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Theorem isidlc 26627
 Description: The predicate "is an ideal of the commutative ring ." (Contributed by Jeff Madsen, 10-Jun-2010.)
Hypotheses
Ref Expression
idlval.1
idlval.2
idlval.3
idlval.4 GId
Assertion
Ref Expression
isidlc CRingOps
Distinct variable groups:   ,,,   ,   ,,,   ,
Allowed substitution hints:   (,,)   (,,)   ()   (,,)

Proof of Theorem isidlc
StepHypRef Expression
1 crngorngo 26612 . . 3 CRingOps
2 idlval.1 . . . 4
3 idlval.2 . . . 4
4 idlval.3 . . . 4
5 idlval.4 . . . 4 GId
62, 3, 4, 5isidl 26626 . . 3
71, 6syl 16 . 2 CRingOps
8 ssel2 3345 . . . . . . . 8
92, 3, 4crngocom 26613 . . . . . . . . . . . . . . 15 CRingOps
109eleq1d 2504 . . . . . . . . . . . . . 14 CRingOps
1110biimprd 216 . . . . . . . . . . . . 13 CRingOps
12113expa 1154 . . . . . . . . . . . 12 CRingOps
1312pm4.71d 617 . . . . . . . . . . 11 CRingOps
1413bicomd 194 . . . . . . . . . 10 CRingOps
1514ralbidva 2723 . . . . . . . . 9 CRingOps
1615anbi2d 686 . . . . . . . 8 CRingOps
178, 16sylan2 462 . . . . . . 7 CRingOps
1817anassrs 631 . . . . . 6 CRingOps
1918ralbidva 2723 . . . . 5 CRingOps
2019adantrr 699 . . . 4 CRingOps
2120pm5.32da 624 . . 3 CRingOps
22 df-3an 939 . . 3
23 df-3an 939 . . 3
2421, 22, 233bitr4g 281 . 2 CRingOps
257, 24bitrd 246 1 CRingOps
 Colors of variables: wff set class Syntax hints:   wi 4   wb 178   wa 360   w3a 937   wceq 1653   wcel 1726  wral 2707   wss 3322   crn 4881  cfv 5456  (class class class)co 6083  c1st 6349  c2nd 6350  GIdcgi 21777  crngo 21965  CRingOpsccring 26607  cidl 26619 This theorem is referenced by:  prnc  26679 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-iota 5420  df-fun 5458  df-fv 5464  df-ov 6086  df-1st 6351  df-2nd 6352  df-rngo 21966  df-com2 22001  df-crngo 26608  df-idl 26622
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