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Theorem istgp 21691
 Description: The predicate "is a topological group". Definition 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 3758 . . 3 (𝐺 ∈ (Grp ∩ TopMnd) ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ TopMnd))
21anbi1i 727 . 2 ((𝐺 ∈ (Grp ∩ TopMnd) ∧ 𝐼 ∈ (𝐽 Cn 𝐽)) ↔ ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopMnd) ∧ 𝐼 ∈ (𝐽 Cn 𝐽)))
3 fvex 6113 . . . . 5 (TopOpen‘𝑓) ∈ V
43a1i 11 . . . 4 (𝑓 = 𝐺 → (TopOpen‘𝑓) ∈ V)
5 simpl 472 . . . . . . 7 ((𝑓 = 𝐺𝑗 = (TopOpen‘𝑓)) → 𝑓 = 𝐺)
65fveq2d 6107 . . . . . 6 ((𝑓 = 𝐺𝑗 = (TopOpen‘𝑓)) → (invg𝑓) = (invg𝐺))
7 istgp.2 . . . . . 6 𝐼 = (invg𝐺)
86, 7syl6eqr 2662 . . . . 5 ((𝑓 = 𝐺𝑗 = (TopOpen‘𝑓)) → (invg𝑓) = 𝐼)
9 id 22 . . . . . . 7 (𝑗 = (TopOpen‘𝑓) → 𝑗 = (TopOpen‘𝑓))
10 fveq2 6103 . . . . . . . 8 (𝑓 = 𝐺 → (TopOpen‘𝑓) = (TopOpen‘𝐺))
11 istgp.1 . . . . . . . 8 𝐽 = (TopOpen‘𝐺)
1210, 11syl6eqr 2662 . . . . . . 7 (𝑓 = 𝐺 → (TopOpen‘𝑓) = 𝐽)
139, 12sylan9eqr 2666 . . . . . 6 ((𝑓 = 𝐺𝑗 = (TopOpen‘𝑓)) → 𝑗 = 𝐽)
1413, 13oveq12d 6567 . . . . 5 ((𝑓 = 𝐺𝑗 = (TopOpen‘𝑓)) → (𝑗 Cn 𝑗) = (𝐽 Cn 𝐽))
158, 14eleq12d 2682 . . . 4 ((𝑓 = 𝐺𝑗 = (TopOpen‘𝑓)) → ((invg𝑓) ∈ (𝑗 Cn 𝑗) ↔ 𝐼 ∈ (𝐽 Cn 𝐽)))
164, 15sbcied 3439 . . 3 (𝑓 = 𝐺 → ([(TopOpen‘𝑓) / 𝑗](invg𝑓) ∈ (𝑗 Cn 𝑗) ↔ 𝐼 ∈ (𝐽 Cn 𝐽)))
17 df-tgp 21687 . . 3 TopGrp = {𝑓 ∈ (Grp ∩ TopMnd) ∣ [(TopOpen‘𝑓) / 𝑗](invg𝑓) ∈ (𝑗 Cn 𝑗)}
1816, 17elrab2 3333 . 2 (𝐺 ∈ TopGrp ↔ (𝐺 ∈ (Grp ∩ TopMnd) ∧ 𝐼 ∈ (𝐽 Cn 𝐽)))
19 df-3an 1033 . 2 ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopMnd ∧ 𝐼 ∈ (𝐽 Cn 𝐽)) ↔ ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopMnd) ∧ 𝐼 ∈ (𝐽 Cn 𝐽)))
202, 18, 193bitr4i 291 1 (𝐺 ∈ TopGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ TopMnd ∧ 𝐼 ∈ (𝐽 Cn 𝐽)))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  Vcvv 3173  [wsbc 3402   ∩ cin 3539  ‘cfv 5804  (class class class)co 6549  TopOpenctopn 15905  Grpcgrp 17245  invgcminusg 17246   Cn ccn 20838  TopMndctmd 21684  TopGrpctgp 21685 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-nul 4717 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-ov 6552  df-tgp 21687 This theorem is referenced by:  tgpgrp  21692  tgptmd  21693  tgpinv  21699  istgp2  21705  oppgtgp  21712  symgtgp  21715  subgtgp  21719  prdstgpd  21738  tlmtgp  21809  nrgtdrg  22307
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