . . . . . . . 8 ((𝐴𝑉𝑥 ∈ (Base‘𝐺)) → ((invg𝐺)‘𝑥) = 𝑥)
18973feqmptd 6726 . . . . . . . 8 ((𝐴𝑉𝑥 ∈ (Base‘𝐺)) → 𝑥 = (𝑦𝐴 ↦ (𝑥𝑦)))
190188, 189eqtrd 2853 . . . . . . 7 ((𝐴𝑉𝑥 ∈ (Base‘𝐺)) → ((invg𝐺)‘𝑥) = (𝑦𝐴 ↦ (𝑥𝑦)))
191190mpteq2dva 5152 . . . . . 6 (𝐴𝑉 → (𝑥 ∈ (Base‘𝐺) ↦ ((invg𝐺)‘𝑥)) = (𝑥 ∈ (Base‘𝐺) ↦ (𝑦𝐴 ↦ (𝑥𝑦))))
192186, 191eqtrd 2853 . . . . 5 (𝐴𝑉 → (invg𝐺) = (𝑥 ∈ (Base‘𝐺) ↦ (𝑦𝐴 ↦ (𝑥𝑦))))
19343oveq2d 7161 . . . . 5 (𝐴𝑉 → ((TopOpen‘𝐺) Cn (𝒫 𝐴ko 𝒫 𝐴)) = ((TopOpen‘𝐺) Cn (∏t‘(𝐴 × {𝒫 𝐴}))))
194182, 192, 1933eltr4d 2925 . . . 4 (𝐴𝑉 → (invg𝐺) ∈ ((TopOpen‘𝐺) Cn (𝒫 𝐴ko 𝒫 𝐴)))
195 frn 6513 . . . . . 6 ((invg𝐺):(Base‘𝐺)⟶(Base‘𝐺) → ran (invg𝐺) ⊆ (Base‘𝐺))
1962, 184, 1953syl 18 . . . . 5 (𝐴𝑉 → ran (invg𝐺) ⊆ (Base‘𝐺))
197 cnrest2 21822 . . . . 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)))))
19830, 196, 33, 197syl3anc 1363 . . . 4 (𝐴𝑉 → ((invg𝐺) ∈ ((TopOpen‘𝐺) Cn (𝒫 𝐴ko 𝒫 𝐴)) ↔ (invg𝐺) ∈ ((TopOpen‘𝐺) Cn ((𝒫 𝐴ko 𝒫 𝐴) ↾t (Base‘𝐺)))))
199194, 198mpbid 233 . . 3 (𝐴𝑉 → (invg𝐺) ∈ ((TopOpen‘𝐺) Cn ((𝒫 𝐴ko 𝒫 𝐴) ↾t (Base‘𝐺))))
20045oveq2d 7161 . . 3 (𝐴𝑉 → ((TopOpen‘𝐺) Cn ((𝒫 𝐴ko 𝒫 𝐴) ↾t (Base‘𝐺))) = ((TopOpen‘𝐺) Cn (TopOpen‘𝐺)))
201199, 200eleqtrd 2912 . 2 (𝐴𝑉 → (invg𝐺) ∈ ((TopOpen‘𝐺) Cn (TopOpen‘𝐺)))
20221, 183istgp 22613 . 2 (𝐺 ∈ TopGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ TopMnd ∧ (invg𝐺) ∈ ((TopOpen‘𝐺) Cn (TopOpen‘𝐺))))
2032, 66, 201, 202syl3anbrc 1335 1 (𝐴𝑉𝐺 ∈ TopGrp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  wral 3135  wrex 3136  {crab 3139  Vcvv 3492  wss 3933  𝒫 cpw 4535  {csn 4557   cuni 4830  cmpt 5137   × cxp 5546  ccnv 5547  dom cdm 5548  ran crn 5549  cima 5551  ccom 5552  Fun wfun 6342   Fn wfn 6343  wf 6344  1-1-ontowf1o 6347  cfv 6348  (class class class)co 7145  cmpo 7147  m cmap 8395  Basecbs 16471  +gcplusg 16553  t crest 16682  TopOpenctopn 16683  tcpt 16700  +𝑓cplusf 17837  Mndcmnd 17899  Grpcgrp 18041  invgcminusg 18042  SymGrpcsymg 18433  Topctop 21429  TopOnctopon 21446  TopSpctps 21468   Cn ccn 21760   CnP ccnp 21761  Compccmp 21922  Locally clly 22000  𝑛-Locally cnlly 22001   ×t ctx 22096  ko cxko 22097  TopMndctmd 22606  TopGrpctgp 22607
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-iin 4913  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-2o 8092  df-oadd 8095  df-er 8278  df-map 8397  df-ixp 8450  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-fi 8863  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-nn 11627  df-2 11688  df-3 11689  df-4 11690  df-5 11691  df-6 11692  df-7 11693  df-8 11694  df-9 11695  df-n0 11886  df-z 11970  df-uz 12232  df-fz 12881  df-struct 16473  df-ndx 16474  df-slot 16475  df-base 16477  df-plusg 16566  df-tset 16572  df-rest 16684  df-topn 16685  df-0g 16703  df-topgen 16705  df-pt 16706  df-plusf 17839  df-mgm 17840  df-sgrp 17889  df-mnd 17900  df-grp 18044  df-minusg 18045  df-symg 18434  df-top 21430  df-topon 21447  df-topsp 21469  df-bases 21482  df-ntr 21556  df-nei 21634  df-cn 21763  df-cnp 21764  df-cmp 21923  df-lly 22002  df-nlly 22003  df-tx 22098  df-xko 22099  df-tmd 22608  df-tgp 22609
This theorem is referenced by: (None)
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