Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 4168 | . . 3 ⊢ (𝑅 ∈ (TopRing ∩ DivRing) ↔ (𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing)) | |
2 | 1 | anbi1i 625 | . 2 ⊢ ((𝑅 ∈ (TopRing ∩ DivRing) ∧ (𝑀 ↾s 𝑈) ∈ TopGrp) ↔ ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing) ∧ (𝑀 ↾s 𝑈) ∈ TopGrp)) |
3 | fveq2 6669 | . . . . . 6 ⊢ (𝑟 = 𝑅 → (mulGrp‘𝑟) = (mulGrp‘𝑅)) | |
4 | istrg.1 | . . . . . 6 ⊢ 𝑀 = (mulGrp‘𝑅) | |
5 | 3, 4 | syl6eqr 2874 | . . . . 5 ⊢ (𝑟 = 𝑅 → (mulGrp‘𝑟) = 𝑀) |
6 | fveq2 6669 | . . . . . 6 ⊢ (𝑟 = 𝑅 → (Unit‘𝑟) = (Unit‘𝑅)) | |
7 | istdrg.1 | . . . . . 6 ⊢ 𝑈 = (Unit‘𝑅) | |
8 | 6, 7 | syl6eqr 2874 | . . . . 5 ⊢ (𝑟 = 𝑅 → (Unit‘𝑟) = 𝑈) |
9 | 5, 8 | oveq12d 7173 | . . . 4 ⊢ (𝑟 = 𝑅 → ((mulGrp‘𝑟) ↾s (Unit‘𝑟)) = (𝑀 ↾s 𝑈)) |
10 | 9 | eleq1d 2897 | . . 3 ⊢ (𝑟 = 𝑅 → (((mulGrp‘𝑟) ↾s (Unit‘𝑟)) ∈ TopGrp ↔ (𝑀 ↾s 𝑈) ∈ TopGrp)) |
11 | df-tdrg 22768 | . . 3 ⊢ TopDRing = {𝑟 ∈ (TopRing ∩ DivRing) ∣ ((mulGrp‘𝑟) ↾s (Unit‘𝑟)) ∈ TopGrp} | |
12 | 10, 11 | elrab2 3682 | . 2 ⊢ (𝑅 ∈ TopDRing ↔ (𝑅 ∈ (TopRing ∩ DivRing) ∧ (𝑀 ↾s 𝑈) ∈ TopGrp)) |
13 | df-3an 1085 | . 2 ⊢ ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing ∧ (𝑀 ↾s 𝑈) ∈ TopGrp) ↔ ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing) ∧ (𝑀 ↾s 𝑈) ∈ TopGrp)) | |
14 | 2, 12, 13 | 3bitr4i 305 | 1 ⊢ (𝑅 ∈ TopDRing ↔ (𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing ∧ (𝑀 ↾s 𝑈) ∈ TopGrp)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ∩ cin 3934 ‘cfv 6354 (class class class)co 7155 ↾s cress 16483 mulGrpcmgp 19238 Unitcui 19388 DivRingcdr 19501 TopGrpctgp 22678 TopRingctrg 22763 TopDRingctdrg 22764 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-br 5066 df-iota 6313 df-fv 6362 df-ov 7158 df-tdrg 22768 |
This theorem is referenced by: tdrgunit 22774 tdrgtrg 22780 tdrgdrng 22781 istdrg2 22785 nrgtdrg 23301 |
Copyright terms: Public domain | W3C validator |