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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 2738 | . . 3 ⊢ (1st ‘𝑅) = (1st ‘𝑅) | |
2 | eqid 2738 | . . 3 ⊢ (GId‘(1st ‘𝑅)) = (GId‘(1st ‘𝑅)) | |
3 | 1, 2 | isprrngo 36208 | . 2 ⊢ (𝑅 ∈ PrRing ↔ (𝑅 ∈ RingOps ∧ {(GId‘(1st ‘𝑅))} ∈ (PrIdl‘𝑅))) |
4 | 3 | simplbi 498 | 1 ⊢ (𝑅 ∈ PrRing → 𝑅 ∈ RingOps) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 {csn 4561 ‘cfv 6433 1st c1st 7829 GIdcgi 28852 RingOpscrngo 36052 PrIdlcpridl 36166 PrRingcprrng 36204 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2709 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-rab 3073 df-v 3434 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5075 df-iota 6391 df-fv 6441 df-prrngo 36206 |
This theorem is referenced by: isdmn2 36213 |
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