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Theorem istgp 21639
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 3757 . . 3 (𝐺 ∈ (Grp ∩ TopMnd) ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ TopMnd))
21anbi1i 726 . 2 ((𝐺 ∈ (Grp ∩ TopMnd) ∧ 𝐼 ∈ (𝐽 Cn 𝐽)) ↔ ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopMnd) ∧ 𝐼 ∈ (𝐽 Cn 𝐽)))
3 fvex 6098 . . . . 5 (TopOpen‘𝑓) ∈ V
43a1i 11 . . . 4 (𝑓 = 𝐺 → (TopOpen‘𝑓) ∈ V)
5 simpl 471 . . . . . . 7 ((𝑓 = 𝐺𝑗 = (TopOpen‘𝑓)) → 𝑓 = 𝐺)
65fveq2d 6092 . . . . . 6 ((𝑓 = 𝐺𝑗 = (TopOpen‘𝑓)) → (invg𝑓) = (invg𝐺))
7 istgp.2 . . . . . 6 𝐼 = (invg𝐺)
86, 7syl6eqr 2661 . . . . 5 ((𝑓 = 𝐺𝑗 = (TopOpen‘𝑓)) → (invg𝑓) = 𝐼)
9 id 22 . . . . . . 7 (𝑗 = (TopOpen‘𝑓) → 𝑗 = (TopOpen‘𝑓))
10 fveq2 6088 . . . . . . . 8 (𝑓 = 𝐺 → (TopOpen‘𝑓) = (TopOpen‘𝐺))
11 istgp.1 . . . . . . . 8 𝐽 = (TopOpen‘𝐺)
1210, 11syl6eqr 2661 . . . . . . 7 (𝑓 = 𝐺 → (TopOpen‘𝑓) = 𝐽)
139, 12sylan9eqr 2665 . . . . . 6 ((𝑓 = 𝐺𝑗 = (TopOpen‘𝑓)) → 𝑗 = 𝐽)
1413, 13oveq12d 6545 . . . . 5 ((𝑓 = 𝐺𝑗 = (TopOpen‘𝑓)) → (𝑗 Cn 𝑗) = (𝐽 Cn 𝐽))
158, 14eleq12d 2681 . . . 4 ((𝑓 = 𝐺𝑗 = (TopOpen‘𝑓)) → ((invg𝑓) ∈ (𝑗 Cn 𝑗) ↔ 𝐼 ∈ (𝐽 Cn 𝐽)))
164, 15sbcied 3438 . . 3 (𝑓 = 𝐺 → ([(TopOpen‘𝑓) / 𝑗](invg𝑓) ∈ (𝑗 Cn 𝑗) ↔ 𝐼 ∈ (𝐽 Cn 𝐽)))
17 df-tgp 21635 . . 3 TopGrp = {𝑓 ∈ (Grp ∩ TopMnd) ∣ [(TopOpen‘𝑓) / 𝑗](invg𝑓) ∈ (𝑗 Cn 𝑗)}
1816, 17elrab2 3332 . 2 (𝐺 ∈ TopGrp ↔ (𝐺 ∈ (Grp ∩ TopMnd) ∧ 𝐼 ∈ (𝐽 Cn 𝐽)))
19 df-3an 1032 . 2 ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopMnd ∧ 𝐼 ∈ (𝐽 Cn 𝐽)) ↔ ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopMnd) ∧ 𝐼 ∈ (𝐽 Cn 𝐽)))
202, 18, 193bitr4i 290 1 (𝐺 ∈ TopGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ TopMnd ∧ 𝐼 ∈ (𝐽 Cn 𝐽)))
Colors of variables: wff setvar class
Syntax hints:  wb 194  wa 382  w3a 1030   = wceq 1474  wcel 1976  Vcvv 3172  [wsbc 3401  cin 3538  cfv 5790  (class class class)co 6527  TopOpenctopn 15854  Grpcgrp 17194  invgcminusg 17195   Cn ccn 20786  TopMndctmd 21632  TopGrpctgp 21633
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-nul 4712
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-iota 5754  df-fv 5798  df-ov 6530  df-tgp 21635
This theorem is referenced by:  tgpgrp  21640  tgptmd  21641  tgpinv  21647  istgp2  21653  oppgtgp  21660  symgtgp  21663  subgtgp  21667  prdstgpd  21686  tlmtgp  21757  nrgtdrg  22255
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