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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 6887 | . . . 4 ⊢ (𝑟 = 𝑅 → (Unit‘𝑟) = (Unit‘𝑅)) | |
2 | isdrng.u | . . . 4 ⊢ 𝑈 = (Unit‘𝑅) | |
3 | 1, 2 | eqtr4di 2791 | . . 3 ⊢ (𝑟 = 𝑅 → (Unit‘𝑟) = 𝑈) |
4 | fveq2 6887 | . . . . 5 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅)) | |
5 | isdrng.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑅) | |
6 | 4, 5 | eqtr4di 2791 | . . . 4 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵) |
7 | fveq2 6887 | . . . . . 6 ⊢ (𝑟 = 𝑅 → (0g‘𝑟) = (0g‘𝑅)) | |
8 | isdrng.z | . . . . . 6 ⊢ 0 = (0g‘𝑅) | |
9 | 7, 8 | eqtr4di 2791 | . . . . 5 ⊢ (𝑟 = 𝑅 → (0g‘𝑟) = 0 ) |
10 | 9 | sneqd 4638 | . . . 4 ⊢ (𝑟 = 𝑅 → {(0g‘𝑟)} = { 0 }) |
11 | 6, 10 | difeq12d 4121 | . . 3 ⊢ (𝑟 = 𝑅 → ((Base‘𝑟) ∖ {(0g‘𝑟)}) = (𝐵 ∖ { 0 })) |
12 | 3, 11 | eqeq12d 2749 | . 2 ⊢ (𝑟 = 𝑅 → ((Unit‘𝑟) = ((Base‘𝑟) ∖ {(0g‘𝑟)}) ↔ 𝑈 = (𝐵 ∖ { 0 }))) |
13 | df-drng 20305 | . 2 ⊢ DivRing = {𝑟 ∈ Ring ∣ (Unit‘𝑟) = ((Base‘𝑟) ∖ {(0g‘𝑟)})} | |
14 | 12, 13 | elrab2 3684 | 1 ⊢ (𝑅 ∈ DivRing ↔ (𝑅 ∈ Ring ∧ 𝑈 = (𝐵 ∖ { 0 }))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∖ cdif 3943 {csn 4626 ‘cfv 6539 Basecbs 17139 0gc0g 17380 Ringcrg 20046 Unitcui 20157 DivRingcdr 20303 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-rab 3434 df-v 3477 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4321 df-if 4527 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4907 df-br 5147 df-iota 6491 df-fv 6547 df-drng 20305 |
This theorem is referenced by: drngunit 20308 drngui 20309 drngring 20310 isdrng2 20316 drngprop 20317 drngid 20320 opprdrng 20334 drngpropd 20339 issubdrg 20376 imadrhmcl 20400 cntzsdrg 20405 drngdomn 20905 fidomndrng 20910 zringndrg 21021 istdrg2 23663 cvsunit 24628 cphreccllem 24676 isdrng4 32363 sradrng 32618 zrhunitpreima 32895 |
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