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Mathbox for Jeff Madsen |
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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 | β’ (π β PrRing β π β RingOps) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2737 | . . 3 β’ (1st βπ ) = (1st βπ ) | |
2 | eqid 2737 | . . 3 β’ (GIdβ(1st βπ )) = (GIdβ(1st βπ )) | |
3 | 1, 2 | isprrngo 36512 | . 2 β’ (π β PrRing β (π β RingOps β§ {(GIdβ(1st βπ ))} β (PrIdlβπ ))) |
4 | 3 | simplbi 499 | 1 β’ (π β PrRing β π β RingOps) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wcel 2107 {csn 4587 βcfv 6497 1st c1st 7920 GIdcgi 29435 RingOpscrngo 36356 PrIdlcpridl 36470 PrRingcprrng 36508 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2708 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-rab 3409 df-v 3448 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-iota 6449 df-fv 6505 df-prrngo 36510 |
This theorem is referenced by: isdmn2 36517 |
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