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Theorem symgtgp 21818
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 17744 . 2 (𝐴𝑉𝐺 ∈ Grp)
3 grpmnd 17353 . . . 4 (𝐺 ∈ Grp → 𝐺 ∈ Mnd)
42, 3syl 17 . . 3 (𝐴𝑉𝐺 ∈ Mnd)
5 eqid 2621 . . . . . 6 (Base‘𝐺) = (Base‘𝐺)
61, 5symgtopn 17749 . . . . 5 (𝐴𝑉 → ((∏t‘(𝐴 × {𝒫 𝐴})) ↾t (Base‘𝐺)) = (TopOpen‘𝐺))
7 distopon 20714 . . . . . . 7 (𝐴𝑉 → 𝒫 𝐴 ∈ (TopOn‘𝐴))
8 eqid 2621 . . . . . . . 8 (∏t‘(𝐴 × {𝒫 𝐴})) = (∏t‘(𝐴 × {𝒫 𝐴}))
98pttoponconst 21313 . . . . . . 7 ((𝐴𝑉 ∧ 𝒫 𝐴 ∈ (TopOn‘𝐴)) → (∏t‘(𝐴 × {𝒫 𝐴})) ∈ (TopOn‘(𝐴𝑚 𝐴)))
107, 9mpdan 701 . . . . . 6 (𝐴𝑉 → (∏t‘(𝐴 × {𝒫 𝐴})) ∈ (TopOn‘(𝐴𝑚 𝐴)))
111, 5elsymgbas 17726 . . . . . . . 8 (𝐴𝑉 → (𝑥 ∈ (Base‘𝐺) ↔ 𝑥:𝐴1-1-onto𝐴))
12 f1of 6096 . . . . . . . . 9 (𝑥:𝐴1-1-onto𝐴𝑥:𝐴𝐴)
13 elmapg 7818 . . . . . . . . . 10 ((𝐴𝑉𝐴𝑉) → (𝑥 ∈ (𝐴𝑚 𝐴) ↔ 𝑥:𝐴𝐴))
1413anidms 676 . . . . . . . . 9 (𝐴𝑉 → (𝑥 ∈ (𝐴𝑚 𝐴) ↔ 𝑥:𝐴𝐴))
1512, 14syl5ibr 236 . . . . . . . 8 (𝐴𝑉 → (𝑥:𝐴1-1-onto𝐴𝑥 ∈ (𝐴𝑚 𝐴)))
1611, 15sylbid 230 . . . . . . 7 (𝐴𝑉 → (𝑥 ∈ (Base‘𝐺) → 𝑥 ∈ (𝐴𝑚 𝐴)))
1716ssrdv 3590 . . . . . 6 (𝐴𝑉 → (Base‘𝐺) ⊆ (𝐴𝑚 𝐴))
18 resttopon 20878 . . . . . 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 692 . . . . 5 (𝐴𝑉 → ((∏t‘(𝐴 × {𝒫 𝐴})) ↾t (Base‘𝐺)) ∈ (TopOn‘(Base‘𝐺)))
206, 19eqeltrrd 2699 . . . 4 (𝐴𝑉 → (TopOpen‘𝐺) ∈ (TopOn‘(Base‘𝐺)))
21 eqid 2621 . . . . 5 (TopOpen‘𝐺) = (TopOpen‘𝐺)
225, 21istps 20652 . . . 4 (𝐺 ∈ TopSp ↔ (TopOpen‘𝐺) ∈ (TopOn‘(Base‘𝐺)))
2320, 22sylibr 224 . . 3 (𝐴𝑉𝐺 ∈ TopSp)
24 eqid 2621 . . . . . . . 8 (+g𝐺) = (+g𝐺)
251, 5, 24symgplusg 17733 . . . . . . 7 (+g𝐺) = (𝑥 ∈ (Base‘𝐺), 𝑦 ∈ (Base‘𝐺) ↦ (𝑥𝑦))
26 eqid 2621 . . . . . . . 8 ((𝒫 𝐴 ^ko 𝒫 𝐴) ↾t (Base‘𝐺)) = ((𝒫 𝐴 ^ko 𝒫 𝐴) ↾t (Base‘𝐺))
27 distop 20713 . . . . . . . . 9 (𝐴𝑉 → 𝒫 𝐴 ∈ Top)
28 eqid 2621 . . . . . . . . . 10 (𝒫 𝐴 ^ko 𝒫 𝐴) = (𝒫 𝐴 ^ko 𝒫 𝐴)
2928xkotopon 21316 . . . . . . . . 9 ((𝒫 𝐴 ∈ Top ∧ 𝒫 𝐴 ∈ Top) → (𝒫 𝐴 ^ko 𝒫 𝐴) ∈ (TopOn‘(𝒫 𝐴 Cn 𝒫 𝐴)))
3027, 27, 29syl2anc 692 . . . . . . . 8 (𝐴𝑉 → (𝒫 𝐴 ^ko 𝒫 𝐴) ∈ (TopOn‘(𝒫 𝐴 Cn 𝒫 𝐴)))
31 cndis 21008 . . . . . . . . . 10 ((𝐴𝑉 ∧ 𝒫 𝐴 ∈ (TopOn‘𝐴)) → (𝒫 𝐴 Cn 𝒫 𝐴) = (𝐴𝑚 𝐴))
327, 31mpdan 701 . . . . . . . . 9 (𝐴𝑉 → (𝒫 𝐴 Cn 𝒫 𝐴) = (𝐴𝑚 𝐴))
3317, 32sseqtr4d 3623 . . . . . . . 8 (𝐴𝑉 → (Base‘𝐺) ⊆ (𝒫 𝐴 Cn 𝒫 𝐴))
34 disllycmp 21214 . . . . . . . . . 10 (𝐴𝑉 → 𝒫 𝐴 ∈ Locally Comp)
35 llynlly 21193 . . . . . . . . . 10 (𝒫 𝐴 ∈ Locally Comp → 𝒫 𝐴 ∈ 𝑛-Locally Comp)
3634, 35syl 17 . . . . . . . . 9 (𝐴𝑉 → 𝒫 𝐴 ∈ 𝑛-Locally Comp)
37 eqid 2621 . . . . . . . . . 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 21376 . . . . . . . . 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 𝐴 ^ko 𝒫 𝐴) ×t (𝒫 𝐴 ^ko 𝒫 𝐴)) Cn (𝒫 𝐴 ^ko 𝒫 𝐴)))
3927, 36, 27, 38syl3anc 1323 . . . . . . . 8 (𝐴𝑉 → (𝑥 ∈ (𝒫 𝐴 Cn 𝒫 𝐴), 𝑦 ∈ (𝒫 𝐴 Cn 𝒫 𝐴) ↦ (𝑥𝑦)) ∈ (((𝒫 𝐴 ^ko 𝒫 𝐴) ×t (𝒫 𝐴 ^ko 𝒫 𝐴)) Cn (𝒫 𝐴 ^ko 𝒫 𝐴)))
4026, 30, 33, 26, 30, 33, 39cnmpt2res 21393 . . . . . . 7 (𝐴𝑉 → (𝑥 ∈ (Base‘𝐺), 𝑦 ∈ (Base‘𝐺) ↦ (𝑥𝑦)) ∈ ((((𝒫 𝐴 ^ko 𝒫 𝐴) ↾t (Base‘𝐺)) ×t ((𝒫 𝐴 ^ko 𝒫 𝐴) ↾t (Base‘𝐺))) Cn (𝒫 𝐴 ^ko 𝒫 𝐴)))
4125, 40syl5eqel 2702 . . . . . 6 (𝐴𝑉 → (+g𝐺) ∈ ((((𝒫 𝐴 ^ko 𝒫 𝐴) ↾t (Base‘𝐺)) ×t ((𝒫 𝐴 ^ko 𝒫 𝐴) ↾t (Base‘𝐺))) Cn (𝒫 𝐴 ^ko 𝒫 𝐴)))
42 xkopt 21371 . . . . . . . . . . 11 ((𝒫 𝐴 ∈ Top ∧ 𝐴𝑉) → (𝒫 𝐴 ^ko 𝒫 𝐴) = (∏t‘(𝐴 × {𝒫 𝐴})))
