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| 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 3967 | . . 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 6906 | . . . . . 6 ⊢ (𝑟 = 𝑅 → (mulGrp‘𝑟) = (mulGrp‘𝑅)) | |
| 4 | istrg.1 | . . . . . 6 ⊢ 𝑀 = (mulGrp‘𝑅) | |
| 5 | 3, 4 | eqtr4di 2795 | . . . . 5 ⊢ (𝑟 = 𝑅 → (mulGrp‘𝑟) = 𝑀) | 
| 6 | fveq2 6906 | . . . . . 6 ⊢ (𝑟 = 𝑅 → (Unit‘𝑟) = (Unit‘𝑅)) | |
| 7 | istdrg.1 | . . . . . 6 ⊢ 𝑈 = (Unit‘𝑅) | |
| 8 | 6, 7 | eqtr4di 2795 | . . . . 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 2826 | . . 3 ⊢ (𝑟 = 𝑅 → (((mulGrp‘𝑟) ↾s (Unit‘𝑟)) ∈ TopGrp ↔ (𝑀 ↾s 𝑈) ∈ TopGrp)) | 
| 11 | df-tdrg 24169 | . . 3 ⊢ TopDRing = {𝑟 ∈ (TopRing ∩ DivRing) ∣ ((mulGrp‘𝑟) ↾s (Unit‘𝑟)) ∈ TopGrp} | |
| 12 | 10, 11 | elrab2 3695 | . 2 ⊢ (𝑅 ∈ TopDRing ↔ (𝑅 ∈ (TopRing ∩ DivRing) ∧ (𝑀 ↾s 𝑈) ∈ TopGrp)) | 
| 13 | df-3an 1089 | . 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 1087 = wceq 1540 ∈ wcel 2108 ∩ cin 3950 ‘cfv 6561 (class class class)co 7431 ↾s cress 17274 mulGrpcmgp 20137 Unitcui 20355 DivRingcdr 20729 TopGrpctgp 24079 TopRingctrg 24164 TopDRingctdrg 24165 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-iota 6514 df-fv 6569 df-ov 7434 df-tdrg 24169 | 
| This theorem is referenced by: tdrgunit 24175 tdrgtrg 24181 tdrgdrng 24182 istdrg2 24186 nrgtdrg 24714 | 
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