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Theorem rngodi 21965
 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
rngodi

Proof of Theorem rngodi
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 21960 . . . 4
54simprd 450 . . 3
65simpld 446 . 2
7 simp2 958 . . . . . 6
87ralimi 2773 . . . . 5
98ralimi 2773 . . . 4
109ralimi 2773 . . 3
11 oveq1 6080 . . . . 5
12 oveq1 6080 . . . . . 6
13 oveq1 6080 . . . . . 6
1412, 13oveq12d 6091 . . . . 5
1511, 14eqeq12d 2449 . . . 4
16 oveq1 6080 . . . . . 6
1716oveq2d 6089 . . . . 5
18 oveq2 6081 . . . . . 6
1918oveq1d 6088 . . . . 5
2017, 19eqeq12d 2449 . . . 4
21 oveq2 6081 . . . . . 6
2221oveq2d 6089 . . . . 5
23 oveq2 6081 . . . . . 6
2423oveq2d 6089 . . . . 5
2522, 24eqeq12d 2449 . . . 4
2615, 20, 25rspc3v 3053 . . 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 2697  wrex 2698   cxp 4868   crn 4871  wf 5442  cfv 5446  (class class class)co 6073  c1st 6339  c2nd 6340  cablo 21861  crngo 21955 This theorem is referenced by:  rngorz  21982  rngonegmn1r  26557  rngosubdi  26560 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 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693 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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fv 5454  df-ov 6076  df-1st 6341  df-2nd 6342  df-rngo 21956
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