4327, 42mpancom 702 . . . . . . . . . 10 (𝐴𝑉 → (𝒫 𝐴 ^ko 𝒫 𝐴) = (∏t‘(𝐴 × {𝒫 𝐴})))
4443oveq1d 6622 . . . . . . . . 9 (𝐴𝑉 → ((𝒫 𝐴 ^ko 𝒫 𝐴) ↾t (Base‘𝐺)) = ((∏t‘(𝐴 × {𝒫 𝐴})) ↾t (Base‘𝐺)))
4544, 6eqtrd 2655 . . . . . . . 8 (𝐴𝑉 → ((𝒫 𝐴 ^ko 𝒫 𝐴) ↾t (Base‘𝐺)) = (TopOpen‘𝐺))
4645, 45oveq12d 6625 . . . . . . 7 (𝐴𝑉 → (((𝒫 𝐴 ^ko 𝒫 𝐴) ↾t (Base‘𝐺)) ×t ((𝒫 𝐴 ^ko 𝒫 𝐴) ↾t (Base‘𝐺))) = ((TopOpen‘𝐺) ×t (TopOpen‘𝐺)))
4746oveq1d 6622 . . . . . 6 (𝐴𝑉 → ((((𝒫 𝐴 ^ko 𝒫 𝐴) ↾t (Base‘𝐺)) ×t ((𝒫 𝐴 ^ko 𝒫 𝐴) ↾t (Base‘𝐺))) Cn (𝒫 𝐴 ^ko 𝒫 𝐴)) = (((TopOpen‘𝐺) ×t (TopOpen‘𝐺)) Cn (𝒫 𝐴 ^ko 𝒫 𝐴)))
4841, 47eleqtrd 2700 . . . . 5 (𝐴𝑉 → (+g𝐺) ∈ (((TopOpen‘𝐺) ×t (TopOpen‘𝐺)) Cn (𝒫 𝐴 ^ko 𝒫 𝐴)))
49 vex 3189 . . . . . . . . . . . 12 𝑥 ∈ V
50 vex 3189 . . . . . . . . . . . 12 𝑦 ∈ V
5149, 50coex 7068 . . . . . . . . . . 11 (𝑥𝑦) ∈ V
5225, 51fnmpt2i 7187 . . . . . . . . . 10 (+g𝐺) Fn ((Base‘𝐺) × (Base‘𝐺))
53 eqid 2621 . . . . . . . . . . 11 (+𝑓𝐺) = (+𝑓𝐺)
545, 24, 53plusfeq 17173 . . . . . . . . . 10 ((+g𝐺) Fn ((Base‘𝐺) × (Base‘𝐺)) → (+𝑓𝐺) = (+g𝐺))
5552, 54ax-mp 5 . . . . . . . . 9 (+𝑓𝐺) = (+g𝐺)
5655eqcomi 2630 . . . . . . . 8 (+g𝐺) = (+𝑓𝐺)
575, 56grpplusf 17358 . . . . . . 7 (𝐺 ∈ Grp → (+g𝐺):((Base‘𝐺) × (Base‘𝐺))⟶(Base‘𝐺))
58 frn 6012 . . . . . . 7 ((+g𝐺):((Base‘𝐺) × (Base‘𝐺))⟶(Base‘𝐺) → ran (+g𝐺) ⊆ (Base‘𝐺))
592, 57, 583syl 18 . . . . . 6 (𝐴𝑉 → ran (+g𝐺) ⊆ (Base‘𝐺))
60 cnrest2 21003 . . . . . 6 (((𝒫 𝐴 ^ko 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 𝐴 ^ko 𝒫 𝐴)) ↔ (+g𝐺) ∈ (((TopOpen‘𝐺) ×t (TopOpen‘𝐺)) Cn ((𝒫 𝐴 ^ko 𝒫 𝐴) ↾t (Base‘𝐺)))))
6130, 59, 33, 60syl3anc 1323 . . . . 5 (𝐴𝑉 → ((+g𝐺) ∈ (((TopOpen‘𝐺) ×t (TopOpen‘𝐺)) Cn (𝒫 𝐴 ^ko 𝒫 𝐴)) ↔ (+g𝐺) ∈ (((TopOpen‘𝐺) ×t (TopOpen‘𝐺)) Cn ((𝒫 𝐴 ^ko 𝒫 𝐴) ↾t (Base‘𝐺)))))
6248, 61mpbid 222 . . . 4 (𝐴𝑉 → (+g𝐺) ∈ (((TopOpen‘𝐺) ×t (TopOpen‘𝐺)) Cn ((𝒫 𝐴 ^ko 𝒫 𝐴) ↾t (Base‘𝐺))))
6345oveq2d 6623 . . . 4 (𝐴𝑉 → (((TopOpen‘𝐺) ×t (TopOpen‘𝐺)) Cn ((𝒫 𝐴 ^ko 𝒫 𝐴) ↾t (Base‘𝐺))) = (((TopOpen‘𝐺) ×t (TopOpen‘𝐺)) Cn (TopOpen‘𝐺)))
6462, 63eleqtrd 2700 . . 3 (𝐴𝑉 → (+g𝐺) ∈ (((TopOpen‘𝐺) ×t (TopOpen‘𝐺)) Cn (TopOpen‘𝐺)))
6556, 21istmd 21791 . . 3 (𝐺 ∈ TopMnd ↔ (𝐺 ∈ Mnd ∧ 𝐺 ∈ TopSp ∧ (+g𝐺) ∈ (((TopOpen‘𝐺) ×t (TopOpen‘𝐺)) Cn (TopOpen‘𝐺))))
664, 23, 64, 65syl3anbrc 1244 . 2 (𝐴𝑉𝐺 ∈ TopMnd)
67 id 22 . . . . . 6 (𝐴𝑉𝐴𝑉)
68 fconst6g 6053 . . . . . . 7 (𝒫 𝐴 ∈ Top → (𝐴 × {𝒫 𝐴}):𝐴⟶Top)
6927, 68syl 17 . . . . . 6 (𝐴𝑉 → (𝐴 × {𝒫 𝐴}):𝐴⟶Top)
7011biimpa 501 . . . . . . . . . . . 12 ((𝐴𝑉𝑥 ∈ (Base‘𝐺)) → 𝑥:𝐴1-1-onto𝐴)
71 f1ocnv 6108 . . . . . . . . . . . 12 (𝑥:𝐴1-1-onto𝐴𝑥:𝐴1-1-onto𝐴)
72 f1of 6096 . . . . . . . . . . . 12 (𝑥:𝐴1-1-onto𝐴𝑥:𝐴𝐴)
7370, 71, 723syl 18 . . . . . . . . . . 11 ((𝐴𝑉𝑥 ∈ (Base‘𝐺)) → 𝑥:𝐴𝐴)
7473ffvelrnda 6317 . . . . . . . . . 10 (((𝐴𝑉𝑥 ∈ (Base‘𝐺)) ∧ 𝑦𝐴) → (𝑥𝑦) ∈ 𝐴)
7574an32s 845 . . . . . . . . 9 (((𝐴𝑉𝑦𝐴) ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑥𝑦) ∈ 𝐴)
76 eqid 2621 . . . . . . . . 9 (𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦)) = (𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦))
7775, 76fmptd 6343 . . . . . . . 8 ((𝐴𝑉𝑦𝐴) → (𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦)):(Base‘𝐺)⟶𝐴)
7877adantr 481 . . . . . . . . . 10 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → (𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦)):(Base‘𝐺)⟶𝐴)
