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Mathbox for Jeff Madsen |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > pridlnr | Structured version Visualization version GIF version |
Description: A prime ideal is a proper ideal. (Contributed by Jeff Madsen, 19-Jun-2010.) |
Ref | Expression |
---|---|
pridlnr.1 | β’ πΊ = (1st βπ ) |
prdilnr.2 | β’ π = ran πΊ |
Ref | Expression |
---|---|
pridlnr | β’ ((π β RingOps β§ π β (PrIdlβπ )) β π β π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pridlnr.1 | . . . 4 β’ πΊ = (1st βπ ) | |
2 | eqid 2732 | . . . 4 β’ (2nd βπ ) = (2nd βπ ) | |
3 | prdilnr.2 | . . . 4 β’ π = ran πΊ | |
4 | 1, 2, 3 | ispridl 36897 | . . 3 β’ (π β RingOps β (π β (PrIdlβπ ) β (π β (Idlβπ ) β§ π β π β§ βπ β (Idlβπ )βπ β (Idlβπ )(βπ₯ β π βπ¦ β π (π₯(2nd βπ )π¦) β π β (π β π β¨ π β π))))) |
5 | 3anan12 1096 | . . 3 β’ ((π β (Idlβπ ) β§ π β π β§ βπ β (Idlβπ )βπ β (Idlβπ )(βπ₯ β π βπ¦ β π (π₯(2nd βπ )π¦) β π β (π β π β¨ π β π))) β (π β π β§ (π β (Idlβπ ) β§ βπ β (Idlβπ )βπ β (Idlβπ )(βπ₯ β π βπ¦ β π (π₯(2nd βπ )π¦) β π β (π β π β¨ π β π))))) | |
6 | 4, 5 | bitrdi 286 | . 2 β’ (π β RingOps β (π β (PrIdlβπ ) β (π β π β§ (π β (Idlβπ ) β§ βπ β (Idlβπ )βπ β (Idlβπ )(βπ₯ β π βπ¦ β π (π₯(2nd βπ )π¦) β π β (π β π β¨ π β π)))))) |
7 | 6 | simprbda 499 | 1 β’ ((π β RingOps β§ π β (PrIdlβπ )) β π β π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 396 β¨ wo 845 β§ w3a 1087 = wceq 1541 β wcel 2106 β wne 2940 βwral 3061 β wss 3948 ran crn 5677 βcfv 6543 (class class class)co 7408 1st c1st 7972 2nd c2nd 7973 RingOpscrngo 36757 Idlcidl 36870 PrIdlcpridl 36871 |
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-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
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-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 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-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-iota 6495 df-fun 6545 df-fv 6551 df-ov 7411 df-pridl 36874 |
This theorem is referenced by: (None) |
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