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Mathbox for Jeff Madsen |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > pridlidl | Structured version Visualization version GIF version |
Description: A prime ideal is an ideal. (Contributed by Jeff Madsen, 19-Jun-2010.) |
Ref | Expression |
---|---|
pridlidl | β’ ((π β RingOps β§ π β (PrIdlβπ )) β π β (Idlβπ )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2731 | . . . 4 β’ (1st βπ ) = (1st βπ ) | |
2 | eqid 2731 | . . . 4 β’ (2nd βπ ) = (2nd βπ ) | |
3 | eqid 2731 | . . . 4 β’ ran (1st βπ ) = ran (1st βπ ) | |
4 | 1, 2, 3 | ispridl 37206 | . . 3 β’ (π β RingOps β (π β (PrIdlβπ ) β (π β (Idlβπ ) β§ π β ran (1st βπ ) 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 | 3anass 1094 | . . 3 β’ ((π β (Idlβπ ) β§ π β ran (1st βπ ) 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 (Idlβπ ) β§ (π β ran (1st βπ ) 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 (Idlβπ ) β§ (π β ran (1st βπ ) 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 498 | 1 β’ ((π β RingOps β§ π β (PrIdlβπ )) β π β (Idlβπ )) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 β¨ wo 844 β§ w3a 1086 β wcel 2105 β wne 2939 βwral 3060 β wss 3949 ran crn 5678 βcfv 6544 (class class class)co 7412 1st c1st 7976 2nd c2nd 7977 RingOpscrngo 37066 Idlcidl 37179 PrIdlcpridl 37180 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5300 ax-nul 5307 ax-pr 5428 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-iota 6496 df-fun 6546 df-fv 6552 df-ov 7415 df-pridl 37183 |
This theorem is referenced by: (None) |
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