79 cnveq 5258 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑓𝑥 = 𝑓)
8079fveq1d 6152 . . . . . . . . . . . . . . 15 (𝑥 = 𝑓 → (𝑥𝑦) = (𝑓𝑦))
81 fvex 6160 . . . . . . . . . . . . . . 15 (𝑓𝑦) ∈ V
8280, 76, 81fvmpt 6241 . . . . . . . . . . . . . 14 (𝑓 ∈ (Base‘𝐺) → ((𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦))‘𝑓) = (𝑓𝑦))
8382ad2antlr 762 . . . . . . . . . . . . 13 ((((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ 𝑡 ∈ 𝒫 𝐴) → ((𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦))‘𝑓) = (𝑓𝑦))
8483eleq1d 2683 . . . . . . . . . . . 12 ((((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ 𝑡 ∈ 𝒫 𝐴) → (((𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦))‘𝑓) ∈ 𝑡 ↔ (𝑓𝑦) ∈ 𝑡))
85 eqid 2621 . . . . . . . . . . . . . . . . . 18 (𝑢 ∈ (Base‘𝐺) ↦ (𝑢‘(𝑓𝑦))) = (𝑢 ∈ (Base‘𝐺) ↦ (𝑢‘(𝑓𝑦)))
8685mptiniseg 5590 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ V → ((𝑢 ∈ (Base‘𝐺) ↦ (𝑢‘(𝑓𝑦))) “ {𝑦}) = {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(𝑓𝑦)) = 𝑦})
8750, 86ax-mp 5 . . . . . . . . . . . . . . . 16 ((𝑢 ∈ (Base‘𝐺) ↦ (𝑢‘(𝑓𝑦))) “ {𝑦}) = {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(𝑓𝑦)) = 𝑦}
88 eqid 2621 . . . . . . . . . . . . . . . . . . 19 ((∏t‘(𝐴 × {𝒫 𝐴})) ↾t (Base‘𝐺)) = ((∏t‘(𝐴 × {𝒫 𝐴})) ↾t (Base‘𝐺))
8910ad2antrr 761 . . . . . . . . . . . . . . . . . . 19 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → (∏t‘(𝐴 × {𝒫 𝐴})) ∈ (TopOn‘(𝐴𝑚 𝐴)))
9017ad2antrr 761 . . . . . . . . . . . . . . . . . . 19 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → (Base‘𝐺) ⊆ (𝐴𝑚 𝐴))
91 toponuni 20641 . . . . . . . . . . . . . . . . . . . . 21 ((∏t‘(𝐴 × {𝒫 𝐴})) ∈ (TopOn‘(𝐴𝑚 𝐴)) → (𝐴𝑚 𝐴) = (∏t‘(𝐴 × {𝒫 𝐴})))
92 mpteq1 4699 . . . . . . . . . . . . . . . . . . . . 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 789 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → 𝐴𝑉)
9569ad2antrr 761 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → (𝐴 × {𝒫 𝐴}):𝐴⟶Top)
961, 5elsymgbas 17726 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝐴𝑉 → (𝑓 ∈ (Base‘𝐺) ↔ 𝑓:𝐴1-1-onto𝐴))
9796adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝐴𝑉𝑦𝐴) → (𝑓 ∈ (Base‘𝐺) ↔ 𝑓:𝐴1-1-onto𝐴))
9897biimpa 501 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → 𝑓:𝐴1-1-onto𝐴)
99 f1ocnv 6108 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑓:𝐴1-1-onto𝐴𝑓:𝐴1-1-onto𝐴)
100 f1of 6096 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑓:𝐴1-1-onto𝐴𝑓:𝐴𝐴)
10198, 99, 1003syl 18 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → 𝑓:𝐴𝐴)
102 simplr 791 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → 𝑦𝐴)
103101, 102ffvelrnd 6318 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → (𝑓𝑦) ∈ 𝐴)
104 eqid 2621 . . . . . . . . . . . . . . . . . . . . . . 23 (∏t‘(𝐴 × {𝒫 𝐴})) = (∏t‘(𝐴 × {𝒫 𝐴}))
105104, 8ptpjcn 21327 . . . . . . . . . . . . . . . . . . . . . 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 1323 . . . . . . . . . . . . . . . . . . . . 21 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → (𝑢 (∏t‘(𝐴 × {𝒫 𝐴})) ↦ (𝑢‘(𝑓𝑦))) ∈ ((∏t‘(𝐴 × {𝒫 𝐴})) Cn ((𝐴 × {𝒫 𝐴})‘(𝑓𝑦))))
10727ad2antrr 761 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → 𝒫 𝐴 ∈ Top)
