Mathbox for Jeff Madsen |
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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 6663 | . . . . . . 7 ⊢ (𝑟 = 𝑅 → (1st ‘𝑟) = (1st ‘𝑅)) | |
2 | isprrng.1 | . . . . . . 7 ⊢ 𝐺 = (1st ‘𝑅) | |
3 | 1, 2 | syl6eqr 2871 | . . . . . 6 ⊢ (𝑟 = 𝑅 → (1st ‘𝑟) = 𝐺) |
4 | 3 | fveq2d 6667 | . . . . 5 ⊢ (𝑟 = 𝑅 → (GId‘(1st ‘𝑟)) = (GId‘𝐺)) |
5 | isprrng.2 | . . . . 5 ⊢ 𝑍 = (GId‘𝐺) | |
6 | 4, 5 | syl6eqr 2871 | . . . 4 ⊢ (𝑟 = 𝑅 → (GId‘(1st ‘𝑟)) = 𝑍) |
7 | 6 | sneqd 4569 | . . 3 ⊢ (𝑟 = 𝑅 → {(GId‘(1st ‘𝑟))} = {𝑍}) |
8 | fveq2 6663 | . . 3 ⊢ (𝑟 = 𝑅 → (PrIdl‘𝑟) = (PrIdl‘𝑅)) | |
9 | 7, 8 | eleq12d 2904 | . 2 ⊢ (𝑟 = 𝑅 → ({(GId‘(1st ‘𝑟))} ∈ (PrIdl‘𝑟) ↔ {𝑍} ∈ (PrIdl‘𝑅))) |
10 | df-prrngo 35207 | . 2 ⊢ PrRing = {𝑟 ∈ RingOps ∣ {(GId‘(1st ‘𝑟))} ∈ (PrIdl‘𝑟)} | |
11 | 9, 10 | elrab2 3680 | 1 ⊢ (𝑅 ∈ PrRing ↔ (𝑅 ∈ RingOps ∧ {𝑍} ∈ (PrIdl‘𝑅))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 {csn 4557 ‘cfv 6348 1st c1st 7676 GIdcgi 28194 RingOpscrngo 35053 PrIdlcpridl 35167 PrRingcprrng 35205 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-iota 6307 df-fv 6356 df-prrngo 35207 |
This theorem is referenced by: prrngorngo 35210 smprngopr 35211 isdmn3 35233 |
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