Nearby theorems
|
|
Mathbox for Jeff Madsen |
< Previous
Next >
Nearby theorems |
|
| 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 4122 | . . . 4 β’ (π΄ β (π β π) β Β¬ π΄ β π) | |
| 2 | 1 | 3ad2ant1 1130 | . . 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 481 | . 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 4121 | . . 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 37450 | . . . 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 861 | . . 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 1413 | . 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 395 β§ w3a 1084 = wceq 1533 β wcel 2098 β cdif 3940 ran crn 5670 βcfv 6536 (class class class)co 7404 1st c1st 7969 2nd c2nd 7970 CRingOpsccring 37372 PrIdlcpridl 37387 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-1st 7971 df-2nd 7972 df-grpo 30251 df-gid 30252 df-ginv 30253 df-ablo 30303 df-ass 37222 df-exid 37224 df-mgmOLD 37228 df-sgrOLD 37240 df-mndo 37246 df-rngo 37274 df-com2 37369 df-crngo 37373 df-idl 37389 df-pridl 37390 df-igen 37439 |
| This theorem is referenced by: pridlc3 37452 |
| Copyright terms: Public domain | W3C validator |