108 fvconst2g 6424 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝒫 𝐴 ∈ Top ∧ (𝑓𝑦) ∈ 𝐴) → ((𝐴 × {𝒫 𝐴})‘(𝑓𝑦)) = 𝒫 𝐴)
109107, 103, 108syl2anc 692 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → ((𝐴 × {𝒫 𝐴})‘(𝑓𝑦)) = 𝒫 𝐴)
110109oveq2d 6623 . . . . . . . . . . . . . . . . . . . . 21 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → ((∏t‘(𝐴 × {𝒫 𝐴})) Cn ((𝐴 × {𝒫 𝐴})‘(𝑓𝑦))) = ((∏t‘(𝐴 × {𝒫 𝐴})) Cn 𝒫 𝐴))
111106, 110eleqtrd 2700 . . . . . . . . . . . . . . . . . . . 20 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → (𝑢 (∏t‘(𝐴 × {𝒫 𝐴})) ↦ (𝑢‘(𝑓𝑦))) ∈ ((∏t‘(𝐴 × {𝒫 𝐴})) Cn 𝒫 𝐴))
11293, 111eqeltrd 2698 . . . . . . . . . . . . . . . . . . 19 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → (𝑢 ∈ (𝐴𝑚 𝐴) ↦ (𝑢‘(𝑓𝑦))) ∈ ((∏t‘(𝐴 × {𝒫 𝐴})) Cn 𝒫 𝐴))
11388, 89, 90, 112cnmpt1res 21392 . . . . . . . . . . . . . . . . . 18 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → (𝑢 ∈ (Base‘𝐺) ↦ (𝑢‘(𝑓𝑦))) ∈ (((∏t‘(𝐴 × {𝒫 𝐴})) ↾t (Base‘𝐺)) Cn 𝒫 𝐴))
1146oveq1d 6622 . . . . . . . . . . . . . . . . . . 19 (𝐴𝑉 → (((∏t‘(𝐴 × {𝒫 𝐴})) ↾t (Base‘𝐺)) Cn 𝒫 𝐴) = ((TopOpen‘𝐺) Cn 𝒫 𝐴))
115114ad2antrr 761 . . . . . . . . . . . . . . . . . 18 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → (((∏t‘(𝐴 × {𝒫 𝐴})) ↾t (Base‘𝐺)) Cn 𝒫 𝐴) = ((TopOpen‘𝐺) Cn 𝒫 𝐴))
116113, 115eleqtrd 2700 . . . . . . . . . . . . . . . . 17 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → (𝑢 ∈ (Base‘𝐺) ↦ (𝑢‘(𝑓𝑦))) ∈ ((TopOpen‘𝐺) Cn 𝒫 𝐴))
117 snelpwi 4875 . . . . . . . . . . . . . . . . . 18 (𝑦𝐴 → {𝑦} ∈ 𝒫 𝐴)
118117ad2antlr 762 . . . . . . . . . . . . . . . . 17 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → {𝑦} ∈ 𝒫 𝐴)
119 cnima 20982 . . . . . . . . . . . . . . . . 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 692 . . . . . . . . . . . . . . . 16 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → ((𝑢 ∈ (Base‘𝐺) ↦ (𝑢‘(𝑓𝑦))) “ {𝑦}) ∈ (TopOpen‘𝐺))
12187, 120syl5eqelr 2703 . . . . . . . . . . . . . . 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 simplr 791 . . . . . . . . . . . . . . 15 ((((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ (𝑡 ∈ 𝒫 𝐴 ∧ (𝑓𝑦) ∈ 𝑡)) → 𝑓 ∈ (Base‘𝐺))
12498adantr 481 . . . . . . . . . . . . . . . 16 ((((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ (𝑡 ∈ 𝒫 𝐴 ∧ (𝑓𝑦) ∈ 𝑡)) → 𝑓:𝐴1-1-onto𝐴)
125 simpllr 798 . . . . . . . . . . . . . . . 16 ((((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ (𝑡 ∈ 𝒫 𝐴 ∧ (𝑓𝑦) ∈ 𝑡)) → 𝑦𝐴)
126 f1ocnvfv2 6490 . . . . . . . . . . . . . . . 16 ((𝑓:𝐴1-1-onto𝐴𝑦𝐴) → (𝑓‘(𝑓𝑦)) = 𝑦)
127124, 125, 126syl2anc 692 . . . . . . . . . . . . . . 15 ((((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ (𝑡 ∈ 𝒫 𝐴 ∧ (𝑓𝑦) ∈ 𝑡)) → (𝑓‘(𝑓𝑦)) = 𝑦)
128 fveq1 6149 . . . . . . . . . . . . . . . . 17 (𝑢 = 𝑓 → (𝑢‘(𝑓𝑦)) = (𝑓‘(𝑓𝑦)))
129128eqeq1d 2623 . . . . . . . . . . . . . . . 16 (𝑢 = 𝑓 → ((𝑢‘(𝑓𝑦)) = 𝑦 ↔ (𝑓‘(𝑓𝑦)) = 𝑦))
130129elrab 3347 . . . . . . . . . . . . . . 15 (𝑓 ∈ {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(𝑓𝑦)) = 𝑦} ↔ (𝑓 ∈ (Base‘𝐺) ∧ (𝑓‘(𝑓𝑦)) = 𝑦))
131123, 127, 130sylanbrc 697 . . . . . . . . . . . . . 14 ((((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ (𝑡 ∈ 𝒫 𝐴 ∧ (𝑓𝑦) ∈ 𝑡)) → 𝑓 ∈ {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(𝑓𝑦)) = 𝑦})
