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Theorem rngodir 25366
 Description: Distributive law for the multiplication operation of a ring (right-distributivity). (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 25360 . . . 4
54simprd 463 . . 3
65simpld 459 . 2
7 simp3 999 . . . . . 6
87ralimi 2836 . . . . 5
98ralimi 2836 . . . 4
109ralimi 2836 . . 3
11 oveq1 6288 . . . . . 6
1211oveq1d 6296 . . . . 5
13 oveq1 6288 . . . . . 6
1413oveq1d 6296 . . . . 5
1512, 14eqeq12d 2465 . . . 4
16 oveq2 6289 . . . . . 6
1716oveq1d 6296 . . . . 5
18 oveq1 6288 . . . . . 6
1918oveq2d 6297 . . . . 5
2017, 19eqeq12d 2465 . . . 4
21 oveq2 6289 . . . . 5
22 oveq2 6289 . . . . . 6
23 oveq2 6289 . . . . . 6
2422, 23oveq12d 6299 . . . . 5
2521, 24eqeq12d 2465 . . . 4
2615, 20, 25rspc3v 3208 . . 3
2710, 26syl5 32 . 2
286, 27mpan9 469 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 369   w3a 974   wceq 1383   wcel 1804  wral 2793  wrex 2794   cxp 4987   crn 4990  wf 5574  cfv 5578  (class class class)co 6281  c1st 6783  c2nd 6784  cablo 25261  crngo 25355 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-sbc 3314  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-fv 5586  df-ov 6284  df-1st 6785  df-2nd 6786  df-rngo 25356 This theorem is referenced by:  rngo2  25368  rngolz  25381  rngonegmn1l  30328  rngosubdir  30333  prnc  30440
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