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Mirrors > Home > MPE Home > Th. List > istdrg2 | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
istdrg2.m | ⊢ 𝑀 = (mulGrp‘𝑅) |
istdrg2.b | ⊢ 𝐵 = (Base‘𝑅) |
istdrg2.z | ⊢ 0 = (0g‘𝑅) |
Ref | Expression |
---|---|
istdrg2 | ⊢ (𝑅 ∈ TopDRing ↔ (𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing ∧ (𝑀 ↾s (𝐵 ∖ { 0 })) ∈ TopGrp)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | istdrg2.m | . . 3 ⊢ 𝑀 = (mulGrp‘𝑅) | |
2 | eqid 2610 | . . 3 ⊢ (Unit‘𝑅) = (Unit‘𝑅) | |
3 | 1, 2 | istdrg 21779 | . 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 18574 | . . . . . . . 8 ⊢ (𝑅 ∈ DivRing ↔ (𝑅 ∈ Ring ∧ (Unit‘𝑅) = (𝐵 ∖ { 0 }))) |
7 | 6 | simprbi 479 | . . . . . . 7 ⊢ (𝑅 ∈ DivRing → (Unit‘𝑅) = (𝐵 ∖ { 0 })) |
8 | 7 | adantl 481 | . . . . . 6 ⊢ ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing) → (Unit‘𝑅) = (𝐵 ∖ { 0 })) |
9 | 8 | oveq2d 6565 | . . . . 5 ⊢ ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing) → (𝑀 ↾s (Unit‘𝑅)) = (𝑀 ↾s (𝐵 ∖ { 0 }))) |
10 | 9 | eleq1d 2672 | . . . 4 ⊢ ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing) → ((𝑀 ↾s (Unit‘𝑅)) ∈ TopGrp ↔ (𝑀 ↾s (𝐵 ∖ { 0 })) ∈ TopGrp)) |
11 | 10 | pm5.32i 667 | . . 3 ⊢ (((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing) ∧ (𝑀 ↾s (Unit‘𝑅)) ∈ TopGrp) ↔ ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing) ∧ (𝑀 ↾s (𝐵 ∖ { 0 })) ∈ TopGrp)) |
12 | df-3an 1033 | . . 3 ⊢ ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing ∧ (𝑀 ↾s (Unit‘𝑅)) ∈ TopGrp) ↔ ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing) ∧ (𝑀 ↾s (Unit‘𝑅)) ∈ TopGrp)) | |
13 | df-3an 1033 | . . 3 ⊢ ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing ∧ (𝑀 ↾s (𝐵 ∖ { 0 })) ∈ TopGrp) ↔ ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing) ∧ (𝑀 ↾s (𝐵 ∖ { 0 })) ∈ TopGrp)) | |
14 | 11, 12, 13 | 3bitr4i 291 | . 2 ⊢ ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing ∧ (𝑀 ↾s (Unit‘𝑅)) ∈ TopGrp) ↔ (𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing ∧ (𝑀 ↾s (𝐵 ∖ { 0 })) ∈ TopGrp)) |
15 | 3, 14 | bitri 263 | 1 ⊢ (𝑅 ∈ TopDRing ↔ (𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing ∧ (𝑀 ↾s (𝐵 ∖ { 0 })) ∈ TopGrp)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ∖ cdif 3537 {csn 4125 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 ↾s cress 15696 0gc0g 15923 mulGrpcmgp 18312 Ringcrg 18370 Unitcui 18462 DivRingcdr 18570 TopGrpctgp 21685 TopRingctrg 21769 TopDRingctdrg 21770 |
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 |
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-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 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-drng 18572 df-tdrg 21774 |
This theorem is referenced by: (None) |
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