132 ssrab2 3668 . . . . . . . . . . . . . . . . . 18 {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(𝑓𝑦)) = 𝑦} ⊆ (Base‘𝐺)
133132a1i 11 . . . . . . . . . . . . . . . . 17 ((((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ (𝑡 ∈ 𝒫 𝐴 ∧ (𝑓𝑦) ∈ 𝑡)) → {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(𝑓𝑦)) = 𝑦} ⊆ (Base‘𝐺))
13411ad3antrrr 765 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ (𝑡 ∈ 𝒫 𝐴 ∧ (𝑓𝑦) ∈ 𝑡)) → (𝑥 ∈ (Base‘𝐺) ↔ 𝑥:𝐴1-1-onto𝐴))
135134biimpa 501 . . . . . . . . . . . . . . . . . . . . 21 (((((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ (𝑡 ∈ 𝒫 𝐴 ∧ (𝑓𝑦) ∈ 𝑡)) ∧ 𝑥 ∈ (Base‘𝐺)) → 𝑥:𝐴1-1-onto𝐴)
136103ad2antrr 761 . . . . . . . . . . . . . . . . . . . . 21 (((((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ (𝑡 ∈ 𝒫 𝐴 ∧ (𝑓𝑦) ∈ 𝑡)) ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑓𝑦) ∈ 𝐴)
137 f1ocnvfv 6491 . . . . . . . . . . . . . . . . . . . . 21 ((𝑥:𝐴1-1-onto𝐴 ∧ (𝑓𝑦) ∈ 𝐴) → ((𝑥‘(𝑓𝑦)) = 𝑦 → (𝑥𝑦) = (𝑓𝑦)))
138135, 136, 137syl2anc 692 . . . . . . . . . . . . . . . . . . . 20 (((((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ (𝑡 ∈ 𝒫 𝐴 ∧ (𝑓𝑦) ∈ 𝑡)) ∧ 𝑥 ∈ (Base‘𝐺)) → ((𝑥‘(𝑓𝑦)) = 𝑦 → (𝑥𝑦) = (𝑓𝑦)))
139 simplrr 800 . . . . . . . . . . . . . . . . . . . . 21 (((((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ (𝑡 ∈ 𝒫 𝐴 ∧ (𝑓𝑦) ∈ 𝑡)) ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑓𝑦) ∈ 𝑡)
140 eleq1 2686 . . . . . . . . . . . . . . . . . . . . 21 ((𝑥𝑦) = (𝑓𝑦) → ((𝑥𝑦) ∈ 𝑡 ↔ (𝑓𝑦) ∈ 𝑡))
141139, 140syl5ibrcom 237 . . . . . . . . . . . . . . . . . . . 20 (((((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ (𝑡 ∈ 𝒫 𝐴 ∧ (𝑓𝑦) ∈ 𝑡)) ∧ 𝑥 ∈ (Base‘𝐺)) → ((𝑥𝑦) = (𝑓𝑦) → (𝑥𝑦) ∈ 𝑡))
142138, 141syld 47 . . . . . . . . . . . . . . . . . . 19 (((((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ (𝑡 ∈ 𝒫 𝐴 ∧ (𝑓𝑦) ∈ 𝑡)) ∧ 𝑥 ∈ (Base‘𝐺)) → ((𝑥‘(𝑓𝑦)) = 𝑦 → (𝑥𝑦) ∈ 𝑡))
143142ralrimiva 2960 . . . . . . . . . . . . . . . . . 18 ((((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ (𝑡 ∈ 𝒫 𝐴 ∧ (𝑓𝑦) ∈ 𝑡)) → ∀𝑥 ∈ (Base‘𝐺)((𝑥‘(𝑓𝑦)) = 𝑦 → (𝑥𝑦) ∈ 𝑡))
144 fveq1 6149 . . . . . . . . . . . . . . . . . . . 20 (𝑢 = 𝑥 → (𝑢‘(𝑓𝑦)) = (𝑥‘(𝑓𝑦)))
145144eqeq1d 2623 . . . . . . . . . . . . . . . . . . 19 (𝑢 = 𝑥 → ((𝑢‘(𝑓𝑦)) = 𝑦 ↔ (𝑥‘(𝑓𝑦)) = 𝑦))
146145ralrab 3351 . . . . . . . . . . . . . . . . . 18 (∀𝑥 ∈ {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(𝑓𝑦)) = 𝑦} (𝑥𝑦) ∈ 𝑡 ↔ ∀𝑥 ∈ (Base‘𝐺)((𝑥‘(𝑓𝑦)) = 𝑦 → (𝑥𝑦) ∈ 𝑡))
147143, 146sylibr 224 . . . . . . . . . . . . . . . . 17 ((((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ (𝑡 ∈ 𝒫 𝐴 ∧ (𝑓𝑦) ∈ 𝑡)) → ∀𝑥 ∈ {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(𝑓𝑦)) = 𝑦} (𝑥𝑦) ∈ 𝑡)
148 ssrab 3661 . . . . . . . . . . . . . . . . 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))
149133, 147, 148sylanbrc 697 . . . . . . . . . . . . . . . 16 ((((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ (𝑡 ∈ 𝒫 𝐴 ∧ (𝑓𝑦) ∈ 𝑡)) → {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(𝑓𝑦)) = 𝑦} ⊆ {𝑥 ∈ (Base‘𝐺) ∣ (𝑥𝑦) ∈ 𝑡})
15076mptpreima 5589 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦)) “ 𝑡) = {𝑥 ∈ (Base‘𝐺) ∣ (𝑥𝑦) ∈ 𝑡}
151149, 150syl6sseqr 3633 . . . . . . . . . . . . . . 15 ((((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ (𝑡 ∈ 𝒫 𝐴 ∧ (𝑓𝑦) ∈ 𝑡)) → {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(𝑓𝑦)) = 𝑦} ⊆ ((𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦)) “ 𝑡))
