Theorem istdrg2 24168

Description: A topological-ring division ring is a topological division ring iff the group of nonzero elements is a topological group. (Contributed by Mario Carneiro, 5-Oct-2015.)

Hypotheses

Ref	Expression
istdrg2.m	⊢ 𝑀 = (mulGrp‘𝑅)
istdrg2.b	⊢ 𝐵 = (Base‘𝑅)
istdrg2.z	⊢ 0 = (0g‘𝑅)

Assertion

Ref	Expression
istdrg2	⊢ (𝑅 ∈ TopDRing ↔ (𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing ∧ (𝑀 ↾s (𝐵 ∖ { 0 })) ∈ TopGrp))

Proof of Theorem istdrg2

Step	Hyp	Ref	Expression
1		istdrg2.m	. . 3 ⊢ 𝑀 = (mulGrp‘𝑅)
2		eqid 2740	. . 3 ⊢ (Unit‘𝑅) = (Unit‘𝑅)
3	1, 2	istdrg 24156	. 2 ⊢ (𝑅 ∈ TopDRing ↔ (𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing ∧ (𝑀 ↾s (Unit‘𝑅)) ∈ TopGrp))
4		istdrg2.b	. . . . . . . . 9 ⊢ 𝐵 = (Base‘𝑅)
5		istdrg2.z	. . . . . . . . 9 ⊢ 0 = (0g‘𝑅)
6	4, 2, 5	isdrng 20712	. . . . . . . 8 ⊢ (𝑅 ∈ DivRing ↔ (𝑅 ∈ Ring ∧ (Unit‘𝑅) = (𝐵 ∖ { 0 })))
7	6	simprbi 498	. . . . . . 7 ⊢ (𝑅 ∈ DivRing → (Unit‘𝑅) = (𝐵 ∖ { 0 }))
8	7	adantl 482	. . . . . 6 ⊢ ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing) → (Unit‘𝑅) = (𝐵 ∖ { 0 }))
9	8	oveq2d 7379	. . . . 5 ⊢ ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing) → (𝑀 ↾s (Unit‘𝑅)) = (𝑀 ↾s (𝐵 ∖ { 0 })))
10	9	eleq1d 2825	. . . 4 ⊢ ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing) → ((𝑀 ↾s (Unit‘𝑅)) ∈ TopGrp ↔ (𝑀 ↾s (𝐵 ∖ { 0 })) ∈ TopGrp))
11	10	pm5.32i 579	. . 3 ⊢ (((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing) ∧ (𝑀 ↾s (Unit‘𝑅)) ∈ TopGrp) ↔ ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing) ∧ (𝑀 ↾s (𝐵 ∖ { 0 })) ∈ TopGrp))
12		df-3an 1094	. . 3 ⊢ ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing ∧ (𝑀 ↾s (Unit‘𝑅)) ∈ TopGrp) ↔ ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing) ∧ (𝑀 ↾s (Unit‘𝑅)) ∈ TopGrp))
13		df-3an 1094	. . 3 ⊢ ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing ∧ (𝑀 ↾s (𝐵 ∖ { 0 })) ∈ TopGrp) ↔ ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing) ∧ (𝑀 ↾s (𝐵 ∖ { 0 })) ∈ TopGrp))
14	11, 12, 13	3bitr4i 304	. 2 ⊢ ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing ∧ (𝑀 ↾s (Unit‘𝑅)) ∈ TopGrp) ↔ (𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing ∧ (𝑀 ↾s (𝐵 ∖ { 0 })) ∈ TopGrp))
15	3, 14	bitri 276	1 ⊢ (𝑅 ∈ TopDRing ↔ (𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing ∧ (𝑀 ↾s (𝐵 ∖ { 0 })) ∈ TopGrp))

Syntax hints:   ↔ wb 207   ∧ wa 396   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119   ∖ cdif 3887  {csn 4562  ‘cfv 6492  (class class class)co 7363  Basecbs 17177   ↾_s cress 17198  0_gc0g 17400  mulGrpcmgp 20119  Ringcrg 20212  Unitcui 20333  DivRingcdr 20708  TopGrpctgp 24061  TopRingctrg 24146  TopDRingctdrg 24147

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-iota 6448  df-fv 6500  df-ov 7366  df-drng 20710  df-tdrg 24151

This theorem is referenced by: (None)