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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 6906 | . . . 4 ⊢ (𝑟 = 𝑅 → (Unit‘𝑟) = (Unit‘𝑅)) | |
| 2 | isdrng.u | . . . 4 ⊢ 𝑈 = (Unit‘𝑅) | |
| 3 | 1, 2 | eqtr4di 2795 | . . 3 ⊢ (𝑟 = 𝑅 → (Unit‘𝑟) = 𝑈) |
| 4 | fveq2 6906 | . . . . 5 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅)) | |
| 5 | isdrng.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑅) | |
| 6 | 4, 5 | eqtr4di 2795 | . . . 4 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵) |
| 7 | fveq2 6906 | . . . . . 6 ⊢ (𝑟 = 𝑅 → (0g‘𝑟) = (0g‘𝑅)) | |
| 8 | isdrng.z | . . . . . 6 ⊢ 0 = (0g‘𝑅) | |
| 9 | 7, 8 | eqtr4di 2795 | . . . . 5 ⊢ (𝑟 = 𝑅 → (0g‘𝑟) = 0 ) |
| 10 | 9 | sneqd 4638 | . . . 4 ⊢ (𝑟 = 𝑅 → {(0g‘𝑟)} = { 0 }) |
| 11 | 6, 10 | difeq12d 4127 | . . 3 ⊢ (𝑟 = 𝑅 → ((Base‘𝑟) ∖ {(0g‘𝑟)}) = (𝐵 ∖ { 0 })) |
| 12 | 3, 11 | eqeq12d 2753 | . 2 ⊢ (𝑟 = 𝑅 → ((Unit‘𝑟) = ((Base‘𝑟) ∖ {(0g‘𝑟)}) ↔ 𝑈 = (𝐵 ∖ { 0 }))) |
| 13 | df-drng 20731 | . 2 ⊢ DivRing = {𝑟 ∈ Ring ∣ (Unit‘𝑟) = ((Base‘𝑟) ∖ {(0g‘𝑟)})} | |
| 14 | 12, 13 | elrab2 3695 | 1 ⊢ (𝑅 ∈ DivRing ↔ (𝑅 ∈ Ring ∧ 𝑈 = (𝐵 ∖ { 0 }))) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∖ cdif 3948 {csn 4626 ‘cfv 6561 Basecbs 17247 0gc0g 17484 Ringcrg 20230 Unitcui 20355 DivRingcdr 20729 |
| 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-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-drng 20731 |
| This theorem is referenced by: drngunit 20734 drngui 20735 drngring 20736 isdrng2 20743 drngprop 20744 drngid 20746 drngdomn 20749 opprdrng 20764 drngpropd 20769 fidomndrng 20774 issubdrg 20781 imadrhmcl 20798 cntzsdrg 20803 zringndrg 21479 istdrg2 24186 cvsunit 25164 cphreccllem 25212 isdrng4 33298 sradrng 33633 assafld 33688 zrhunitpreima 33977 aks5lem7 42201 |
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