152 funmpt 5886 . . . . . . . . . . . . . . . 16 Fun (𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦))
153 fvex 6160 . . . . . . . . . . . . . . . . . 18 (𝑥𝑦) ∈ V
154153, 76dmmpti 5982 . . . . . . . . . . . . . . . . 17 dom (𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦)) = (Base‘𝐺)
155133, 154syl6sseqr 3633 . . . . . . . . . . . . . . . 16 ((((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ (𝑡 ∈ 𝒫 𝐴 ∧ (𝑓𝑦) ∈ 𝑡)) → {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(𝑓𝑦)) = 𝑦} ⊆ dom (𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦)))
156 funimass3 6291 . . . . . . . . . . . . . . . 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)))
157152, 155, 156sylancr 694 . . . . . . . . . . . . . . 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)))
158151, 157mpbird 247 . . . . . . . . . . . . . 14 ((((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ (𝑡 ∈ 𝒫 𝐴 ∧ (𝑓𝑦) ∈ 𝑡)) → ((𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦)) “ {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(𝑓𝑦)) = 𝑦}) ⊆ 𝑡)
159 eleq2 2687 . . . . . . . . . . . . . . . 16 (𝑣 = {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(𝑓𝑦)) = 𝑦} → (𝑓𝑣𝑓 ∈ {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(𝑓𝑦)) = 𝑦}))
160 imaeq2 5423 . . . . . . . . . . . . . . . . 17 (𝑣 = {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(𝑓𝑦)) = 𝑦} → ((𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦)) “ 𝑣) = ((𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦)) “ {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(𝑓𝑦)) = 𝑦}))
161160sseq1d 3613 . . . . . . . . . . . . . . . 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))
162159, 161anbi12d 746 . . . . . . . . . . . . . . 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)))
163162rspcev 3295 . . . . . . . . . . . . . 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))
164122, 131, 158, 163syl12anc 1321 . . . . . . . . . . . . 13 ((((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ (𝑡 ∈ 𝒫 𝐴 ∧ (𝑓𝑦) ∈ 𝑡)) → ∃𝑣 ∈ (TopOpen‘𝐺)(𝑓𝑣 ∧ ((𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦)) “ 𝑣) ⊆ 𝑡))
165164expr 642 . . . . . . . . . . . 12 ((((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ 𝑡 ∈ 𝒫 𝐴) → ((𝑓𝑦) ∈ 𝑡 → ∃𝑣 ∈ (TopOpen‘𝐺)(𝑓𝑣 ∧ ((𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦)) “ 𝑣) ⊆ 𝑡)))
16684, 165sylbid 230 . . . . . . . . . . 11 ((((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ 𝑡 ∈ 𝒫 𝐴) → (((𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦))‘𝑓) ∈ 𝑡 → ∃𝑣 ∈ (TopOpen‘𝐺)(𝑓𝑣 ∧ ((𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦)) “ 𝑣) ⊆ 𝑡)))
167166ralrimiva 2960 . . . . . . . . . 10 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → ∀𝑡 ∈ 𝒫 𝐴(((𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦))‘𝑓) ∈ 𝑡 → ∃𝑣 ∈ (TopOpen‘𝐺)(𝑓𝑣 ∧ ((𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦)) “ 𝑣) ⊆ 𝑡)))
16820ad2antrr 761 . . . . . . . . . . 11 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → (TopOpen‘𝐺) ∈ (TopOn‘(Base‘𝐺)))
1697ad2antrr 761 . . . . . . . . . . 11 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → 𝒫 𝐴 ∈ (TopOn‘𝐴))
