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Theorem istgp 24101
Description: The predicate "is a topological group". Definition 1 of [BourbakiTop1] p. III.1. (Contributed by FL, 18-Apr-2010.) (Revised by Mario Carneiro, 13-Aug-2015.)
Hypotheses
Ref Expression
istgp.1 𝐽 = (TopOpen‘𝐺)
istgp.2 𝐼 = (invg𝐺)
Assertion
Ref Expression
istgp (𝐺 ∈ TopGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ TopMnd ∧ 𝐼 ∈ (𝐽 Cn 𝐽)))

Proof of Theorem istgp
Dummy variables 𝑓 𝑗 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elin 3979 . . 3 (𝐺 ∈ (Grp ∩ TopMnd) ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ TopMnd))
21anbi1i 624 . 2 ((𝐺 ∈ (Grp ∩ TopMnd) ∧ 𝐼 ∈ (𝐽 Cn 𝐽)) ↔ ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopMnd) ∧ 𝐼 ∈ (𝐽 Cn 𝐽)))
3 fvexd 6922 . . . 4 (𝑓 = 𝐺 → (TopOpen‘𝑓) ∈ V)
4 simpl 482 . . . . . . 7 ((𝑓 = 𝐺𝑗 = (TopOpen‘𝑓)) → 𝑓 = 𝐺)
54fveq2d 6911 . . . . . 6 ((𝑓 = 𝐺𝑗 = (TopOpen‘𝑓)) → (invg𝑓) = (invg𝐺))
6 istgp.2 . . . . . 6 𝐼 = (invg𝐺)
75, 6eqtr4di 2793 . . . . 5 ((𝑓 = 𝐺𝑗 = (TopOpen‘𝑓)) → (invg𝑓) = 𝐼)
8 id 22 . . . . . . 7 (𝑗 = (TopOpen‘𝑓) → 𝑗 = (TopOpen‘𝑓))
9 fveq2 6907 . . . . . . . 8 (𝑓 = 𝐺 → (TopOpen‘𝑓) = (TopOpen‘𝐺))
10 istgp.1 . . . . . . . 8 𝐽 = (TopOpen‘𝐺)
119, 10eqtr4di 2793 . . . . . . 7 (𝑓 = 𝐺 → (TopOpen‘𝑓) = 𝐽)
128, 11sylan9eqr 2797 . . . . . 6 ((𝑓 = 𝐺𝑗 = (TopOpen‘𝑓)) → 𝑗 = 𝐽)
1312, 12oveq12d 7449 . . . . 5 ((𝑓 = 𝐺𝑗 = (TopOpen‘𝑓)) → (𝑗 Cn 𝑗) = (𝐽 Cn 𝐽))
147, 13eleq12d 2833 . . . 4 ((𝑓 = 𝐺𝑗 = (TopOpen‘𝑓)) → ((invg𝑓) ∈ (𝑗 Cn 𝑗) ↔ 𝐼 ∈ (𝐽 Cn 𝐽)))
153, 14sbcied 3837 . . 3 (𝑓 = 𝐺 → ([(TopOpen‘𝑓) / 𝑗](invg𝑓) ∈ (𝑗 Cn 𝑗) ↔ 𝐼 ∈ (𝐽 Cn 𝐽)))
16 df-tgp 24097 . . 3 TopGrp = {𝑓 ∈ (Grp ∩ TopMnd) ∣ [(TopOpen‘𝑓) / 𝑗](invg𝑓) ∈ (𝑗 Cn 𝑗)}
1715, 16elrab2 3698 . 2 (𝐺 ∈ TopGrp ↔ (𝐺 ∈ (Grp ∩ TopMnd) ∧ 𝐼 ∈ (𝐽 Cn 𝐽)))
18 df-3an 1088 . 2 ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopMnd ∧ 𝐼 ∈ (𝐽 Cn 𝐽)) ↔ ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopMnd) ∧ 𝐼 ∈ (𝐽 Cn 𝐽)))
192, 17, 183bitr4i 303 1 (𝐺 ∈ TopGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ TopMnd ∧ 𝐼 ∈ (𝐽 Cn 𝐽)))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  Vcvv 3478  [wsbc 3791  cin 3962  cfv 6563  (class class class)co 7431  TopOpenctopn 17468  Grpcgrp 18964  invgcminusg 18965   Cn ccn 23248  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-ext 2706  ax-nul 5312
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571  df-ov 7434  df-tgp 24097
This theorem is referenced by:  tgpgrp  24102  tgptmd  24103  tgpinv  24109  istgp2  24115  oppgtgp  24122  subgtgp  24129  symgtgp  24130  prdstgpd  24149  tlmtgp  24220  nrgtdrg  24730
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