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| Mirrors > Home > MPE Home > Th. List > Mathboxes > pridlc2 | Structured version Visualization version GIF version | ||
| Description: Property of a prime ideal in a commutative ring. (Contributed by Jeff Madsen, 17-Jun-2011.) |
| Ref | Expression |
|---|---|
| ispridlc.1 | β’ πΊ = (1st βπ ) |
| ispridlc.2 | β’ π» = (2nd βπ ) |
| ispridlc.3 | β’ π = ran πΊ |
| Ref | Expression |
|---|---|
| pridlc2 | β’ (((π β CRingOps β§ π β (PrIdlβπ )) β§ (π΄ β (π β π) β§ π΅ β π β§ (π΄π»π΅) β π)) β π΅ β π) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eldifn 4127 | . . . 4 β’ (π΄ β (π β π) β Β¬ π΄ β π) | |
| 2 | 1 | 3ad2ant1 1133 | . . 3 β’ ((π΄ β (π β π) β§ π΅ β π β§ (π΄π»π΅) β π) β Β¬ π΄ β π) |
| 3 | 2 | adantl 482 | . 2 β’ (((π β CRingOps β§ π β (PrIdlβπ )) β§ (π΄ β (π β π) β§ π΅ β π β§ (π΄π»π΅) β π)) β Β¬ π΄ β π) |
| 4 | eldifi 4126 | . . 3 β’ (π΄ β (π β π) β π΄ β π) | |
| 5 | ispridlc.1 | . . . . 5 β’ πΊ = (1st βπ ) | |
| 6 | ispridlc.2 | . . . . 5 β’ π» = (2nd βπ ) | |
| 7 | ispridlc.3 | . . . . 5 β’ π = ran πΊ | |
| 8 | 5, 6, 7 | pridlc 36934 | . . . 4 β’ (((π β CRingOps β§ π β (PrIdlβπ )) β§ (π΄ β π β§ π΅ β π β§ (π΄π»π΅) β π)) β (π΄ β π β¨ π΅ β π)) |
| 9 | 8 | ord 862 | . . 3 β’ (((π β CRingOps β§ π β (PrIdlβπ )) β§ (π΄ β π β§ π΅ β π β§ (π΄π»π΅) β π)) β (Β¬ π΄ β π β π΅ β π)) |
| 10 | 4, 9 | syl3anr1 1416 | . 2 β’ (((π β CRingOps β§ π β (PrIdlβπ )) β§ (π΄ β (π β π) β§ π΅ β π β§ (π΄π»π΅) β π)) β (Β¬ π΄ β π β π΅ β π)) |
| 11 | 3, 10 | mpd 15 | 1 β’ (((π β CRingOps β§ π β (PrIdlβπ )) β§ (π΄ β (π β π) β§ π΅ β π β§ (π΄π»π΅) β π)) β π΅ β π) |
| Colors of variables: wff setvar class |
| Syntax hints: Β¬ wn 3 β wi 4 β§ wa 396 β§ w3a 1087 = wceq 1541 β wcel 2106 β cdif 3945 ran crn 5677 βcfv 6543 (class class class)co 7408 1st c1st 7972 2nd c2nd 7973 CRingOpsccring 36856 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-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 |
| 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-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 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-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-1st 7974 df-2nd 7975 df-grpo 29741 df-gid 29742 df-ginv 29743 df-ablo 29793 df-ass 36706 df-exid 36708 df-mgmOLD 36712 df-sgrOLD 36724 df-mndo 36730 df-rngo 36758 df-com2 36853 df-crngo 36857 df-idl 36873 df-pridl 36874 df-igen 36923 |
| This theorem is referenced by: pridlc3 36936 |
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