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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 | β’ πΊ = (1st βπ ) |
isprrng.2 | β’ π = (GIdβπΊ) |
Ref | Expression |
---|---|
isprrngo | β’ (π β PrRing β (π β RingOps β§ {π} β (PrIdlβπ ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6891 | . . . . . . 7 β’ (π = π β (1st βπ) = (1st βπ )) | |
2 | isprrng.1 | . . . . . . 7 β’ πΊ = (1st βπ ) | |
3 | 1, 2 | eqtr4di 2790 | . . . . . 6 β’ (π = π β (1st βπ) = πΊ) |
4 | 3 | fveq2d 6895 | . . . . 5 β’ (π = π β (GIdβ(1st βπ)) = (GIdβπΊ)) |
5 | isprrng.2 | . . . . 5 β’ π = (GIdβπΊ) | |
6 | 4, 5 | eqtr4di 2790 | . . . 4 β’ (π = π β (GIdβ(1st βπ)) = π) |
7 | 6 | sneqd 4640 | . . 3 β’ (π = π β {(GIdβ(1st βπ))} = {π}) |
8 | fveq2 6891 | . . 3 β’ (π = π β (PrIdlβπ) = (PrIdlβπ )) | |
9 | 7, 8 | eleq12d 2827 | . 2 β’ (π = π β ({(GIdβ(1st βπ))} β (PrIdlβπ) β {π} β (PrIdlβπ ))) |
10 | df-prrngo 37002 | . 2 β’ PrRing = {π β RingOps β£ {(GIdβ(1st βπ))} β (PrIdlβπ)} | |
11 | 9, 10 | elrab2 3686 | 1 β’ (π β PrRing β (π β RingOps β§ {π} β (PrIdlβπ ))) |
Colors of variables: wff setvar class |
Syntax hints: β wb 205 β§ wa 396 = wceq 1541 β wcel 2106 {csn 4628 βcfv 6543 1st c1st 7975 GIdcgi 29781 RingOpscrngo 36848 PrIdlcpridl 36962 PrRingcprrng 37000 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-rab 3433 df-v 3476 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-iota 6495 df-fv 6551 df-prrngo 37002 |
This theorem is referenced by: prrngorngo 37005 smprngopr 37006 isdmn3 37028 |
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