170 simpr 477 . . . . . . . . . . 11 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → 𝑓 ∈ (Base‘𝐺))
171 iscnp 20954 . . . . . . . . . . 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)))))
172168, 169, 170, 171syl3anc 1323 . . . . . . . . . 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)))))
17378, 167, 172mpbir2and 956 . . . . . . . . 9 (((𝐴𝑉𝑦𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → (𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦)) ∈ (((TopOpen‘𝐺) CnP 𝒫 𝐴)‘𝑓))
174173ralrimiva 2960 . . . . . . . 8 ((𝐴𝑉𝑦𝐴) → ∀𝑓 ∈ (Base‘𝐺)(𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦)) ∈ (((TopOpen‘𝐺) CnP 𝒫 𝐴)‘𝑓))
175 cncnp 20997 . . . . . . . . . 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))))
17620, 7, 175syl2anc 692 . . . . . . . . 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))))
177176adantr 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))))
17877, 174, 177mpbir2and 956 . . . . . . 7 ((𝐴𝑉𝑦𝐴) → (𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦)) ∈ ((TopOpen‘𝐺) Cn 𝒫 𝐴))
179 fvconst2g 6424 . . . . . . . . 9 ((𝒫 𝐴 ∈ Top ∧ 𝑦𝐴) → ((𝐴 × {𝒫 𝐴})‘𝑦) = 𝒫 𝐴)
18027, 179sylan 488 . . . . . . . 8 ((𝐴𝑉𝑦𝐴) → ((𝐴 × {𝒫 𝐴})‘𝑦) = 𝒫 𝐴)
181180oveq2d 6623 . . . . . . 7 ((𝐴𝑉𝑦𝐴) → ((TopOpen‘𝐺) Cn ((𝐴 × {𝒫 𝐴})‘𝑦)) = ((TopOpen‘𝐺) Cn 𝒫 𝐴))
182178, 181eleqtrrd 2701 . . . . . 6 ((𝐴𝑉𝑦𝐴) → (𝑥 ∈ (Base‘𝐺) ↦ (𝑥𝑦)) ∈ ((TopOpen‘𝐺) Cn ((𝐴 × {𝒫 𝐴})‘𝑦)))
1838, 20, 67, 69, 182ptcn 21343 . . . . 5 (𝐴𝑉 → (𝑥 ∈ (Base‘𝐺) ↦ (𝑦𝐴 ↦ (𝑥𝑦))) ∈ ((TopOpen‘𝐺) Cn (∏t‘(𝐴 × {𝒫 𝐴}))))
184 eqid 2621 . . . . . . . . 9 (invg𝐺) = (invg𝐺)
1855, 184grpinvf 17390 . . . . . . . 8 (𝐺 ∈ Grp → (invg𝐺):(Base‘𝐺)⟶(Base‘𝐺))
1862, 185syl 17 . . . . . . 7 (𝐴𝑉 → (invg𝐺):(Base‘𝐺)⟶(Base‘𝐺))
187186feqmptd 6208 . . . . . 6 (𝐴𝑉 → (invg𝐺) = (𝑥 ∈ (Base‘𝐺) ↦ ((invg𝐺)‘𝑥)))
1881, 5, 184symginv 17746 . . . . . . . . 9 (𝑥 ∈ (Base‘𝐺) → ((invg𝐺)‘𝑥) = 𝑥)
189188adantl 482 . . . . . . . 8 ((𝐴𝑉𝑥 ∈ (Base‘𝐺)) → ((invg𝐺)‘𝑥) = 𝑥)
19073feqmptd 6208 . . . . . . . 8 ((𝐴𝑉𝑥 ∈ (Base‘𝐺)) → 𝑥 = (𝑦𝐴 ↦ (𝑥𝑦)))
191189, 190eqtrd 2655 . . . . . . 7 ((𝐴𝑉𝑥 ∈ (Base‘𝐺)) → ((invg𝐺)‘𝑥) = (𝑦𝐴 ↦ (𝑥𝑦)))
192191mpteq2dva 4706 . . . . . 6 (𝐴𝑉 → (𝑥 ∈ (Base‘𝐺) ↦ ((invg𝐺)‘𝑥)) = (𝑥 ∈ (Base‘𝐺) ↦ (𝑦𝐴 ↦ (𝑥𝑦))))
193187, 192eqtrd 2655 . . . . 5 (𝐴𝑉 → (invg𝐺) = (𝑥 ∈ (Base‘𝐺) ↦ (𝑦𝐴 ↦ (𝑥𝑦))))
19443oveq2d 6623 . . . . 5 (𝐴𝑉 → ((TopOpen‘𝐺) Cn (𝒫 𝐴 ^ko 𝒫 𝐴)) = ((TopOpen‘𝐺) Cn (∏t‘(𝐴 × {𝒫 𝐴}))))
195183, 193, 1943eltr4d 2713 . . . 4 (𝐴𝑉 → (invg𝐺) ∈ ((TopOpen‘𝐺) Cn (𝒫 𝐴 ^ko 𝒫 𝐴)))
196 frn 6012 . . . . . 6 ((invg𝐺):(Base‘𝐺)⟶(Base‘𝐺) → ran (invg𝐺) ⊆ (Base‘𝐺))
1972, 185, 1963syl 18 . . . . 5 (𝐴𝑉 → ran (invg𝐺) ⊆ (Base‘𝐺))
198 cnrest2 21003 . . . . 5 (((𝒫 𝐴 ^ko 𝒫 𝐴) ∈ (TopOn‘(𝒫 𝐴 Cn 𝒫 𝐴)) ∧ ran (invg𝐺) ⊆ (Base‘𝐺) ∧ (Base‘𝐺) ⊆ (𝒫 𝐴 Cn 𝒫 𝐴)) → ((invg𝐺) ∈ ((TopOpen‘𝐺) Cn (𝒫 𝐴 ^ko 𝒫 𝐴)) ↔ (invg𝐺) ∈ ((TopOpen‘𝐺) Cn ((𝒫 𝐴 ^ko 𝒫 𝐴) ↾t (Base‘𝐺)))))
19930, 197, 33, 198syl3anc 1323 . . . 4 (𝐴𝑉 → ((invg𝐺) ∈ ((TopOpen‘𝐺) Cn (𝒫 𝐴 ^ko 𝒫 𝐴)) ↔ (invg𝐺) ∈ ((TopOpen‘𝐺) Cn ((𝒫 𝐴 ^ko 𝒫 𝐴) ↾t (Base‘𝐺)))))
200195, 199mpbid 222 . . 3 (𝐴𝑉 → (invg𝐺) ∈ ((TopOpen‘𝐺) Cn ((𝒫 𝐴 ^ko 𝒫 𝐴) ↾t (Base‘𝐺))))
20145oveq2d 6623 . . 3 (𝐴𝑉 → ((TopOpen‘𝐺) Cn ((𝒫 𝐴 ^ko 𝒫 𝐴) ↾t (Base‘𝐺))) = ((TopOpen‘𝐺) Cn (TopOpen‘𝐺)))
