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Mirrors > Home > MPE Home > Th. List > Mathboxes > prrngorngo | Structured version Visualization version GIF version |
Description: A prime ring is a ring. (Contributed by Jeff Madsen, 10-Jun-2010.) |
Ref | Expression |
---|---|
prrngorngo | ⊢ (𝑅 ∈ PrRing → 𝑅 ∈ RingOps) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2821 | . . 3 ⊢ (1st ‘𝑅) = (1st ‘𝑅) | |
2 | eqid 2821 | . . 3 ⊢ (GId‘(1st ‘𝑅)) = (GId‘(1st ‘𝑅)) | |
3 | 1, 2 | isprrngo 35322 | . 2 ⊢ (𝑅 ∈ PrRing ↔ (𝑅 ∈ RingOps ∧ {(GId‘(1st ‘𝑅))} ∈ (PrIdl‘𝑅))) |
4 | 3 | simplbi 500 | 1 ⊢ (𝑅 ∈ PrRing → 𝑅 ∈ RingOps) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2110 {csn 4560 ‘cfv 6349 1st c1st 7681 GIdcgi 28261 RingOpscrngo 35166 PrIdlcpridl 35280 PrRingcprrng 35318 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-iota 6308 df-fv 6357 df-prrngo 35320 |
This theorem is referenced by: isdmn2 35327 |
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