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 = (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 3758	. . 3 ⊢ (𝐺 ∈ (Grp ∩ TopMnd) ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ TopMnd))
2	1	anbi1i 727	. 2 ⊢ ((𝐺 ∈ (Grp ∩ TopMnd) ∧ 𝐼 ∈ (𝐽 Cn 𝐽)) ↔ ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopMnd) ∧ 𝐼 ∈ (𝐽 Cn 𝐽)))
3		fvex 6113	. . . . 5 ⊢ (TopOpen‘𝑓) ∈ V
4	3	a1i 11	. . . 4 ⊢ (𝑓 = 𝐺 → (TopOpen‘𝑓) ∈ V)
5		simpl 472	. . . . . . 7 ⊢ ((𝑓 = 𝐺 ∧ 𝑗 = (TopOpen‘𝑓)) → 𝑓 = 𝐺)
6	5	fveq2d 6107	. . . . . 6 ⊢ ((𝑓 = 𝐺 ∧ 𝑗 = (TopOpen‘𝑓)) → (invg‘𝑓) = (invg‘𝐺))
7		istgp.2	. . . . . 6 ⊢ 𝐼 = (invg‘𝐺)
8	6, 7	syl6eqr 2662	. . . . 5 ⊢ ((𝑓 = 𝐺 ∧ 𝑗 = (TopOpen‘𝑓)) → (invg‘𝑓) = 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)
12	10, 11	syl6eqr 2662	. . . . . . 7 ⊢ (𝑓 = 𝐺 → (TopOpen‘𝑓) = 𝐽)
13	9, 12	sylan9eqr 2666	. . . . . 6 ⊢ ((𝑓 = 𝐺 ∧ 𝑗 = (TopOpen‘𝑓)) → 𝑗 = 𝐽)
14	13, 13	oveq12d 6567	. . . . 5 ⊢ ((𝑓 = 𝐺 ∧ 𝑗 = (TopOpen‘𝑓)) → (𝑗 Cn 𝑗) = (𝐽 Cn 𝐽))
15	8, 14	eleq12d 2682	. . . 4 ⊢ ((𝑓 = 𝐺 ∧ 𝑗 = (TopOpen‘𝑓)) → ((invg‘𝑓) ∈ (𝑗 Cn 𝑗) ↔ 𝐼 ∈ (𝐽 Cn 𝐽)))
16	4, 15	sbcied 3439	. . 3 ⊢ (𝑓 = 𝐺 → ([(TopOpen‘𝑓) / 𝑗](invg‘𝑓) ∈ (𝑗 Cn 𝑗) ↔ 𝐼 ∈ (𝐽 Cn 𝐽)))
17		df-tgp 21687	. . 3 ⊢ TopGrp = {𝑓 ∈ (Grp ∩ TopMnd) ∣ [(TopOpen‘𝑓) / 𝑗](invg‘𝑓) ∈ (𝑗 Cn 𝑗)}
18	16, 17	elrab2 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)))
20	2, 18, 19	3bitr4i 291	1 ⊢ (𝐺 ∈ TopGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ TopMnd ∧ 𝐼 ∈ (𝐽 Cn 𝐽)))

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  inv_gcminusg 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