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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 6901 | . . . . . . 7 ⊢ (𝑟 = 𝑅 → (1st ‘𝑟) = (1st ‘𝑅)) | |
2 | isprrng.1 | . . . . . . 7 ⊢ 𝐺 = (1st ‘𝑅) | |
3 | 1, 2 | eqtr4di 2791 | . . . . . 6 ⊢ (𝑟 = 𝑅 → (1st ‘𝑟) = 𝐺) |
4 | 3 | fveq2d 6905 | . . . . 5 ⊢ (𝑟 = 𝑅 → (GId‘(1st ‘𝑟)) = (GId‘𝐺)) |
5 | isprrng.2 | . . . . 5 ⊢ 𝑍 = (GId‘𝐺) | |
6 | 4, 5 | eqtr4di 2791 | . . . 4 ⊢ (𝑟 = 𝑅 → (GId‘(1st ‘𝑟)) = 𝑍) |
7 | 6 | sneqd 4642 | . . 3 ⊢ (𝑟 = 𝑅 → {(GId‘(1st ‘𝑟))} = {𝑍}) |
8 | fveq2 6901 | . . 3 ⊢ (𝑟 = 𝑅 → (PrIdl‘𝑟) = (PrIdl‘𝑅)) | |
9 | 7, 8 | eleq12d 2831 | . 2 ⊢ (𝑟 = 𝑅 → ({(GId‘(1st ‘𝑟))} ∈ (PrIdl‘𝑟) ↔ {𝑍} ∈ (PrIdl‘𝑅))) |
10 | df-prrngo 37995 | . 2 ⊢ PrRing = {𝑟 ∈ RingOps ∣ {(GId‘(1st ‘𝑟))} ∈ (PrIdl‘𝑟)} | |
11 | 9, 10 | elrab2 3698 | 1 ⊢ (𝑅 ∈ PrRing ↔ (𝑅 ∈ RingOps ∧ {𝑍} ∈ (PrIdl‘𝑅))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1535 ∈ wcel 2104 {csn 4630 ‘cfv 6558 1st c1st 8005 GIdcgi 30500 RingOpscrngo 37841 PrIdlcpridl 37955 PrRingcprrng 37993 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1963 ax-7 2003 ax-8 2106 ax-9 2114 ax-ext 2704 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1087 df-tru 1538 df-fal 1548 df-ex 1775 df-sb 2061 df-clab 2711 df-cleq 2725 df-clel 2812 df-rab 3433 df-v 3479 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4915 df-br 5150 df-iota 6510 df-fv 6566 df-prrngo 37995 |
This theorem is referenced by: prrngorngo 37998 smprngopr 37999 isdmn3 38021 |
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