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 4128 | . . . 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 480 | . 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 4127 | . . 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 37577 | . . . 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 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 394 β§ w3a 1084 = wceq 1533 β wcel 2098 β cdif 3946 ran crn 5683 βcfv 6553 (class class class)co 7426 1st c1st 7997 2nd c2nd 7998 CRingOpsccring 37499 PrIdlcpridl 37514 |
| 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 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 |
| This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-1st 7999 df-2nd 8000 df-grpo 30323 df-gid 30324 df-ginv 30325 df-ablo 30375 df-ass 37349 df-exid 37351 df-mgmOLD 37355 df-sgrOLD 37367 df-mndo 37373 df-rngo 37401 df-com2 37496 df-crngo 37500 df-idl 37516 df-pridl 37517 df-igen 37566 |
| This theorem is referenced by: pridlc3 37579 |
| Copyright terms: Public domain | W3C validator |