Mathbox for Jeff Madsen < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  iscrngo2 Structured version   Unicode version

Theorem iscrngo2 26610
 Description: The predicate "is a commutative ring". (Contributed by Jeff Madsen, 8-Jun-2010.)
Hypotheses
Ref Expression
iscring2.1
iscring2.2
iscring2.3
Assertion
Ref Expression
iscrngo2 CRingOps
Distinct variable groups:   ,,   ,,
Allowed substitution hints:   (,)   (,)

Proof of Theorem iscrngo2
StepHypRef Expression
1 iscrngo 26609 . 2 CRingOps
2 relrngo 21967 . . . . 5
3 1st2nd 6395 . . . . 5
42, 3mpan 653 . . . 4
5 eleq1 2498 . . . . 5
6 iscring2.3 . . . . . . . 8
7 iscring2.1 . . . . . . . . 9
87rneqi 5098 . . . . . . . 8
96, 8eqtri 2458 . . . . . . 7
109raleqi 2910 . . . . . 6
11 iscring2.2 . . . . . . . . . 10
1211oveqi 6096 . . . . . . . . 9
1311oveqi 6096 . . . . . . . . 9
1412, 13eqeq12i 2451 . . . . . . . 8
159, 14raleqbii 2737 . . . . . . 7
1615ralbii 2731 . . . . . 6
17 fvex 5744 . . . . . . 7
18 fvex 5744 . . . . . . 7
19 iscom2 22002 . . . . . . 7
2017, 18, 19mp2an 655 . . . . . 6
2110, 16, 203bitr4ri 271 . . . . 5
225, 21syl6bb 254 . . . 4
234, 22syl 16 . . 3
2423pm5.32i 620 . 2
251, 24bitri 242 1 CRingOps
 Colors of variables: wff set class Syntax hints:   wb 178   wa 360   wceq 1653   wcel 1726  wral 2707  cvv 2958  cop 3819   crn 4881   wrel 4885  cfv 5456  (class class class)co 6083  c1st 6349  c2nd 6350  crngo 21965  ccm2 22000  CRingOpsccring 26607 This theorem is referenced by:  crngocom  26613  crngohomfo  26618 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-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
 Copyright terms: Public domain W3C validator