![]() |
Mathbox for Jeff Madsen |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > isprrngo | Structured version Visualization version GIF version |
Description: The predicate "is a prime ring". (Contributed by Jeff Madsen, 10-Jun-2010.) |
Ref | Expression |
---|---|
isprrng.1 | ⊢ 𝐺 = (1st ‘𝑅) |
isprrng.2 | ⊢ 𝑍 = (GId‘𝐺) |
Ref | Expression |
---|---|
isprrngo | ⊢ (𝑅 ∈ PrRing ↔ (𝑅 ∈ RingOps ∧ {𝑍} ∈ (PrIdl‘𝑅))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6645 | . . . . . . 7 ⊢ (𝑟 = 𝑅 → (1st ‘𝑟) = (1st ‘𝑅)) | |
2 | isprrng.1 | . . . . . . 7 ⊢ 𝐺 = (1st ‘𝑅) | |
3 | 1, 2 | eqtr4di 2851 | . . . . . 6 ⊢ (𝑟 = 𝑅 → (1st ‘𝑟) = 𝐺) |
4 | 3 | fveq2d 6649 | . . . . 5 ⊢ (𝑟 = 𝑅 → (GId‘(1st ‘𝑟)) = (GId‘𝐺)) |
5 | isprrng.2 | . . . . 5 ⊢ 𝑍 = (GId‘𝐺) | |
6 | 4, 5 | eqtr4di 2851 | . . . 4 ⊢ (𝑟 = 𝑅 → (GId‘(1st ‘𝑟)) = 𝑍) |
7 | 6 | sneqd 4537 | . . 3 ⊢ (𝑟 = 𝑅 → {(GId‘(1st ‘𝑟))} = {𝑍}) |
8 | fveq2 6645 | . . 3 ⊢ (𝑟 = 𝑅 → (PrIdl‘𝑟) = (PrIdl‘𝑅)) | |
9 | 7, 8 | eleq12d 2884 | . 2 ⊢ (𝑟 = 𝑅 → ({(GId‘(1st ‘𝑟))} ∈ (PrIdl‘𝑟) ↔ {𝑍} ∈ (PrIdl‘𝑅))) |
10 | df-prrngo 35486 | . 2 ⊢ PrRing = {𝑟 ∈ RingOps ∣ {(GId‘(1st ‘𝑟))} ∈ (PrIdl‘𝑟)} | |
11 | 9, 10 | elrab2 3631 | 1 ⊢ (𝑅 ∈ PrRing ↔ (𝑅 ∈ RingOps ∧ {𝑍} ∈ (PrIdl‘𝑅))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 {csn 4525 ‘cfv 6324 1st c1st 7669 GIdcgi 28273 RingOpscrngo 35332 PrIdlcpridl 35446 PrRingcprrng 35484 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-rab 3115 df-v 3443 df-un 3886 df-in 3888 df-ss 3898 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-iota 6283 df-fv 6332 df-prrngo 35486 |
This theorem is referenced by: prrngorngo 35489 smprngopr 35490 isdmn3 35512 |
Copyright terms: Public domain | W3C validator |