Theorem istgp 23992

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 = (invg‘𝐺)

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.

Step	Hyp	Ref	Expression
1		elin 3913	. . 3 ⊢ (𝐺 ∈ (Grp ∩ TopMnd) ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ TopMnd))
2	1	anbi1i 624	. 2 ⊢ ((𝐺 ∈ (Grp ∩ TopMnd) ∧ 𝐼 ∈ (𝐽 Cn 𝐽)) ↔ ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopMnd) ∧ 𝐼 ∈ (𝐽 Cn 𝐽)))
3		fvexd 6837	. . . 4 ⊢ (𝑓 = 𝐺 → (TopOpen‘𝑓) ∈ V)
4		simpl 482	. . . . . . 7 ⊢ ((𝑓 = 𝐺 ∧ 𝑗 = (TopOpen‘𝑓)) → 𝑓 = 𝐺)
5	4	fveq2d 6826	. . . . . 6 ⊢ ((𝑓 = 𝐺 ∧ 𝑗 = (TopOpen‘𝑓)) → (invg‘𝑓) = (invg‘𝐺))
6		istgp.2	. . . . . 6 ⊢ 𝐼 = (invg‘𝐺)
7	5, 6	eqtr4di 2784	. . . . 5 ⊢ ((𝑓 = 𝐺 ∧ 𝑗 = (TopOpen‘𝑓)) → (invg‘𝑓) = 𝐼)
8		id 22	. . . . . . 7 ⊢ (𝑗 = (TopOpen‘𝑓) → 𝑗 = (TopOpen‘𝑓))
9		fveq2 6822	. . . . . . . 8 ⊢ (𝑓 = 𝐺 → (TopOpen‘𝑓) = (TopOpen‘𝐺))
10		istgp.1	. . . . . . . 8 ⊢ 𝐽 = (TopOpen‘𝐺)
11	9, 10	eqtr4di 2784	. . . . . . 7 ⊢ (𝑓 = 𝐺 → (TopOpen‘𝑓) = 𝐽)
12	8, 11	sylan9eqr 2788	. . . . . 6 ⊢ ((𝑓 = 𝐺 ∧ 𝑗 = (TopOpen‘𝑓)) → 𝑗 = 𝐽)
13	12, 12	oveq12d 7364	. . . . 5 ⊢ ((𝑓 = 𝐺 ∧ 𝑗 = (TopOpen‘𝑓)) → (𝑗 Cn 𝑗) = (𝐽 Cn 𝐽))
14	7, 13	eleq12d 2825	. . . 4 ⊢ ((𝑓 = 𝐺 ∧ 𝑗 = (TopOpen‘𝑓)) → ((invg‘𝑓) ∈ (𝑗 Cn 𝑗) ↔ 𝐼 ∈ (𝐽 Cn 𝐽)))
15	3, 14	sbcied 3780	. . 3 ⊢ (𝑓 = 𝐺 → ([(TopOpen‘𝑓) / 𝑗](invg‘𝑓) ∈ (𝑗 Cn 𝑗) ↔ 𝐼 ∈ (𝐽 Cn 𝐽)))
16		df-tgp 23988	. . 3 ⊢ TopGrp = {𝑓 ∈ (Grp ∩ TopMnd) ∣ [(TopOpen‘𝑓) / 𝑗](invg‘𝑓) ∈ (𝑗 Cn 𝑗)}
17	15, 16	elrab2 3645	. 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)))
19	2, 17, 18	3bitr4i 303	1 ⊢ (𝐺 ∈ TopGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ TopMnd ∧ 𝐼 ∈ (𝐽 Cn 𝐽)))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111  Vcvv 3436  [wsbc 3736   ∩ cin 3896  ‘cfv 6481  (class class class)co 7346  TopOpenctopn 17325  Grpcgrp 18846  inv_gcminusg 18847   Cn ccn 23139  TopMndctmd 23985  TopGrpctgp 23986

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-nul 5242

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-rab 3396  df-v 3438  df-sbc 3737  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-iota 6437  df-fv 6489  df-ov 7349  df-tgp 23988

This theorem is referenced by:  tgpgrp  23993  tgptmd  23994  tgpinv  24000  istgp2  24006  oppgtgp  24013  subgtgp  24020  symgtgp  24021  prdstgpd  24040  tlmtgp  24111  nrgtdrg  24608