MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  istgp Structured version   Visualization version   GIF version

Theorem istgp 22369
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 4090 . . 3 (𝐺 ∈ (Grp ∩ TopMnd) ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ TopMnd))
21anbi1i 623 . 2 ((𝐺 ∈ (Grp ∩ TopMnd) ∧ 𝐼 ∈ (𝐽 Cn 𝐽)) ↔ ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopMnd) ∧ 𝐼 ∈ (𝐽 Cn 𝐽)))
3 fvexd 6553 . . . 4 (𝑓 = 𝐺 → (TopOpen‘𝑓) ∈ V)
4 simpl 483 . . . . . . 7 ((𝑓 = 𝐺𝑗 = (TopOpen‘𝑓)) → 𝑓 = 𝐺)
54fveq2d 6542 . . . . . 6 ((𝑓 = 𝐺𝑗 = (TopOpen‘𝑓)) → (invg𝑓) = (invg𝐺))
6 istgp.2 . . . . . 6 𝐼 = (invg𝐺)
75, 6syl6eqr 2849 . . . . 5 ((𝑓 = 𝐺𝑗 = (TopOpen‘𝑓)) → (invg𝑓) = 𝐼)
8 id 22 . . . . . . 7 (𝑗 = (TopOpen‘𝑓) → 𝑗 = (TopOpen‘𝑓))
9 fveq2 6538 . . . . . . . 8 (𝑓 = 𝐺 → (TopOpen‘𝑓) = (TopOpen‘𝐺))
10 istgp.1 . . . . . . . 8 𝐽 = (TopOpen‘𝐺)
119, 10syl6eqr 2849 . . . . . . 7 (𝑓 = 𝐺 → (TopOpen‘𝑓) = 𝐽)
128, 11sylan9eqr 2853 . . . . . 6 ((𝑓 = 𝐺𝑗 = (TopOpen‘𝑓)) → 𝑗 = 𝐽)
1312, 12oveq12d 7034 . . . . 5 ((𝑓 = 𝐺𝑗 = (TopOpen‘𝑓)) → (𝑗 Cn 𝑗) = (𝐽 Cn 𝐽))
147, 13eleq12d 2877 . . . 4 ((𝑓 = 𝐺𝑗 = (TopOpen‘𝑓)) → ((invg𝑓) ∈ (𝑗 Cn 𝑗) ↔ 𝐼 ∈ (𝐽 Cn 𝐽)))
153, 14sbcied 3743 . . 3 (𝑓 = 𝐺 → ([(TopOpen‘𝑓) / 𝑗](invg𝑓) ∈ (𝑗 Cn 𝑗) ↔ 𝐼 ∈ (𝐽 Cn 𝐽)))
16 df-tgp 22365 . . 3 TopGrp = {𝑓 ∈ (Grp ∩ TopMnd) ∣ [(TopOpen‘𝑓) / 𝑗](invg𝑓) ∈ (𝑗 Cn 𝑗)}
1715, 16elrab2 3621 . 2 (𝐺 ∈ TopGrp ↔ (𝐺 ∈ (Grp ∩ TopMnd) ∧ 𝐼 ∈ (𝐽 Cn 𝐽)))
18 df-3an 1082 . 2 ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopMnd ∧ 𝐼 ∈ (𝐽 Cn 𝐽)) ↔ ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopMnd) ∧ 𝐼 ∈ (𝐽 Cn 𝐽)))
192, 17, 183bitr4i 304 1 (𝐺 ∈ TopGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ TopMnd ∧ 𝐼 ∈ (𝐽 Cn 𝐽)))
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396  w3a 1080   = wceq 1522  wcel 2081  Vcvv 3437  [wsbc 3706  cin 3858  cfv 6225  (class class class)co 7016  TopOpenctopn 16524  Grpcgrp 17861  invgcminusg 17862   Cn ccn 21516  TopMndctmd 22362  TopGrpctgp 22363
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-ext 2769  ax-nul 5101
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ral 3110  df-rex 3111  df-rab 3114  df-v 3439  df-sbc 3707  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-sn 4473  df-pr 4475  df-op 4479  df-uni 4746  df-br 4963  df-iota 6189  df-fv 6233  df-ov 7019  df-tgp 22365
This theorem is referenced by:  tgpgrp  22370  tgptmd  22371  tgpinv  22377  istgp2  22383  oppgtgp  22390  symgtgp  22393  subgtgp  22397  prdstgpd  22416  tlmtgp  22487  nrgtdrg  22985
  Copyright terms: Public domain W3C validator