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Mirrors > Home > MPE Home > Th. List > isdrng | Structured version Visualization version GIF version |
Description: The predicate "is a division ring". (Contributed by NM, 18-Oct-2012.) (Revised by Mario Carneiro, 2-Dec-2014.) |
Ref | Expression |
---|---|
isdrng.b | ⊢ 𝐵 = (Base‘𝑅) |
isdrng.u | ⊢ 𝑈 = (Unit‘𝑅) |
isdrng.z | ⊢ 0 = (0g‘𝑅) |
Ref | Expression |
---|---|
isdrng | ⊢ (𝑅 ∈ DivRing ↔ (𝑅 ∈ Ring ∧ 𝑈 = (𝐵 ∖ { 0 }))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6664 | . . . 4 ⊢ (𝑟 = 𝑅 → (Unit‘𝑟) = (Unit‘𝑅)) | |
2 | isdrng.u | . . . 4 ⊢ 𝑈 = (Unit‘𝑅) | |
3 | 1, 2 | syl6eqr 2874 | . . 3 ⊢ (𝑟 = 𝑅 → (Unit‘𝑟) = 𝑈) |
4 | fveq2 6664 | . . . . 5 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅)) | |
5 | isdrng.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑅) | |
6 | 4, 5 | syl6eqr 2874 | . . . 4 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵) |
7 | fveq2 6664 | . . . . . 6 ⊢ (𝑟 = 𝑅 → (0g‘𝑟) = (0g‘𝑅)) | |
8 | isdrng.z | . . . . . 6 ⊢ 0 = (0g‘𝑅) | |
9 | 7, 8 | syl6eqr 2874 | . . . . 5 ⊢ (𝑟 = 𝑅 → (0g‘𝑟) = 0 ) |
10 | 9 | sneqd 4572 | . . . 4 ⊢ (𝑟 = 𝑅 → {(0g‘𝑟)} = { 0 }) |
11 | 6, 10 | difeq12d 4099 | . . 3 ⊢ (𝑟 = 𝑅 → ((Base‘𝑟) ∖ {(0g‘𝑟)}) = (𝐵 ∖ { 0 })) |
12 | 3, 11 | eqeq12d 2837 | . 2 ⊢ (𝑟 = 𝑅 → ((Unit‘𝑟) = ((Base‘𝑟) ∖ {(0g‘𝑟)}) ↔ 𝑈 = (𝐵 ∖ { 0 }))) |
13 | df-drng 19498 | . 2 ⊢ DivRing = {𝑟 ∈ Ring ∣ (Unit‘𝑟) = ((Base‘𝑟) ∖ {(0g‘𝑟)})} | |
14 | 12, 13 | elrab2 3682 | 1 ⊢ (𝑅 ∈ DivRing ↔ (𝑅 ∈ Ring ∧ 𝑈 = (𝐵 ∖ { 0 }))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∖ cdif 3932 {csn 4560 ‘cfv 6349 Basecbs 16477 0gc0g 16707 Ringcrg 19291 Unitcui 19383 DivRingcdr 19496 |
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-rex 3144 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 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-iota 6308 df-fv 6357 df-drng 19498 |
This theorem is referenced by: drngunit 19501 drngui 19502 drngring 19503 isdrng2 19506 drngprop 19507 drngid 19510 opprdrng 19520 drngpropd 19523 issubdrg 19554 cntzsdrg 19575 drngdomn 20070 fidomndrng 20074 zringndrg 20631 istdrg2 22780 cvsunit 23729 cphreccllem 23776 sradrng 30983 zrhunitpreima 31214 |
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