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Theorem istgp 24085
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 3967 . . 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 6921 . . . 4 (𝑓 = 𝐺 → (TopOpen‘𝑓) ∈ V)
4 simpl 482 . . . . . . 7 ((𝑓 = 𝐺𝑗 = (TopOpen‘𝑓)) → 𝑓 = 𝐺)
54fveq2d 6910 . . . . . 6 ((𝑓 = 𝐺𝑗 = (TopOpen‘𝑓)) → (invg𝑓) = (invg𝐺))
6 istgp.2 . . . . . 6 𝐼 = (invg𝐺)
75, 6eqtr4di 2795 . . . . 5 ((𝑓 = 𝐺𝑗 = (TopOpen‘𝑓)) → (invg𝑓) = 𝐼)
8 id 22 . . . . . . 7 (𝑗 = (TopOpen‘𝑓) → 𝑗 = (TopOpen‘𝑓))
9 fveq2 6906 . . . . . . . 8 (𝑓 = 𝐺 → (TopOpen‘𝑓) = (TopOpen‘𝐺))
10 istgp.1 . . . . . . . 8 𝐽 = (TopOpen‘𝐺)
119, 10eqtr4di 2795 . . . . . . 7 (𝑓 = 𝐺 → (TopOpen‘𝑓) = 𝐽)
128, 11sylan9eqr 2799 . . . . . 6 ((𝑓 = 𝐺𝑗 = (TopOpen‘𝑓)) → 𝑗 = 𝐽)
1312, 12oveq12d 7449 . . . . 5 ((𝑓 = 𝐺𝑗 = (TopOpen‘𝑓)) → (𝑗 Cn 𝑗) = (𝐽 Cn 𝐽))
147, 13eleq12d 2835 . . . 4 ((𝑓 = 𝐺𝑗 = (TopOpen‘𝑓)) → ((invg𝑓) ∈ (𝑗 Cn 𝑗) ↔ 𝐼 ∈ (𝐽 Cn 𝐽)))
153, 14sbcied 3832 . . 3 (𝑓 = 𝐺 → ([(TopOpen‘𝑓) / 𝑗](invg𝑓) ∈ (𝑗 Cn 𝑗) ↔ 𝐼 ∈ (𝐽 Cn 𝐽)))
16 df-tgp 24081 . . 3 TopGrp = {𝑓 ∈ (Grp ∩ TopMnd) ∣ [(TopOpen‘𝑓) / 𝑗](invg𝑓) ∈ (𝑗 Cn 𝑗)}
1715, 16elrab2 3695 . 2 (𝐺 ∈ TopGrp ↔ (𝐺 ∈ (Grp ∩ TopMnd) ∧ 𝐼 ∈ (𝐽 Cn 𝐽)))
18 df-3an 1089 . 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 1087   = wceq 1540  wcel 2108  Vcvv 3480  [wsbc 3788  cin 3950  cfv 6561  (class class class)co 7431  TopOpenctopn 17466  Grpcgrp 18951  invgcminusg 18952   Cn ccn 23232  TopMndctmd 24078  TopGrpctgp 24079
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-nul 5306
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-iota 6514  df-fv 6569  df-ov 7434  df-tgp 24081
This theorem is referenced by:  tgpgrp  24086  tgptmd  24087  tgpinv  24093  istgp2  24099  oppgtgp  24106  subgtgp  24113  symgtgp  24114  prdstgpd  24133  tlmtgp  24204  nrgtdrg  24714
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