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Theorem rngi 15678
 Description: Properties of a unital ring. (Contributed by NM, 26-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.)
Hypotheses
Ref Expression
rngi.b
rngi.p
rngi.t
Assertion
Ref Expression
rngi

Proof of Theorem rngi
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rngi.b . . . . . . . . . 10
2 eqid 2438 . . . . . . . . . 10 mulGrp mulGrp
3 rngi.p . . . . . . . . . 10
4 rngi.t . . . . . . . . . 10
51, 2, 3, 4isrng 15670 . . . . . . . . 9 mulGrp
65simp3bi 975 . . . . . . . 8
76adantr 453 . . . . . . 7
8 simpr1 964 . . . . . . 7
9 rsp 2768 . . . . . . 7
107, 8, 9sylc 59 . . . . . 6
11 simpr2 965 . . . . . 6
12 rsp 2768 . . . . . 6
1310, 11, 12sylc 59 . . . . 5
14 simpr3 966 . . . . 5
15 rsp 2768 . . . . 5
1613, 14, 15sylc 59 . . . 4
1716simpld 447 . . 3
1817caovdig 6263 . 2
1916simprd 451 . . 3
2019caovdirg 6266 . 2
2118, 20jca 520 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 360   w3a 937   wceq 1653   wcel 1726  wral 2707  cfv 5456  (class class class)co 6083  cbs 13471   cplusg 13531  cmulr 13532  cmnd 14686  cgrp 14687  mulGrpcmgp 15650  crg 15662 This theorem is referenced by:  rngdi  15684  rngdir  15685 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-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-nul 4340 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-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-iota 5420  df-fv 5464  df-ov 6086  df-rng 15665
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