202200, 201eleqtrd 2700 . 2 (𝐴𝑉 → (invg𝐺) ∈ ((TopOpen‘𝐺) Cn (TopOpen‘𝐺)))
20321, 184istgp 21794 . 2 (𝐺 ∈ TopGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ TopMnd ∧ (invg𝐺) ∈ ((TopOpen‘𝐺) Cn (TopOpen‘𝐺))))
2042, 66, 202, 203syl3anbrc 1244 1 (𝐴𝑉𝐺 ∈ TopGrp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wral 2907  wrex 2908  {crab 2911  Vcvv 3186  wss 3556  𝒫 cpw 4132  {csn 4150   cuni 4404  cmpt 4675   × cxp 5074  ccnv 5075  dom cdm 5076  ran crn 5077  cima 5079  ccom 5080  Fun wfun 5843   Fn wfn 5844  wf 5845  1-1-ontowf1o 5848  cfv 5849  (class class class)co 6607  cmpt2 6609  𝑚 cmap 7805  Basecbs 15784  +gcplusg 15865  t crest 16005  TopOpenctopn 16006  tcpt 16023  +𝑓cplusf 17163  Mndcmnd 17218  Grpcgrp 17346  invgcminusg 17347  SymGrpcsymg 17721  Topctop 20620  TopOnctopon 20637  TopSpctps 20650   Cn ccn 20941   CnP ccnp 20942  Compccmp 21102  Locally clly 21180  𝑛-Locally cnlly 21181   ×t ctx 21276   ^ko cxko 21277  TopMndctmd 21787  TopGrpctgp 21788
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4733  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869  ax-un 6905  ax-cnex 9939  ax-resscn 9940  ax-1cn 9941  ax-icn 9942  ax-addcl 9943  ax-addrcl 9944  ax-mulcl 9945  ax-mulrcl 9946  ax-mulcom 9947  ax-addass 9948  ax-mulass 9949  ax-distr 9950  ax-i2m1 9951  ax-1ne0 9952  ax-1rid 9953  ax-rnegex 9954  ax-rrecex 9955  ax-cnre 9956  ax-pre-lttri 9957  ax-pre-lttrn 9958  ax-pre-ltadd 9959  ax-pre-mulgt0 9960
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3419  df-csb 3516  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-pss 3572  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-tp 4155  df-op 4157  df-uni 4405  df-int 4443  df-iun 4489  df-iin 4490  df-br 4616  df-opab 4676  df-mpt 4677  df-tr 4715  df-eprel 4987  df-id 4991  df-po 4997  df-so 4998  df-fr 5035  df-we 5037  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-pred 5641  df-ord 5687  df-on 5688  df-lim 5689  df-suc 5690  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-f1 5854  df-fo 5855  df-f1o 5856  df-fv 5857  df-riota 6568  df-ov 6610  df-oprab 6611  df-mpt2 6612  df-om 7016  df-1st 7116  df-2nd 7117  df-wrecs 7355  df-recs 7416  df-rdg 7454  df-1o 7508  df-2o 7509  df-oadd 7512  df-er 7690  df-map 7807  df-ixp 7856  df-en 7903  df-dom 7904  df-sdom 7905  df-fin 7906  df-fi 8264  df-pnf 10023  df-mnf 10024  df-xr 10025  df-ltxr 10026  df-le 10027  df-sub 10215  df-neg 10216  df-nn 10968  df-2 11026  df-3 11027  df-4 11028  df-5 11029  df-6 11030  df-7 11031  df-8 11032  df-9 11033  df-n0 11240  df-z 11325  df-uz 11635  df-fz 12272  df-struct 15786  df-ndx 15787  df-slot 15788  df-base 15789  df-plusg 15878  df-tset 15884  df-rest 16007  df-topn 16008  df-0g 16026  df-topgen 16028  df-pt 16029  df-plusf 17165  df-mgm 17166  df-sgrp 17208  df-mnd 17219  df-grp 17349  df-minusg 17350  df-symg 17722  df-top 20621  df-topon 20638  df-topsp 20651  df-bases 20664  df-ntr 20737  df-nei 20815  df-cn 20944  df-cnp 20945  df-cmp 21103  df-lly 21182  df-nlly 21183  df-tx 21278  df-xko 21279  df-tmd 21789  df-tgp 21790
This theorem is referenced by: (None)
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