Theorem istgp 24042

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 3905	. . 3 ⊢ (𝐺 ∈ (Grp ∩ TopMnd) ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ TopMnd))
2	1	anbi1i 625	. 2 ⊢ ((𝐺 ∈ (Grp ∩ TopMnd) ∧ 𝐼 ∈ (𝐽 Cn 𝐽)) ↔ ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopMnd) ∧ 𝐼 ∈ (𝐽 Cn 𝐽)))
3		fvexd 6855	. . . 4 ⊢ (𝑓 = 𝐺 → (TopOpen‘𝑓) ∈ V)
4		simpl 482	. . . . . . 7 ⊢ ((𝑓 = 𝐺 ∧ 𝑗 = (TopOpen‘𝑓)) → 𝑓 = 𝐺)
5	4	fveq2d 6844	. . . . . 6 ⊢ ((𝑓 = 𝐺 ∧ 𝑗 = (TopOpen‘𝑓)) → (invg‘𝑓) = (invg‘𝐺))
6		istgp.2	. . . . . 6 ⊢ 𝐼 = (invg‘𝐺)
7	5, 6	eqtr4di 2789	. . . . 5 ⊢ ((𝑓 = 𝐺 ∧ 𝑗 = (TopOpen‘𝑓)) → (invg‘𝑓) = 𝐼)
8		id 22	. . . . . . 7 ⊢ (𝑗 = (TopOpen‘𝑓) → 𝑗 = (TopOpen‘𝑓))
9		fveq2 6840	. . . . . . . 8 ⊢ (𝑓 = 𝐺 → (TopOpen‘𝑓) = (TopOpen‘𝐺))
10		istgp.1	. . . . . . . 8 ⊢ 𝐽 = (TopOpen‘𝐺)
11	9, 10	eqtr4di 2789	. . . . . . 7 ⊢ (𝑓 = 𝐺 → (TopOpen‘𝑓) = 𝐽)
12	8, 11	sylan9eqr 2793	. . . . . 6 ⊢ ((𝑓 = 𝐺 ∧ 𝑗 = (TopOpen‘𝑓)) → 𝑗 = 𝐽)
13	12, 12	oveq12d 7385	. . . . 5 ⊢ ((𝑓 = 𝐺 ∧ 𝑗 = (TopOpen‘𝑓)) → (𝑗 Cn 𝑗) = (𝐽 Cn 𝐽))
14	7, 13	eleq12d 2830	. . . 4 ⊢ ((𝑓 = 𝐺 ∧ 𝑗 = (TopOpen‘𝑓)) → ((invg‘𝑓) ∈ (𝑗 Cn 𝑗) ↔ 𝐼 ∈ (𝐽 Cn 𝐽)))
15	3, 14	sbcied 3772	. . 3 ⊢ (𝑓 = 𝐺 → ([(TopOpen‘𝑓) / 𝑗](invg‘𝑓) ∈ (𝑗 Cn 𝑗) ↔ 𝐼 ∈ (𝐽 Cn 𝐽)))
16		df-tgp 24038	. . 3 ⊢ TopGrp = {𝑓 ∈ (Grp ∩ TopMnd) ∣ [(TopOpen‘𝑓) / 𝑗](invg‘𝑓) ∈ (𝑗 Cn 𝑗)}
17	15, 16	elrab2 3637	. 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)))
19	2, 17, 18	3bitr4i 303	1 ⊢ (𝐺 ∈ TopGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ TopMnd ∧ 𝐼 ∈ (𝐽 Cn 𝐽)))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  Vcvv 3429  [wsbc 3728   ∩ cin 3888  ‘cfv 6498  (class class class)co 7367  TopOpenctopn 17384  Grpcgrp 18909  inv_gcminusg 18910   Cn ccn 23189  TopMndctmd 24035  TopGrpctgp 24036

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708  ax-nul 5241

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-rab 3390  df-v 3431  df-sbc 3729  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-iota 6454  df-fv 6506  df-ov 7370  df-tgp 24038

This theorem is referenced by:  tgpgrp  24043  tgptmd  24044  tgpinv  24050  istgp2  24056  oppgtgp  24063  subgtgp  24070  symgtgp  24071  prdstgpd  24090  tlmtgp  24161  nrgtdrg  24658