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Theorem rngodir 21053
 Description: Distributive law for the multiplication operation of a ring. (Contributed by Steve Rodriguez, 9-Sep-2007.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
ringi.1
ringi.2
ringi.3
Assertion
Ref Expression
rngodir

Proof of Theorem rngodir
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ringi.1 . . . . 5
2 ringi.2 . . . . 5
3 ringi.3 . . . . 5
41, 2, 3rngoi 21047 . . . 4
54simprd 449 . . 3
65simpld 445 . 2
7 simp3 957 . . . . . 6
87ralimi 2618 . . . . 5
98ralimi 2618 . . . 4
109ralimi 2618 . . 3
11 oveq1 5865 . . . . . 6
1211oveq1d 5873 . . . . 5
13 oveq1 5865 . . . . . 6
1413oveq1d 5873 . . . . 5
1512, 14eqeq12d 2297 . . . 4
16 oveq2 5866 . . . . . 6
1716oveq1d 5873 . . . . 5
18 oveq1 5865 . . . . . 6
1918oveq2d 5874 . . . . 5
2017, 19eqeq12d 2297 . . . 4
21 oveq2 5866 . . . . 5
22 oveq2 5866 . . . . . 6
23 oveq2 5866 . . . . . 6
2422, 23oveq12d 5876 . . . . 5
2521, 24eqeq12d 2297 . . . 4
2615, 20, 25rspc3v 2893 . . 3
2710, 26syl5 28 . 2
286, 27mpan9 455 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 358   w3a 934   wceq 1623   wcel 1684  wral 2543  wrex 2544   cxp 4687   crn 4690  wf 5251  cfv 5255  (class class class)co 5858  c1st 6120  c2nd 6121  cablo 20948  crngo 21042 This theorem is referenced by:  rngo2  21055  rngolz  21068  multinv  25422  glmrngo  25482  rngonegmn1l  26580  rngosubdir  26585  prnc  26692 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-ov 5861  df-1st 6122  df-2nd 6123  df-rngo 21043
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