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| 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 6904 | . . . . . . 7 ⊢ (𝑟 = 𝑅 → (1st ‘𝑟) = (1st ‘𝑅)) | |
| 2 | isprrng.1 | . . . . . . 7 ⊢ 𝐺 = (1st ‘𝑅) | |
| 3 | 1, 2 | eqtr4di 2794 | . . . . . 6 ⊢ (𝑟 = 𝑅 → (1st ‘𝑟) = 𝐺) |
| 4 | 3 | fveq2d 6908 | . . . . 5 ⊢ (𝑟 = 𝑅 → (GId‘(1st ‘𝑟)) = (GId‘𝐺)) |
| 5 | isprrng.2 | . . . . 5 ⊢ 𝑍 = (GId‘𝐺) | |
| 6 | 4, 5 | eqtr4di 2794 | . . . 4 ⊢ (𝑟 = 𝑅 → (GId‘(1st ‘𝑟)) = 𝑍) |
| 7 | 6 | sneqd 4636 | . . 3 ⊢ (𝑟 = 𝑅 → {(GId‘(1st ‘𝑟))} = {𝑍}) |
| 8 | fveq2 6904 | . . 3 ⊢ (𝑟 = 𝑅 → (PrIdl‘𝑟) = (PrIdl‘𝑅)) | |
| 9 | 7, 8 | eleq12d 2834 | . 2 ⊢ (𝑟 = 𝑅 → ({(GId‘(1st ‘𝑟))} ∈ (PrIdl‘𝑟) ↔ {𝑍} ∈ (PrIdl‘𝑅))) |
| 10 | df-prrngo 38033 | . 2 ⊢ PrRing = {𝑟 ∈ RingOps ∣ {(GId‘(1st ‘𝑟))} ∈ (PrIdl‘𝑟)} | |
| 11 | 9, 10 | elrab2 3694 | 1 ⊢ (𝑅 ∈ PrRing ↔ (𝑅 ∈ RingOps ∧ {𝑍} ∈ (PrIdl‘𝑅))) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 {csn 4624 ‘cfv 6559 1st c1st 8008 GIdcgi 30499 RingOpscrngo 37879 PrIdlcpridl 37993 PrRingcprrng 38031 |
| 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 2707 |
| 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 2714 df-cleq 2728 df-clel 2815 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4906 df-br 5142 df-iota 6512 df-fv 6567 df-prrngo 38033 |
| This theorem is referenced by: prrngorngo 38036 smprngopr 38037 isdmn3 38059 |
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