Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  rngodir Structured version   Unicode version

Theorem rngodir 21974
 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 21968 . . . 4
54simprd 450 . . 3
65simpld 446 . 2
7 simp3 959 . . . . . 6
87ralimi 2781 . . . . 5
98ralimi 2781 . . . 4
109ralimi 2781 . . 3
11 oveq1 6088 . . . . . 6
1211oveq1d 6096 . . . . 5
13 oveq1 6088 . . . . . 6
1413oveq1d 6096 . . . . 5
1512, 14eqeq12d 2450 . . . 4
16 oveq2 6089 . . . . . 6
1716oveq1d 6096 . . . . 5
18 oveq1 6088 . . . . . 6
1918oveq2d 6097 . . . . 5
2017, 19eqeq12d 2450 . . . 4
21 oveq2 6089 . . . . 5
22 oveq2 6089 . . . . . 6
23 oveq2 6089 . . . . . 6
2422, 23oveq12d 6099 . . . . 5
2521, 24eqeq12d 2450 . . . 4
2615, 20, 25rspc3v 3061 . . 3
2710, 26syl5 30 . 2
286, 27mpan9 456 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 359   w3a 936   wceq 1652   wcel 1725  wral 2705  wrex 2706   cxp 4876   crn 4879  wf 5450  cfv 5454  (class class class)co 6081  c1st 6347  c2nd 6348  cablo 21869  crngo 21963 This theorem is referenced by:  rngo2  21976  rngolz  21989  rngonegmn1l  26565  rngosubdir  26570  prnc  26677 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-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  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-fv 5462  df-ov 6084  df-1st 6349  df-2nd 6350  df-rngo 21964
 Copyright terms: Public domain W3C validator