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Theorem rngoi 21973
 Description: The properties of a unital ring. (Contributed by Steve Rodriguez, 8-Sep-2007.) (Proof shortened by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
ringi.1
ringi.2
ringi.3
Assertion
Ref Expression
rngoi
Distinct variable groups:   ,,,   ,,,   ,,,   ,
Allowed substitution hints:   (,)

Proof of Theorem rngoi
StepHypRef Expression
1 relrngo 21970 . . . . 5
2 1st2nd 6396 . . . . 5
31, 2mpan 653 . . . 4
4 ringi.1 . . . . 5
5 ringi.2 . . . . 5
64, 5opeq12i 3991 . . . 4
73, 6syl6reqr 2489 . . 3
8 id 21 . . 3
97, 8eqeltrd 2512 . 2
10 fvex 5745 . . . 4
115, 10eqeltri 2508 . . 3
12 ringi.3 . . . 4
1312isrngo 21971 . . 3
1411, 13ax-mp 5 . 2
159, 14sylib 190 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 178   wa 360   w3a 937   wceq 1653   wcel 1726  wral 2707  wrex 2708  cvv 2958  cop 3819   cxp 4879   crn 4882   wrel 4886  wf 5453  cfv 5457  (class class class)co 6084  c1st 6350  c2nd 6351  cablo 21874  crngo 21968 This theorem is referenced by:  rngosm  21974  rngoid  21976  rngoideu  21977  rngodi  21978  rngodir  21979  rngoass  21980  rngoablo  21982  rngorn1eq  22013  rngomndo  22014 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704 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 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-fv 5465  df-ov 6087  df-1st 6352  df-2nd 6353  df-rngo 21969
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