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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 6907 | . . . 4 ⊢ (𝑟 = 𝑅 → (Unit‘𝑟) = (Unit‘𝑅)) | |
2 | isdrng.u | . . . 4 ⊢ 𝑈 = (Unit‘𝑅) | |
3 | 1, 2 | eqtr4di 2793 | . . 3 ⊢ (𝑟 = 𝑅 → (Unit‘𝑟) = 𝑈) |
4 | fveq2 6907 | . . . . 5 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅)) | |
5 | isdrng.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑅) | |
6 | 4, 5 | eqtr4di 2793 | . . . 4 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵) |
7 | fveq2 6907 | . . . . . 6 ⊢ (𝑟 = 𝑅 → (0g‘𝑟) = (0g‘𝑅)) | |
8 | isdrng.z | . . . . . 6 ⊢ 0 = (0g‘𝑅) | |
9 | 7, 8 | eqtr4di 2793 | . . . . 5 ⊢ (𝑟 = 𝑅 → (0g‘𝑟) = 0 ) |
10 | 9 | sneqd 4643 | . . . 4 ⊢ (𝑟 = 𝑅 → {(0g‘𝑟)} = { 0 }) |
11 | 6, 10 | difeq12d 4137 | . . 3 ⊢ (𝑟 = 𝑅 → ((Base‘𝑟) ∖ {(0g‘𝑟)}) = (𝐵 ∖ { 0 })) |
12 | 3, 11 | eqeq12d 2751 | . 2 ⊢ (𝑟 = 𝑅 → ((Unit‘𝑟) = ((Base‘𝑟) ∖ {(0g‘𝑟)}) ↔ 𝑈 = (𝐵 ∖ { 0 }))) |
13 | df-drng 20748 | . 2 ⊢ DivRing = {𝑟 ∈ Ring ∣ (Unit‘𝑟) = ((Base‘𝑟) ∖ {(0g‘𝑟)})} | |
14 | 12, 13 | elrab2 3698 | 1 ⊢ (𝑅 ∈ DivRing ↔ (𝑅 ∈ Ring ∧ 𝑈 = (𝐵 ∖ { 0 }))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∖ cdif 3960 {csn 4631 ‘cfv 6563 Basecbs 17245 0gc0g 17486 Ringcrg 20251 Unitcui 20372 DivRingcdr 20746 |
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-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-drng 20748 |
This theorem is referenced by: drngunit 20751 drngui 20752 drngring 20753 isdrng2 20760 drngprop 20761 drngid 20763 drngdomn 20766 opprdrng 20781 drngpropd 20786 fidomndrng 20791 issubdrg 20798 imadrhmcl 20815 cntzsdrg 20820 zringndrg 21497 istdrg2 24202 cvsunit 25178 cphreccllem 25226 isdrng4 33279 sradrng 33613 assafld 33665 zrhunitpreima 33939 aks5lem7 42182 |
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