Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 6672 | . . . 4 ⊢ (𝑟 = 𝑅 → (Unit‘𝑟) = (Unit‘𝑅)) | |
2 | isdrng.u | . . . 4 ⊢ 𝑈 = (Unit‘𝑅) | |
3 | 1, 2 | syl6eqr 2876 | . . 3 ⊢ (𝑟 = 𝑅 → (Unit‘𝑟) = 𝑈) |
4 | fveq2 6672 | . . . . 5 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅)) | |
5 | isdrng.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑅) | |
6 | 4, 5 | syl6eqr 2876 | . . . 4 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵) |
7 | fveq2 6672 | . . . . . 6 ⊢ (𝑟 = 𝑅 → (0g‘𝑟) = (0g‘𝑅)) | |
8 | isdrng.z | . . . . . 6 ⊢ 0 = (0g‘𝑅) | |
9 | 7, 8 | syl6eqr 2876 | . . . . 5 ⊢ (𝑟 = 𝑅 → (0g‘𝑟) = 0 ) |
10 | 9 | sneqd 4581 | . . . 4 ⊢ (𝑟 = 𝑅 → {(0g‘𝑟)} = { 0 }) |
11 | 6, 10 | difeq12d 4102 | . . 3 ⊢ (𝑟 = 𝑅 → ((Base‘𝑟) ∖ {(0g‘𝑟)}) = (𝐵 ∖ { 0 })) |
12 | 3, 11 | eqeq12d 2839 | . 2 ⊢ (𝑟 = 𝑅 → ((Unit‘𝑟) = ((Base‘𝑟) ∖ {(0g‘𝑟)}) ↔ 𝑈 = (𝐵 ∖ { 0 }))) |
13 | df-drng 19506 | . 2 ⊢ DivRing = {𝑟 ∈ Ring ∣ (Unit‘𝑟) = ((Base‘𝑟) ∖ {(0g‘𝑟)})} | |
14 | 12, 13 | elrab2 3685 | 1 ⊢ (𝑅 ∈ DivRing ↔ (𝑅 ∈ Ring ∧ 𝑈 = (𝐵 ∖ { 0 }))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∖ cdif 3935 {csn 4569 ‘cfv 6357 Basecbs 16485 0gc0g 16715 Ringcrg 19299 Unitcui 19391 DivRingcdr 19504 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-iota 6316 df-fv 6365 df-drng 19506 |
This theorem is referenced by: drngunit 19509 drngui 19510 drngring 19511 isdrng2 19514 drngprop 19515 drngid 19518 opprdrng 19528 drngpropd 19531 issubdrg 19562 cntzsdrg 19583 drngdomn 20078 fidomndrng 20082 zringndrg 20639 istdrg2 22788 cvsunit 23737 cphreccllem 23784 sradrng 30990 zrhunitpreima 31221 |
Copyright terms: Public domain | W3C validator |