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Theorem rngoidl 26634
 Description: A ring is an ideal. (Contributed by Jeff Madsen, 10-Jun-2010.)
Hypotheses
Ref Expression
rngidl.1
rngidl.2
Assertion
Ref Expression
rngoidl

Proof of Theorem rngoidl
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssid 3367 . . 3
21a1i 11 . 2
3 rngidl.1 . . 3
4 rngidl.2 . . 3
5 eqid 2436 . . 3 GId GId
63, 4, 5rngo0cl 21986 . 2 GId
73, 4rngogcl 21979 . . . . . 6
873expa 1153 . . . . 5
98ralrimiva 2789 . . . 4
10 eqid 2436 . . . . . . . . 9
113, 10, 4rngocl 21970 . . . . . . . 8
12113com23 1159 . . . . . . 7
133, 10, 4rngocl 21970 . . . . . . 7
1412, 13jca 519 . . . . . 6
15143expa 1153 . . . . 5
1615ralrimiva 2789 . . . 4
179, 16jca 519 . . 3
1817ralrimiva 2789 . 2
193, 10, 4, 5isidl 26624 . 2 GId
202, 6, 18, 19mpbir3and 1137 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 359   w3a 936   wceq 1652   wcel 1725  wral 2705   wss 3320   crn 4879  cfv 5454  (class class class)co 6081  c1st 6347  c2nd 6348  GIdcgi 21775  crngo 21963  cidl 26617 This theorem is referenced by:  divrngidl  26638  igenval  26671  igenidl  26673 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 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701 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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-fo 5460  df-fv 5462  df-ov 6084  df-1st 6349  df-2nd 6350  df-riota 6549  df-grpo 21779  df-gid 21780  df-ablo 21870  df-rngo 21964  df-idl 26620
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