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Theorem ispridl2 26629
 Description: A condition that shows an ideal is prime. For commutative rings, this is often taken to be the definition. See ispridlc 26661 for the equivalence in the commutative case. (Contributed by Jeff Madsen, 19-Jun-2010.)
Hypotheses
Ref Expression
ispridl2.1
ispridl2.2
ispridl2.3
Assertion
Ref Expression
ispridl2
Distinct variable groups:   ,,   ,,   ,,
Allowed substitution hints:   (,)   (,)

Proof of Theorem ispridl2
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ispridl2.1 . . . . . . . . . . . . . 14
2 ispridl2.3 . . . . . . . . . . . . . 14
31, 2idlss 26607 . . . . . . . . . . . . 13
4 ssralv 3399 . . . . . . . . . . . . 13
53, 4syl 16 . . . . . . . . . . . 12
65adantrr 698 . . . . . . . . . . 11
71, 2idlss 26607 . . . . . . . . . . . . 13
8 ssralv 3399 . . . . . . . . . . . . . 14
98ralimdv 2777 . . . . . . . . . . . . 13
107, 9syl 16 . . . . . . . . . . . 12
1110adantrl 697 . . . . . . . . . . 11
126, 11syld 42 . . . . . . . . . 10
1312adantlr 696 . . . . . . . . 9
14 r19.26-2 2831 . . . . . . . . . . 11
15 pm3.35 571 . . . . . . . . . . . . . 14
1615ralimi 2773 . . . . . . . . . . . . 13
1716ralimi 2773 . . . . . . . . . . . 12
18 2ralor 2869 . . . . . . . . . . . . . 14
1918biimpi 187 . . . . . . . . . . . . 13
20 dfss3 3330 . . . . . . . . . . . . . 14
21 dfss3 3330 . . . . . . . . . . . . . 14
2220, 21orbi12i 508 . . . . . . . . . . . . 13
2319, 22sylibr 204 . . . . . . . . . . . 12
2417, 23syl 16 . . . . . . . . . . 11
2514, 24sylbir 205 . . . . . . . . . 10
2625expcom 425 . . . . . . . . 9
2713, 26syl6 31 . . . . . . . 8
2827ralrimdvva 2793 . . . . . . 7
2928ex 424 . . . . . 6
3029adantrd 455 . . . . 5
3130imdistand 674 . . . 4
32 df-3an 938 . . . 4
33 df-3an 938 . . . 4
3431, 32, 333imtr4g 262 . . 3
35 ispridl2.2 . . . 4
361, 35, 2ispridl 26625 . . 3
3734, 36sylibrd 226 . 2
3837imp 419 1
 Colors of variables: wff set class Syntax hints:   wi 4   wo 358   wa 359   w3a 936   wceq 1652   wcel 1725   wne 2598  wral 2697   wss 3312   crn 4871  cfv 5446  (class class class)co 6073  c1st 6339  c2nd 6340  crngo 21955  cidl 26598  cpridl 26599 This theorem is referenced by:  ispridlc  26661 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-iota 5410  df-fun 5448  df-fv 5454  df-ov 6076  df-idl 26601  df-pridl 26602
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