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Mirrors > Home > MPE Home > Th. List > istdrg | Structured version Visualization version GIF version |
Description: Express the predicate "𝑅 is a topological ring". (Contributed by Mario Carneiro, 5-Oct-2015.) |
Ref | Expression |
---|---|
istrg.1 | ⊢ 𝑀 = (mulGrp‘𝑅) |
istdrg.1 | ⊢ 𝑈 = (Unit‘𝑅) |
Ref | Expression |
---|---|
istdrg | ⊢ (𝑅 ∈ TopDRing ↔ (𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing ∧ (𝑀 ↾s 𝑈) ∈ TopGrp)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elin 3979 | . . 3 ⊢ (𝑅 ∈ (TopRing ∩ DivRing) ↔ (𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing)) | |
2 | 1 | anbi1i 624 | . 2 ⊢ ((𝑅 ∈ (TopRing ∩ DivRing) ∧ (𝑀 ↾s 𝑈) ∈ TopGrp) ↔ ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing) ∧ (𝑀 ↾s 𝑈) ∈ TopGrp)) |
3 | fveq2 6907 | . . . . . 6 ⊢ (𝑟 = 𝑅 → (mulGrp‘𝑟) = (mulGrp‘𝑅)) | |
4 | istrg.1 | . . . . . 6 ⊢ 𝑀 = (mulGrp‘𝑅) | |
5 | 3, 4 | eqtr4di 2793 | . . . . 5 ⊢ (𝑟 = 𝑅 → (mulGrp‘𝑟) = 𝑀) |
6 | fveq2 6907 | . . . . . 6 ⊢ (𝑟 = 𝑅 → (Unit‘𝑟) = (Unit‘𝑅)) | |
7 | istdrg.1 | . . . . . 6 ⊢ 𝑈 = (Unit‘𝑅) | |
8 | 6, 7 | eqtr4di 2793 | . . . . 5 ⊢ (𝑟 = 𝑅 → (Unit‘𝑟) = 𝑈) |
9 | 5, 8 | oveq12d 7449 | . . . 4 ⊢ (𝑟 = 𝑅 → ((mulGrp‘𝑟) ↾s (Unit‘𝑟)) = (𝑀 ↾s 𝑈)) |
10 | 9 | eleq1d 2824 | . . 3 ⊢ (𝑟 = 𝑅 → (((mulGrp‘𝑟) ↾s (Unit‘𝑟)) ∈ TopGrp ↔ (𝑀 ↾s 𝑈) ∈ TopGrp)) |
11 | df-tdrg 24185 | . . 3 ⊢ TopDRing = {𝑟 ∈ (TopRing ∩ DivRing) ∣ ((mulGrp‘𝑟) ↾s (Unit‘𝑟)) ∈ TopGrp} | |
12 | 10, 11 | elrab2 3698 | . 2 ⊢ (𝑅 ∈ TopDRing ↔ (𝑅 ∈ (TopRing ∩ DivRing) ∧ (𝑀 ↾s 𝑈) ∈ TopGrp)) |
13 | df-3an 1088 | . 2 ⊢ ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing ∧ (𝑀 ↾s 𝑈) ∈ TopGrp) ↔ ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing) ∧ (𝑀 ↾s 𝑈) ∈ TopGrp)) | |
14 | 2, 12, 13 | 3bitr4i 303 | 1 ⊢ (𝑅 ∈ TopDRing ↔ (𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing ∧ (𝑀 ↾s 𝑈) ∈ TopGrp)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ∩ cin 3962 ‘cfv 6563 (class class class)co 7431 ↾s cress 17274 mulGrpcmgp 20152 Unitcui 20372 DivRingcdr 20746 TopGrpctgp 24095 TopRingctrg 24180 TopDRingctdrg 24181 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-iota 6516 df-fv 6571 df-ov 7434 df-tdrg 24185 |
This theorem is referenced by: tdrgunit 24191 tdrgtrg 24197 tdrgdrng 24198 istdrg2 24202 nrgtdrg 24730 |
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